synchronization of complex networks has aroused many interest in nonlinear science since the discoveries of the small - world and scale - free properties in many real and man - made systems . in this study ,one important issue is to explore the inter - dependent relationship between the collective behaviors of the complex systems and their underlying topologies . in particular ,many efforts have been paid to the construction of optimal networks , and a number of factors which have important affections to the synchronizability of complex networks have been gradually disclosed .now it is known that random networks , due to their small average distances , are generally more synchronizable than regular networks ; and scale - free networks , with weighted and asymmetric couplings , can be more synchronizable than homogeneous networks . in these studies , the standard method employed for synchronization analysis is the master stability function ( msf ) , where the network synchronizability is estimated by an eigenratio calculated from the coupling matrix , and system which has a smaller eigenratio is believed to be more synchronizable than that of larger eigenratio .inspired by this , to improve the network synchronizability , the only task seems to be upgrading the coupling matrix so as to decrease the eigenratio , either by changing the network topology or by adjusting the coupling scheme netsyn : motter , netsyn : bocca .the msf method , while bringing great convenience to the analysis , overlooks the temporal , local property of the system and reflects only partial information about the system dynamics .specifically , from msf we only know ultimately whether the network is globally synchronizable or unsynchronizable , but do not know how the global synchronization is reached if the network is synchronizable , or what s the pattern and how it evolves if the network is unsynchronizable .these evolution details , or the transient behavior in system development , contain rich information about the system dynamics and may give additional insights to the organization of complex systems .for instance , the recent studies about synchronization transition have shown that , in the unsynchronizable states , heterogeneous networks are more synchronizable ( have a higher degree of coherence ) than homogeneous networks at small couplings , while at larger couplings the opposite happens .this crossover phenomenon of network synchronizability are difficult to understood if we only look at the final state of the system , but are straightforward if we look at the transient behaviors of their evolutions . besides revealing the synchronization mechanisms , the transient behavior of network synchronizationcan also be used to detect the topological scales and hierarchical structures in the real systems , e.g. , the detection of cluster structures in social and biological networks . however , despite of its theoretical and practical significance , the study of transient dynamics of complex networks is still at its infancy and many questions remain open , say for example , the pattern evolution of unsynchronizable complex networks .pattern formation in unsynchronizable but near - synchronization networks has been an important issue in studying the collective behavior of regular networks . by setting the coupling strength nearby the synchronization bifurcation point , the system state shares both the dynamical properties of the synchronizable and unsynchronizable states : a state of high coherence but is not synchronized .the bifacial dynamical property makes this state a natural choice in investigating the transition process of networks synchronization .previously studies about regular networks , say for example the lattices , have shown that , when the coupling strength is slightly out of the synchronizable region , although global synchronization is unreachable , nodes are still synchronized in a partial sense .that is , nodes are self - organized into a number of synchronous clusters .the distribution of these clusters , also called the synchronous pattern , is determined by a set of factors such as the coupling strength , the system size and the coupling scheme . as the coupling strength leaves away from the bifurcation point ,the pattern structure becomes more and more complicated and the system coherence will be decreased , and finally reaches the turbulence state .it is worthy of note that the patterns arisen in regular networks have two common properties : spatially symmetric and temporally stationary .more specifically , the contents of each cluster are fixed and the clusters are of translation symmetry in space . for this reason , we say that _ the patterns formed in regular networks are symmetric and stationary_. these two properties , as have been discussed in the previous studies , are rooted in the symmetric topology of the regular networks .this makes it interesting to ask the following question : how about the patterns in unsynchronizable complex networks ?different to the regular networks , in complex networks we are not able to find any symmetry from their topologies .the asymmetric topology , according to the pattern analysis developed in studying regular networks syn : partial , will induce two significant changes to the patterns : 1 ) the synchronous clusters , if they exist , will be asymmetric ; and 2 ) all the possible patterns , including the one of global synchronization , are linearly unstable under small perturbations . in other words ,the patterns formed in complex networks are expected to be asymmetric and nonstationary . _our mission of this paper is just to understand and characterize the nonstationary patterns arisen in the development of complex networks .specifically , we are trying to investigate the following questions : 1 ) is there any pattern arises during the system evolution ?2 ) the pattern is stationary or nonstationary ? if nonstationary , how is it evolving and how is it reflected from the system dynamics ?3 ) what happens to the pattern properties during the transition of network synchronization ? and 4 ) how the coupling strength and bifurcation type affect the pattern properties ? by investigating these dynamical and statistical properties , we wish to have a global understanding to the dynamics of unsynchronizable complex networks .our main findings are : 1 ) for coupling strength immediately outside of the synchronizable region , the system dynamics undergoes the process of on - off intermittency .that is , most of the time the system stays on the global synchronization state ( the `` off '' state ) but , once in a while , it develops into a breaking state ( the `` on '' state ) which is composed by a giant cluster and a few number of small clusters ( hereafter we call it the giant - cluster state ) . as the system develops , the giant cluster changes its shape by absorbing or emitting the small clusters , leading to the `` off '' or `` on '' states , respectively ; 2 ) the few active nodes which escape from the giant cluster with the high frequencies are coupling - strength independent but are bifurcation - type dependent .that is , in the neighboring region of a fixed bifurcation point , the locations of these active nodes do not change with the coupling strength ; if we change the coupling strength from nearby another bifurcation point ( the two bifurcation points will be explained later ) , their locations will be totally changed ; 3 ) as coupling strength leaves away from the bifurcation point , the giant cluster is gradually dissolved and more small clusters are generated from it .eventually , the giant cluster disappears and the pattern is composed by only the small clusters ( hereafter we call it the scattering cluster state ) .during the course of system evolution , each small cluster may either increase its size by integrating with other small clusters or decrease its size by breaking to even small clusters , but it can never reach to the global synchronization state ; 4 ) besides the giant cluster , the giant- and scattering - cluster states are also distinct in their small clusters . for giant - cluster statethe size of the small clusters follows a power - law distribution , while for scattering - cluster state it follows a gaussian distribution .the rest of the paper is going to be arranged as follows . in sec .ii we will give our model of coupled map network and , based on the method of msf , point out the two bifurcation points and the transition areas that we are going to study with . in sec .iii we will employ the method of finite - time lyapunov exponent to predict and describe the intermittent system dynamics in the bifurcation regions .direct simulations about on - off intermittency will be presented in sec .iv . by introducing the method of temporal phase synchronization , in sec .v we will investigate in detail the dynamical and statistical properties of the nonstationary pattern .meanwhile , properties of the giant- and scattering states will be compared and the transition between the two states will be conducted . in sec .vi we will discuss the phenomenon of active nodes and investigate their dependence to the network properties .discussions and conclusions about pattern evolution in complex networks will be presented in sec .our model of coupled map network is of the following form .\label{net}\]]where is a -dimensional map representing the local dynamics on node , is a global coupling parameter , is laplacian matrix representing the couplings , and is a coupling function . to facilitate our analysis , we adopt the following coupling scheme : and , with the degree of node and adjacent matrix of the network : if node and are connected and otherwise . in comparison with the traditional coupling schemes ,one important advantage we benefit from this coupling scheme is that the synchronizability of the network , i.e. the eigenratio of the coupling matrix described in eq .[ [ cm ] ] , can be easily adjusted by the parameter , while the network topology is kept unchanged .this advantage brings many convenience in network selection since for a given network topology , even though it is unsynchronizable under the traditional schemes , can now be synchronizable by adjusting in eq . [ [ cm ] ] .this convenience is of particular importance when our studies of network dynamics are focused on the bifurcation regions , where the network synchronizability should be deliberately arranged in order to demonstrate both the two types of bifurcations .we note that the adoption of eq .[ cm ] is only for the purpose of convenient analysis , the findings we are going to report are general and can also be observed by other coupling schemes given the network is properly prepared . in practice , we use logistic map as the local dynamics and adopt as the coupling function .we first locate the two bifurcation points of global synchronization .the linear stability of the global synchronization state is determined by the corresponding variational equations , which can be diagonalized into blocks of form y(t ) , \label{linearstability}\]]with and the jacobian matrices of the corresponding vector functions evaluated at , and represents the different modes that are transverse to the synchronous manifold .we have for the block , , and are the eigenvalues of matrix .the largest lyapunov exponent of eq .[ [ linearstability ] ] , known as the master stability function ( msf ) , determines the linear stability of the synchronous manifold .in particular , the synchronous manifold is stable if only for each .the set of lyapunov exponents govern the stability of the synchronous manifold in the transverse spaces , and a positive value of represents the loss of the stability in the transverse space of mode .it was found that for a large class of chaotic systems , is only fulfilled within a limit range in the parameter space  .this indicates that , to make the global synchronization state linearly stable , all the eigenvalues should be contained within range , i.e. , . for the logistic map employed here ,it is not difficult to prove that and . therefore , to achieve global synchronization , the coupling matrix should be designed with eigenratio .besides the condition of , to guarantee the synchronization , we also need to set the coupling strength in a proper way : either small or large couplings may deteriorate the synchronization .if , the couplings are too weak to restrict the node trajectories to the synchronous manifold ; while if , the couplings will be too strong and actually act as large perturbations to the synchronization manifold . therefore , to achieve the global synchronization, we also require .the two critical couplings and , which are named as the long - wave ( lw ) and short - wave ( sw ) bifurcations bif : fast respectively in the studies of regular networks , thus stand as the boundaries of the synchronizable region . our studies about network synchronization will be focused on the neighboring regions of the two bifurcation points , i.e. , the region of or .  by the standard ba growth model , we construct a scale - free network of nodes and of average degree . by setting in eq . [ [ cm ] ] , we have and .because of , the network is globally synchronizable .also , because of and , both the two bifurcations can be realized by adjusting the coupling strength within range . in specific , when , we have and , the synchronous manifold loses its stability at the lower boundary of the synchronizable region and lw bifurcation occurs ; and when , we have and , the synchronous manifold loses its stability at the upper boundary of the synchronizable region and sw bifurcation occurs [ fig . 1 ] . in the followingwe will fix the network topology and the parameter , while generating the various patterns by changing the coupling strength nearby the two bifurcation points .before direct simulations , we first give a qualitative description ( prediction ) on the possible system dynamics in bifurcation regions .to concrete our analysis , in the following we will only discuss the situation of sw bifurcation ( ) , while noting that the same phenomena can be found at the lw bifurcation as well ( ) . in preparing the unsynchronizable states ,we only let be slightly puncturing into the unstable region , while keeping all the other exponents still staying in the stable region , i.e. , and for . with this setting , the synchronous manifold is only desynchronized in the transverse space of mode . as such, the system possesses only two positive lyapunov exponents , one is which is associated to the synchronous manifold itself and another one is . noticing that are asymptotic averages , and , as so , they account only for the global stability properties , but do not warrant the possible coherent sets arising in the system evolutions. these coherent sets , for regular networks , refer to the stationary , symmetric patterns to which the system finally develops .while for complex networks , these sets can be the temporal , irregular clusters emerged in the process of system evolution .  in the region of , although global synchronization is unreachable , the system may still keep with the high coherence due to the existence of the synchronous clusters .especially , there could be some moments at which all the trajectories are restrained to a small region in the phase space , very close to the situation of global synchronization .this varying system coherence , however , can not be reflected from the asymptotic value . to characterize the variation, we need to employ some new quantities which are able to capture the temporal behavior of system .one of such quantities is the finite - time lyapunov exponent ( fle ) , a technique developed in studying chaos transition in nonlinear science . in stead of asymptotic average , fle measuresthe diverging rate of nearby trajectories only in a finite time interval . studies are focused on the situation of one - mode desynchronization , the stability of the synchronous manifold and the temporal behavior that it displays are therefore expected to be more reflected from the variation of , the fle that associates with mode . with and , we plot in fig .2 the time evolution of .it is found that , although with a positive asymptotic value about , the instant value of penetrates to the negative region at a high frequency . according to the different signs of , the system evolutionis divided into two types of intervals : the divergent interval and the contractive interval . in the divergent intervals we have and the system dynamicsis temporarily dominated by the divergence of the node trajectories from the synchronous manifold ; while in the contractive intervals we have and the system dynamics is temporarily dominated by the convergence of the node trajectories to the synchronous manifold .the variation of , reflected on the process of pattern evolution , characterizes the travelling property of the system dynamics among the neighboring regions of two different kinds of states : the desynchronization state and the synchronization state . in fig .2 , the minimum value of is about , during this contractive interval the node trajectories will converge to the synchronous manifold by an amount of on average . assuming that before entering this interval the average distance between the node trajectories is ( for logistic map we always have ) , then at the end of this interval the average distance is decreased to , a small value which is usually overshadowed by noise in practice . due to this small distance, the system can be practically regarded as already reached the synchronization state . on the other hand ,if the system enters a divergent interval , the node trajectories will diverge from each other and , at the end of this interval , their average distance will be increased by an order of .this large distance will deteriorate the ordered trajectories ( or the high coherence of the system dynamics ) that achieved during the contractive intervals , and leading to the incoherent , breaking state .the pattern of the breaking state , however , is not unique .depending on the initial conditions and the divergence intervals , the pattern may assume the different configurations .therefore , based on the observations of [ fig .2 ] the dynamics of unsynchronizable networks can be intuitively understood as an intermittent travelling among the synchronization state and the different kinds of desynchronization states .we now investigate the system dynamics by direct simulations . to implement ,we first prepare the system to be staying on the synchronization state .this can be achieved by adopting a large coupling strength from the synchronizable region , i.e. . after synchronization is achieved , we then decrease to a value slightly below the bifurcation point and , in the meantime , an instant small perturbation is added on each node . in practice , we take i.i.d ( independent identically distributed ) noise of strength as the perturbations . after this , we release the system and let it develop according to eq .( [ net ] ) . since ,the synchronization state is unstable and , triggered by the noises , the node trajectories begin to diverge from each other .the divergent trajectories , however , will frequently visit the neighborhood of the synchronous manifold , especially during those contractive intervals of small [ fig . 2 ] .the intermittent system dynamics is plotted in fig .3(a ) , where the average trajectory distance is plotted as a function of time .as we have predicted from lle , the system dynamics indeed undergoes an intermittent process . to characterize the intermittency , we plot in fig .3(b ) the laminar - phase distribution of the sequence plotted in figit is found that the laminar length ( the time interval between two adjacent bursts of amplitude ) and the probability for it to appear follow a power - law scaling . the fitted exponent is about , with a fat tail at large due to the finite simulating time .in chaos theory , intermittent process of laminar - phase exponent is classified as the `` on - off '' intermittency , a typical phenomenon observed in dynamical systems with a symmetric invariant set . on - off intermittencyis also reported in chaos synchronization of regular networks , where the invariant set refers to the synchronous manifold , and the `` off '' state refers to the long stretches that the system dynamics is staying nearby the synchronous manifold and the `` on '' state refers to the short bursts that the system dynamics is staying away from the synchronous manifold .therefore , in terms of laminar - phase distribution , the intermittency we have found in complex networks [ fig .3 ] has no difference to the that of the regular networks , despite of the drastic difference between their topologies .we have also investigated the transition behavior of the averaged distance nearby the bifurcation points . as shown in fig .3(c ) , a linear relation between and is found in the region of .this linear transition of the system performance , again , is consistent with the transition of regular networks .therefore , in terms of complete synchronization , the on - off intermittency we have found in complex networks has no difference to that of the regular networks .to reveal the unique properties of the system dynamics that induced by the complex topology , we go on to investigate the pattern formation of unsynchronizable networks by the method of temporal phase synchronization ( tps ) .tps is defined as follows .let be the time sequence recorded on node , we first transform it into a symbolic sequence according to the following equations    then we divide into short segments of the equal length .regarding each segment as an new element , we therefore have transformed the long , variable sequence into a short , symbolic sequence .if at moment we have , then we say that tps is achieved between the nodes and .the collection of nodes which have the same value of moment are defined as a temporarily synchronous cluster , and all the synchronous clusters constitute the temporarily pattern of the system . during the course of system evolution , the clusters will change their shapes and contents and the pattern will change its configuration .  in comparison with the method of complete synchronization , the advantage we benefit from tps is obvious : it makes the synchronous pattern detectable . with complete synchronization , it is almost impossible for two nodes to have exactly the same variable at the same time . despite the fact that at some moments the system has already reached the high - coherence states ( formed during those contractive intervals in fig .3(a ) ) , with complete synchronization we are not able to distinguish these states from those low - coherence ones quantitatively ( formed during those divergent intervals in fig .( a remedy to this difficulty seems to define the clusters by the method of threshold truncation , i.e. , nodes are regarded as synchronized if the distance between their trajectories is smaller than some small value .however , this definition of synchronization will induce the problem of cluster idenfication , as the same state may generate different patterns if we choose the different reference nodes . ) on the contrary , tps focuses on the loose match ( phase synchronization ) between the node variables over a period of time . by requiring an exact match of the discrete variable , the synchronous pattern is uniquely defined ; while by requiring the match of the long sequences of , the `` synchronous '' nodes are guaranteed with a strong coherence .   with the same set of parameters as in fig .3(a ) , by the method of tps we plot in fig .4 the time evolutions of two basic quantities of pattern evolution : the number of synchronous clusters and the size of the largest cluster .it is found that , similar to the phenomenon in complete synchronization [ fig .3(a ) ] , on - off intermittency is also found in the tps and . in fig .4(a ) it is shown that most of the time the system is broken into only a few number of clusters , or , while occasionally it is broken into a quite large number of clusters , , or united to the synchronization state , . the intermittent pattern evolution is also reflected on the sequence of [ fig .4(b ) ] , where most of the time the size of the giant cluster is about , while occasionally it decreases to some small values of .as we have discussed previously , the main advantage we benefit from tps is in identifying the clusters .this advantage is clearly shown in figs .4(a ) and ( b ) , where for any time instant the two quantities and are uniquely defined . besides cluster identification , we also benefit from tps in quantifying the synchronization degrees . in specific ,the different coherence states shown in fig .3(a ) now can be clearly quantified : high coherence states are those of smaller or larger .specially , the synchronization state now is unambiguously defined as the moments of in fig .4(a ) or , equally , the moments of in fig .we go on to investigate the pattern evolution by statistical analysis .the first statistic we are interested is the laminar - phase distribution of the synchronization state , i.e. the time intervals that in fig .4(a ) or in fig .4(b ) . in its original definition, laminar phase refers to the time intervals that all node trajectories stays within a small distance from the synchronous manifold , therefore the actual value of is varying with the predefined threshold distance .this uncertainty is overcome in tps . as shown in fig .4(a ) , in tps the `` off '' state refers to the moments of specifically . the laminar - phase distribution of is plotted in fig .in consistency with the distribution of complete synchronization [ fig .3 ] , the laminar - phase distribution of also follows a power - law scaling and has the same exponent .therefore the use of tps , while bringing convenience to the pattern analysis , still capture the basic properties of the on - off intermittency .the second statistic we are interested is the size distribution of the largest cluster , an important indicator for the coherence degree of the system . for the sequence plotted in fig .4(b ) , in fig .5(b ) we plot the size distribution of .it is seen that the probability of finding large cluster is much higher than that of small cluster of .in particular , the probability for finding clusters of is about percent and for it is about percent .therefore , in the region of , the distinct feature of the system patterns is the existence of a giant cluster . due to this special feature, we call these states the giant - cluster state .  besides the giant cluster , we are also interested in the properties of the small clusters .we plot in fig .5(c ) the distribution of and in fig .5(d ) the size distribution of the small clusters that surround the giant cluster in the pattern . as shown in fig .5(c ) , the distribution of follows a power - law scaling with the fixed exponent is about .the heterogeneous distribution of indicates that , in the giant - cluster state , the system is usually broken into only a few number of clusters .an interesting finding exists in the size distribution of the small clusters . as shown in fig .5(d ) , in range $ ] a power - law scaling is found between and , with the fitted exponent is about .the distribution of confirms the finding of fig .5(c ) that the small clusters which join or separate from the giant cluster are usually of small size .  combining the findings of fig . 4 and fig . 5 , the picture of pattern evolution in the bifurcation region now becomes clear .generally speaking , the evolution can be divided into two opposite dynamical processes happening around the giant cluster : the separation and integration of the small clusters . during the separation process, the small clusters are escaped from the giant cluster , which weakens the dominant role of the giant cluster and makes the pattern complicated .however , the separated clusters occupy only a small proportion of the nodes [ fig .5(c ) ] , the majority nodes are still attached to the giant cluster , which sustains the synchronization skeleton and keeps the system on the high coherence states . at some rare momentsthe giant cluster may disappears , and the pattern is composed by only small clusters of . at these moments , the synchronization skeleton is broken , the pattern becomes even complicated and the system coherence reaches its minimum . in contrast , during the process of cluster integration , the giant cluster will increase it size by attracting the small clusters , and gradually towards the state of global synchronization .it should be noticed that the separation and integration processes are uneven and are typically occurring at the same time .for instance , during the separation process , while the system evolution is dominated by the separation of new small clusters from the giant cluster , there could be some small clusters rejoin to the giant cluster .as we further decrease the coupling strength from , the picture of pattern evolution will be totally changed . with , we plot in fig .6 the same statistics as in fig .the first observation is the loss of the global synchronization state , as can be found from the time variation of plotted in fig .the loss of global synchronization becomes even clear if we compare fig .6(a ) with fig .4(a ) : in fig .6(a ) , except the moment at , the system can never reach the synchronization state at and very often it is broken into a _ large _ number of small clusters at about  .the fact that the pattern is decomposed into a large number of small clusters is also manifested by the distribution of , as plotted in fig .6(b ) . instead of the power - law distribution found in the giant - cluster state , in the scattering - cluster state follows a gaussian distribution [ fig .6(b ) ] . as further decreases from , the mean value of will shift to the larger values , as indicated by the curve plotted in fig .the second observation is the disappearance of the giant cluster . as shown in fig .6(c ) , the size distribution of the largest cluster also follows a gaussian distribution , with its mean value locates at .the distribution of fig .6(c ) is very different to that of fig .5(b ) , where in fig .5(b ) the largest ( giant ) cluster has size in most of the time .as decreases , the mean value of the largest cluster will shift to small values and the variance of will be decreased , as indicated by the curve plotted in fig .similar to plot of fig .5(d ) , we have also investigated the distribution of , the sizes for all the small clusters appeared in the system evolution [ fig .it is found that the distribution of follows a power - law distribution for , while having an exponential tail for .numerically we find that the exponent of the power - law section , i.e. in range , is about , while the fitted exponent for the exponential section is about .these two exponents , however , are changing with .as decreases , the two exponents will shift to some small values .  combining fig . 5 and fig . 6, we are able to outline the transition process of network synchronization nearby the bifurcation points , i.e. , the transition from the giant - cluster state to the scattering - cluster state as leaves away from . in the region of ,the pattern is composed by a giant cluster and a few number of small clusters , i.e. the giant - cluster state .as decreases from gradually , more and more small clusters will be emitted out from the giant cluster and , as a consequence , both the size of the giant cluster and the fraction of synchronization time will be decreased .then , at about , the giant cluster disappears and the pattern of the system is composed by several larger clusters , of size , together with many small clusters of heterogenous size distribution , i.e. the scattering - cluster state .after that , as decreases from , the clusters shrink their size by breaking into even small clusters , and the pattern becomes even complicated .the detail transition from the giant - cluster state to the scattering - cluster state is presented in fig .7 , where the average number of clusters that the system is broken into , fig .7(a ) , and the average size of the largest cluster , fig .7(b ) , are plotted as a function of the coupling strength in the lw bifurcation region .the transition is found to be smooth and steady , just as we have expected . besides the giant cluster, another difference between the giant - cluster and scattering - cluster states exists in their pattern evolutions . in the giant - cluster state , while the configuration of the giant cluster is continuously updated by emitting or absorbing the small clusters , its main contents are stable and do not change with time .in contrast , in the scattering - cluster state the small clusters integrate with or separate from each other in a random fashion .although occasionally there could be some large clusters show up in the pattern of the scattering - cluster state [ fig .6(d ) ] , these `` large '' clusters , however , are very fragile and will break into small clusters again in a short time .this quick - dissolving property stops the scattering - cluster state from having a high coherence .in the giant - cluster state , most of the nodes are organized into the giant cluster while few nodes , either in forms of small group or isolated node , are separating from or joining to the giant cluster with a high frequency .these active nodes , although are few in amount , play an important role in network synchronization .clearly , a proper characterization of these nodes will deepen our understandings on the system dynamics and give indications to the improvement of network performance .for instance , to improve the synchronizability of the system , we may either remove the few most active nodes from the network , or update their coupling strengths specifically .  in characterizing the active nodes ,the following properties are of general interest : 1 ) what s the dependence of the node activity on the network topology ?can we characterize these nodes by the known network properties such as node degree or betweenness ? 2 ) are their locations sensitive to the coupling strength ? and 3 ) what s the effect of bifurcation type on their locations ? in the following we will explore these questions by numerical simulations .we first try to characterize the active nodes by their topological properties . for the giant - cluster state described in fig .4 , we plot in fig .8(a ) the probability that each node stays in the giant cluster .while the majority nodes stay in the giant cluster with a high probability , few nodes are of unusually small probabilities : percent of the nodes have .one important observation of fig .8(a ) is that the locations of the active nodes are entangled with those of the stable nodes .noticing that in the ba growth model node of higher index in general assume the smaller degree , the observation of fig .8(a ) therefore indicates the independence of the node degree to the node stability , or the inaccuracy of using degree to characterize the node activity .specifically , in fig .8(a ) the most unstable nodes , by a descending order of , are those of degrees , , , , , respectively . except the one of ,all the other nodes have higher degrees .another well - known topological property of complex network is the node betweenness , which counts the number of shortest pathes that pass through each node and actually evaluates the node importance from the global - network point of view .this global - network property , however , is also incapable to characterize the active nodes . in tab .1 we list the detail information about the most active nodes in fig .8(a ) , where the inaccuracy of node degree or node betweenness in characterizing the active nodes are summarized ..for the attaching probability plotted in fig .8(a ) , we list the most unstable nodes and try to characterize them by a set of topological quantities including the node index , the attaching probability , the stability rank , the node degree , the degree rank , the node betweenness , and the betweenness rank . [cols="^,^,^,^,^,^,^",options="header " , ]   [ table_sw ]  previous studies about network synchronization have shown that , while individually it is difficult to predict the dynamical behavior of each node , the average performance of an ensemble of nodes of the same network properties do have some reliable characters .for instance , it has been shown that in complex networks the high - degree nodes are on average more synchronizable than the low - degree ones .regarding to the problem of node activities , it is natural to ask the similar question : are the high - degree nodes more synchronized than the low - degree nodes ? in fig .9 we plot the average attaching probability as a function of degree . still , we can not find a clear dependence of on .it is worthy of note that our studies of active nodes are only focused on the giant - cluster state , and the purpose is to understand their dynamics and reveal their properties . by ensemble average , we may able to improve our prediction of the active nodes , say for example the dependence of on in fig .9 may be smoothed if we average the results over a large number of network realizations .such an improvement , however , comes at the cost of the decreased prediction accuracy due to the increased candidates . taking fig . 9 as an example , although it is noticed that nodes of in general are more active than those of other degrees , only one of them is listed as the most unstable nodes [ tab . 1 ] . in specific , among the total number of nodes which have degree , most of them are tightly attracted to the giant cluster ( percent of them have attaching probabilities ) .therefore , in terms of precise predication , the average method is infeasible in practice .  beside node degree and betweenness ,we have also checked the dependence of the property of node activity to some other well - known network properties such as the clustering coefficient , the modularity , and the assortativity . however , none of them is suitable to characterize the active nodes , their performance is very similar to that of the node degree described in tab . 1 and tabour study thus suggests that , to give a precise prediction to the active nodes , we may need to develop some new quantities .  despite of the amount of studies carried on network synchronization , to the best of our knowledge, we are the first to study the nonstationary pattern in unsynchronizable complex networks . in ref . the authors have discussed the transient process of global synchronization in complex networks , but their study are concentrating on the synchronizable state in which , during the course of system evolution , small clusters integrate into larger clusters monotonically and finally reach the synchronization state .after that , the system will always stay on the synchronization state .our works are also different to the studies of refs . . similar to our works , in these studies the authors also consider the problem of pattern formation in unsynchronizable networks , but their interests are focused on the _ stationary pattern _ of the system .that is , the size and contents of the clusters do not change with time .in contrast , in our studies both the size and contents of the clusters are variable .  in summary ,we have reported and investigated a kind of new phenomena in network synchronization : the nonstationary pattern .that is , the final state of the network settles neither to the synchronization state nor to any stationary state of fixed pattern , the system is travelling among all the possible patterns in an intermittent fashion ( the pattern can be of any configuration , but its probability of showing up is pattern - dependent ) .we attribute this nonstationarity to the asymmetric topology of the complex networks , and its dynamical origin can be understood from the property of the finite - time lyapunov exponent associated to the desynchronized mode .two types of synchronization formats , the complete synchronization and the temporal phase synchronization , have been employed to detect the nonstationary dynamics . for coupling strength immediately out of the stable region , the pattern evolution is characterized by the process of on - off intermittency and the existence of the giant - cluster ; while if the coupling strength is far away from the bifurcation points , the pattern evolution is signatured by the random interactivities among the number of small clusters .a remarkable finding is that , in the giant - cluster state the locations of the active nodes are independent of the coupling strength but are sensitive to the bifurcation types .the active nodes , however , can not be characterized by the currently known network properties , further investigations about their identification are necessary .while we are hoping our studies about nonstationary pattern could give some new understandings to the dynamics of coupled complex systems , we also hope that our findings about unsynchronizable networks could be used to some practical problems where system maintains their normal functions only under the unsynchronizable states , for example the problem of epileptic seizers .s. boccaletti and l.m .pecora , chaos * 16 * , 015101 ( 2006 ) ; a. e. motter , m. a. matas , j. kurths , e. ott , physica d * 224 * , 7 ( 2006 ) ; s. boccaletti , v. latora , y. moreno , m. chavez , and d .- u .hwang , phys .* 424 * , 175 ( 2006 ) .motter , c. zhou , and j. kurths , europohys .lett . * 69 * , 334 ( 2005 ) ; phys .e * 71 * , 016116 ( 2005 ) ; aip conf .proc . * 776 * , 201 ( 2005 ) ; c. zhou , a.e .motter , and j. kurths , phys .* 96 * , 034101 ( 2006 ) .m. zhan , z.g .zheng , g. hu , and x.h .peng , phys .e * 62 * , 3552 ( 2000 ) ; y. zhang , g. hu , h.a .cerdeira , s. chen , t. braun , and y. yao , phys .e * 63 * , 026211 ( 2001 ) ; b. ao and z. zheng , europhys .74 * , 229 ( 2006 ) ; x. zhang , m. fu , j. xiao , and g. hu , phys .e * 74 * , 015202 ( 2006 ) .e. ott and j.c .sommerer , phys .a * 188 * , 39 ( 1994 ) ; m. ding and w. yang , phys .e * 56 * , 4009 ( 1997 ) ; s.h .wang , j. xiao , x.g .wang , b. hu , and g. hu , eur .j. b * 30 * , 571 ( 2002 ) ; j.g .restrepo , e. ott , and b. hunt , phys .* 93 * , 114101 ( 2004 ) .pikovsky , m.g .rosenblum , and j. kurths , _ synchronization : a universal concept in nonlinear science _( cambridge university press , cambridge , uk , 2001 ) ; s. strogatz , _ sync : the emerging science of spontaneous order _ ( hyperion , new york , 2003 ) .

Abstract: pattern formation and evolution in unsynchronizable complex networks are investigated . due to the asymmetric topology , the synchronous patterns formed in complex networks are irregular and nonstationary . for coupling strength immediately out of the synchronizable region , the typical phenomenon is the on - off intermittency of the system dynamics . the patterns appeared in this process are signatured by the coexistence of a giant cluster , which comprises most of the nodes , and a few number of small clusters . the pattern evolution is characterized by the giant cluster irregularly absorbs or emits the small clusters . as the coupling strength leaves away from the synchronization bifurcation point , the giant cluster is gradually dissolved into a number of small clusters , and the system dynamics is characterized by the integration and separation of the small clusters . dynamical mechanisms and statistical properties of the nonstationary pattern evolution are analyzed and conducted , and some scalings are newly revealed . remarkably , it is found that the few active nodes , which escape from the giant cluster with a high frequency , are independent of the coupling strength while are sensitive to the bifurcation types . we hope our findings about nonstationary pattern could give additional understandings to the dynamics of complex systems and have implications to some real problems where systems maintain their normal functions only in the unsynchronizable state .


the concept of templates was introduced in a linear setting .they answered to the computational issue of the polyhedra domain , that is , the number of faces and the number of vertices both explode when performing the code analysis .recently , generalizations of linear templates appeared , such as quadratic lyapunov functions as nonlinear templates .nevertheless , no precise characterization of the templates to use have been developed for program analysis purpose .indeed , depending on the property to show , prefixing a template basis without any rules can lead to unuseful information on the programs .for instance , suppose that we want to show that the values taken by the variables of the program are bounded .then , it is natural to use intervals or norm functions as templates .unfortunately , these functions are not sufficient to show the desired property . in the context of linear systems in optimal control, it is well known that lyapunov functions provide useful templates to bound the variable values .this result can be extended to polynomial systems using polynomial lyapunov functions .the crucial notion behind is that these polynomial functions allow to define sublevel sets which are invariant by the dynamics -in our case , the dynamics being the loop body . in static analysis, lyapunov functions provide inductive invariants , which are precisely the results of computation while using template abstract domains .[ [ related - works . ] ] related works .+ + + + + + + + + + + + + +  template domains were introduced by sankaranarayanan et al . , see also  .the latter authors only considered a finite set of linear templates and did not provide an automatic method to generate templates .linear template domains were generalized to nonlinear quadratic cases by adj et al . in  , where the authors used in practice quadratic lyapunov templates for affine arithmetic programs .these templates are again not automatically generated .roux et al . provide an automatic method to compute floating - point certified lyapunov functions of perturbed affine loop body updates .they use lyapunov functions with squares of coordinate functions as quadratic template bases in case of single loop programs written in affine arithmetic .the extension proposed in  relies on combining polynomial templates with sum - of - squares ( sos ) techniques to certify nonlinear inequalities .proving polynomial inequalities is already np - hard and boils down to show that the infimum of a given polynomial is positive .however , one can obtain lower bounds of the infimum by solving a hierarchy of moment - sos relaxations , introduced by lasserre in  .recent advances in sos optimization allowed to extensively apply these relaxations to various fields , including parametric polynomial optimization , optimal control , combinatorial optimization , _ etc . _ ( see e.g.  for more details ) . in the context of hybrid systems ,certified inductive invariants can be computed by using sos approximations of parametric polynomial optimization problems  . in , the authors develop an sos - based methodology to certify that the trajectories of hybrid systems avoid an unsafe region . recently , ahmadi et al . investigate necessary or sufficient conditions for sos - convex lyapunov functions to stabilize switched systems , either in the linear case or when the switched system is the convex hull of a finite number of nonlinear criteria .  in a static analysis context ,polynomial invariants appear in  , where invariants are given by polynomial inequalities ( of bounded degree ) but the method relies on a reduction to linear inequalities ( the polyhedra domain ) .template polyhedra domains allow to analyze reachability for polynomial systems : in  , the authors propose a method that computes linear templates to improve the accuracy of reachable set approximations , whereas the procedure in  relies on bernstein polynomials and linear programming , with linear templates being fixed in advance .bernstein polynomials also appear in  as template polynomials but there are not generated automatically . in  ,the authors use smt - based techniques to automatically generate templates which are defined as formulas built with arbitrary logical structures and predicate conjunctions .other reductions to systems of polynomial _ equalities _( by contrast with polynomial inequalities , as we consider here ) were studied in and more recently in  .  [ [ contribution - and - methodology . ] ] contribution and methodology .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +  in this paper , we generate polynomial templates by combining the approach of sos approximations extensively used in control theory with template abstract domains originally introduced in static analysis .we focus on analyzing programs composed of a single loop with polynomial conditional branches in the loop body and polynomial assignments . for such programs ,our method consists in computing certificates which yield sufficient conditions that a given property holds .we introduce the notion of _ well - representative _ templates with respect to this property . computing inductive invariant and polynomial templatesboils down to solving a system of functional inequalities . for computational purpose ,we strengthen this system as follows :  1 .we impose that the functions involved in each inequality of the system belong to a convex cone included in the set of nonnegative functions .this allows in turn to define the stronger notion of _ well - representative _ templates .2 .  instantiating to the cone of sos polynomials leads to consider a hierarchy of sos programs , parametrized by the degrees of the polynomial templates .while solving the hierarchy , we extract polynomial template bases and feasible invariant bounds together with ( sos - based ) certificates that the desired property holds .the potential of the method is demonstrated on several `` toy '' nonlinear programs , defined with medium - size polynomial conditionals / assignments , involving at most 4 variables and of degree up to 3 .numerical experiments illustrate the hardness of program analysis in this context , as simple nonlinear examples can already yield unexpected behaviors .[ [ organization - of - the - paper . ] ] organization of the paper .+ + + + + + + + + + + + + + + + + + + + + + + + + +  the paper is organized as follows . in section[ sem - functional ] , we present the programs that we want to analyze and their representation as constrained piecewise discrete - time dynamical system .next , we recall the collecting semantics that we use and finally remind some required background about abstract semantics for generalized template domains .section [ template - generation ] contains the main contribution of the paper , namely the definition of well representative templates and how to generate such templates in practice using sos programming .section [ bench ] provides practical computation examples for program analysis .in this section , we describe the programs which are considered in this paper .next , we explain how to analyze them through their representation as discrete - time dynamical systems .then , we give details about the special properties which can be inferred on such programs . finally , we recall mandatory results for abstract template domains that are used in the sequel of the paper .   in this paper , we are interested in analyzing computer science programs .we focus on programs composed of a single loop with a possibly complicated switch - case type loop body .this loop is supposed to be written as a nested sequence of _ if _ statements .moreover we suppose that the analyzed programs are written in static single assignment ( ssa ) form , that is each variable is initialized at most once .we denote by the vector of the program variables . finally , we consider assignments of variables using only _ parallel assignments _ .tests are either weak inequalities or strict inequalities .we assume that assignments are functions from to and test functions are functions from to . in the program syntax, the notation will be either or .the form of the analyzed program is described in figure [ programstyle ] .    .... x  ; while ( (x)0 and ... and (x)0 ) {  if((x)0 ) {        if((x)0 ) {     x = (x ) ;    }     else {          if((x)0 ) {       x = (x ) ;       }     }  else {      }  } ....+  as depicted in figure [ programstyle ] , an update of the -th condition branch is executed if and only if the conjunction of tests holds .the variable is updated by if the current value of belongs to .consequently , we interpret programs as _ constrained piecewise discrete - time dynamical systems _ ( cpds for short ) .the term _ piecewise _ means that there exists a partition of such that for all , the dynamics of the system is represented by the following relation , for : we assume that the initial condition belongs to some compact set . for the program , is the set where the variables are supposed to be initialized in . since the test entry forthe loop condition can be nontrivial , we add the term _ constrained _ and denotes the set representing the conjunctions of tests for the loop condition .the iterates of the cpds are constrained to live in : if for some step , then the cpds is stopped at this iterate with the terminal value .we define a partition as a family of nonempty sets such that : from equation , for all there exists a unique such that .a set can contain both strict and weak inequalities and characterizes the set of the conjunctions of tests functions .let stands for the vector of tests functions associated to the set .moreover , for , we denote by ( resp . ) the part of corresponding to strict ( resp .weak ) inequalities .finally , we obtain the representation of the set given by equation : we insist on the notation : ( resp . ) means that for all coordinates , ( resp . ) .we suppose that the sets and also admits the representation given by equation  and we denote by the vector of tests functions and by the vector of tests functions .we also decompose and as strict and weak inequality parts denoted respectively by , , and .to sum up , we give a formal definition of cpds .[ pwsdef ] a constrained piecewise discrete - time dynamical system ( cpds ) is the quadruple with :  * is the compact of the possible initial conditions ; * is the set of the constraints which must be respected by the state variable ; * is a partition as defined in equation ; * is the family of the functions from to , w.r.t .the partition satisfying equation .from now on , we associate a cpds representation to each program of the form described at figure [ programstyle ] . since a program admits several cpds representations , we choose one of them , but this arbitrary choice does not change the results provided in this paper . in the sequel , we will often refer to the running example described in example [ running ] .  [ running ]the program below involves four variables and contains an infinite loop with a conditional branch in the loop body .the update of each branch is polynomial .the parameters ( resp . ) are given parameters . during the analysis, we only keep the variables and since and are just memories .  c  .... ; = ; = ; while ( -1 < = 0 ) {  if ( ^2 + ^2 < = 1 ) {     = ;     = ;     = * ^2 + * ^3 ;     = * ^3 + * ^2 ;  }  else {     = ;     = ;     = * ^3 + * ^2 ;     = * ^2 + * ^2 ;  }  } ....  its constrained piecewise discrete - time dynamical system representation corresponds to the quadruple , where the set of initial conditions is : \times [ b_1 , b_2 ] \,,\ ] ] the set in which the variable lies is : the partition verifying equation  is : and the functions relative to the partition are :    the main goal of the paper is to decide automatically if a given property holds for the analyzed program .we are interested in numerical properties and more precisely in properties on the values taken by the -uplet of the variables of the program .hence , in our point - of - view , a property is just the membership of some set . in particular , we study properties which are valid after an arbitrary number of loop iterates .such properties are called _ loop invariants _ of the program .formally , we use the cpds representation of a given program and we say that is a loop invariant of this program if : where is defined at equation  as the state variable at step of the cpds representation of the program .now , let us consider a program of the form described in figure [ programstyle ] and let us denote by the cpds representation of this program .the set of _ reachable values _ is the set of all possible values taken by the state variable along the running of .we define as follows : to prove that a set is a loop invariant of the program is equivalent to prove that .we can rewrite by introducing auxiliary variables , : let us denote by the set of subsets of and introduce the map defined by : we equip with the partial order of inclusion and by the standard component - wise partial order . the infimum is understood in this sense i.e. as the greatest lower bound with respect to this order .the smallest fixed point problem is : it is well - known from tarski s theorem that the solution of this problem exists , is unique and in this case , it corresponds to where are defined in equation .tarski s theorem also states that is the smallest solution of the following problem :   we warn the reader that the construction of is completely determined by the data of the cpds .but for the sake of conciseness , we do not make it explicit on the notations .note also that the map corresponds to a standard transfer function ( or collecting semantics functional ) applied to the cpds representation of a program .since , the transfer function associated to the cpds of example [ running ] is given by :   to prove that a subset is a loop invariant , it suffices to show that satisfies .nevertheless , is still not computable and we use abstract interpretation  to provide safe over - approximations of .next , we use generalized abstract template domains as abstract domains and we construct a safe over - approximation of using a galois connection . in this paper , we consider invariants defined from properties which are encoded with sublevel sets of given functions . a loop invariant is supposed to be the union of sublevel sets of a given function from to .[ funproperty ] given a function from to , we define the sublevel property as follows :   1 .let be a norm on , then is the property `` the values taken by the variables are bounded '' .2 .  let , then is the property `` the values taken by the variable are bounded from above '' .3 .  we can ensure that the set of possible values taken by the program variables avoids an unsafe region with a fixed level sublevel property .for example , if the property to show consists in proving that the square norm of the variable is still greater than 1 , we can set and restrict the sublevel sets to those for which .a sublevel property is called _ sublevel invariant _ when this property is a loop invariant .we describe how to construct template bases , so that we can prove that a sublevel property is a sublevel invariant .the concept of generalized templates was introduced in  .let stands for the set of functions from to .[ templatedef ] a generalized template is a function from to over the vector of variables .templates can be viewed as implicit functional relations on variables to prove certain properties on the analyzed program .we denote by the set of templates .first , we suppose that is given by some oracle and say that forms a template basis . here , we recall the required background about generalized templates ( see  for more details ) .   we replace the classical concrete semantics by meaning of sublevel sets i.e. we have a functional representation of numerical invariants through the functions of . an invariant is determined as the intersection of sublevel sets . the problem is thus reduced to find optimal level sets on each template .let stands for the set of functions from to .[ concretisation ] for , we associate the -sublevel set given by :   in convex analysis , a closed convex set can be represented by its support function i.e. the supremum of linear forms on the set ( e.g.  ) . here, we use the generalization by moreau  ( see also  ) which consists in replacing the linear forms by the functions .  [ ppsupport ] to , we associate the abstract support function denoted by and defined by :   let and be two ordered sets equipped respectively by the order and .let be a map from to and be a map from to .we say that the pair defines a galois connection between and if and only if and are monotonic and the equivalence holds for all and all .we equip with the partial order of real - valued functions i.e. . the set is equipped with the inclusion order .[ galois ] the pair of maps and defines a galois connection between and the set of subsets of .  in the terminology of abstract interpretation , is the abstraction function , and is the concretisation function .the galois connection result provides the correctness of the semantics .we also remind the following property :     now , we are interested in closed elements ( in term of galois connection ) , called -convex elements .[ abstractconvexity ] let , we say that is a -convex function if .a set is a -convex set if .we respectively denote by and the set of -convex functions of and the set of -convex sets of .the family of functions is ordered by the partial order of real - valued functions .the family of sets is ordered by the inclusion order .galois connection allows to construct lattice operations on -convex elements .let and be in .we denote by and the functions defined respectively by , and .we equip with the join operator and the meet operator .similarly , we equip with the join operator and the meet operator .the next theorem follows readily from the fact that the pair of and defines a galois connection ( see e.g.  ) .  [ lattice ] the complete lattices and are isomorphic .since the pair of maps and is a galois connection ( proposition [ galois ] ) , we can construct abstract semantics functional from this pair and the map defined at equation .we obtain a map from to itself defined for and by : since is conditioned by the data of the cpds , it is also the case for . as a corollary of theorem [ lattice ] ,the best abstraction of in the lattice is the smallest fixed point of equation . the infimum is understood in the sense of the order of the component - wise order of the complete lattice .using tarski s theorem , the solution of equation  exists and is unique and is usually called the abstract semantics .this latter solution is optimal but any feasible solution could provide an answer to decide whether a sublevel property is an invariant of the program .[ feasibleinvariantbound ] the function is a feasible invariant bound w.r.t . to the cpds iff it exists such that : in the sequel , we denote by the set of feasible invariant bounds .  from the definition of feasible invariant bound , we state the following proposition .[ basicresults ] let us consider a cpds .the following statements are true :  1 .let be a solution of problem , then is the smallest feasible invariant bound w.r.t . ; 2 .  for all , .for a given program represented by the cpds , we recall that an invariant is to said be an _ inductive invariant _ of this program if for all , the implication holds for the state variable . next , for a given function , we give a simple condition in term of inductive invariants ( up to test functions ) for to be a feasible invariant bound .[ inductive ] let us consider the cpds and .suppose that : then .  from the definition of the operator and proposition [ galois ] , conjunction  holds with for all .we recalled that abstract template domains produce invariants , i.e. -sublevel sets of feasible invariant bounds .it is not surprising since abstract template domains are abstract domains .the main issue is that is supposed to be given .the question is which templates basis can produce a nontrivial ( strictly smaller that ) feasible invariant bound ?this question can be refined when we want to show that some sublevel property is an invariant : which templates basis can ensure that the sublevel property is an invariant of the program ?we propose an answer by considering equation  as a system of equations , where unknowns are the template basis and . given a sublevel , we also impose that and satisfy .this latter constraint leads to the computation of a level for which is an invariant of the program .here , we describe how to certify that a sublevel property is a loop invariant using sum - of - squares ( sos ) approximations . in section [ general ] ,we provide a formal definition of the set of template bases that we shall use to the latter certification .then we describe how to construct template bases so that we can prove sublevel properties ( section [ simplemethods ] ) . in the end , we explain how to compute such bases in practice , by solving a hierarchy of sos programs ( section [ sos ] ) .[ boundwellrepre ] let be a sublevel property and be a cpds .the template basis is well - representative w.r.t . and iff there exists such that .  in the sequel, we fix a cpds and a sublevel property .well - representative template bases explicit the sets of implicit functional relations on the program variables , needed to prove that a sublevel property is an invariant .next , we define a cone structure to strengthen the notion of well - representative bases.  [ convexcones ] a non - empty subset of is a convex cone containing the scalars iff :  1 .  for all , for all , ; 2 .  for all , ; 3 .  for all , ;  in the sequel , we write instead of , for each . for a convex cone containing the scalars , stands for the set of vectors of elements of and stands for the set of tableaux of elements of . for ,we denote the `` row m '' of by and the `` column j '' of by .thus refers to the element of the tableau .we derive a stronger notion of well - representative template bases , namely well - representative template bases this notion is more restrictive , as a well - representative template basis deals with a system of inequalities instead of conjunctions of implications .[ kwell ] a finite template basis is a well - representative template basis w.r.t . and iff there exist , , and for all , there exist , , such that :  1 .initial condition satisfiability : , 2 .`` local '' branch satisfiability : , : 3 .property satisfiability :   for the sake of presentation , let us define for all , for all :   [ strongex ] consider example [ running ] .we are interested in proving the boundedness of the values taken by the variables of the program . for , let consider . recall that \times [ b_1 , b_2],\  x^0={{\mathbb r}^d},\ x^1=\{x\in{\mathbb r}^2\mid x_1 ^ 2+x_2 ^ 2\leq 1\},\quad x^2=\{x\in{\mathbb r}^2\mid -x_1 ^ 2-x_2 ^ 2< -1\ } \in \xin a ] stands for the set of -variate polynomials and  ] . in the sequel, we assume that the data of the cpds representation of some analyzed program are polynomials , that is for all ,  ] , for all ,  ] .we look for a single polynomial template  ] well - representative w.r.t . and ,thus satisfies the three conditions of definition [ kwell ] .one way to strengthen the three conditions of definition [ kwell ] is to take , for all , , then to consider the following _ hierarchy _ of sos constraints , parametrized by the integer : \enspace , w \in { \mathbb r}\enspace , \\\sigma_0 \in \sigma[x ] \enspace , \ \deg \sigma_0 \leq 2 m \enspace , \\ \forall\ , j = 1 , \dots , n_{\mathrm{in}}\enspace , \\sigma_j \in \sigma[x ] \enspace , \\deg ( \sigma_j g_j ) \leq 2 m  \enspace , \\ \\\forall\ , i\in{\mathcal{i}}\enspace , \\sigma^i\in \sigma[x]\enspace , \\deg ( \sigma^i ) \leq 2 m \deg t^i \enspace,\\ \forall\ , i\in{\mathcal{i}}\enspace , \\forall\ , j=1,\ldots , n_i \enspace,\ \mu_j^i\in\sigma[x]\enspace , \ \deg ( \mu_j^i r_j^i )  \leq 2 m \deg t^i \enspace , \\ \forall\ , i\in{\mathcal{i}}\enspace , \\forall\ , j=1,\ldots , n_0\enspace , \\gamma^i\in \sigma[x]\enspace , \\deg ( \gamma_j^i r_j^0 )  \leq 2 m \deg t^i  \enspace , \\ \\\psi \in \sigma[x ] \enspace,\ \deg ( \psi ) \leq 2 m  \enspace .\\\end{array } \right.\ ] ] for an integer , we denote by the set of constraints on the decision variables and depicted at equation .  as objective function, we choose to minimize .the intuition behind this choice is that is enforced to be equal to which defines the level for which is an invariant of the program associated to the cpds .when is the norm , a minimal value ( and thus ) would be the smallest computable bound on the norm of the state variable .thus we synthetize a polynomial template of degree at most by solving the following minimization problem :   hence , computing the polynomial template  ] and for all , the polynomials  ] ( resp . ) the polynomial template ( resp .the upper bound of over ) associated to this solution .let and thus . then and .  as a consequence of the first equality constraint of problem, one has .then , the finite template basis is  ] well - representative template basis w.r.t . and .here , we perform some numerical experiments while solving problem  ( given in section [ sos ] ) on several examples . in section [ benchbound ] , we verify that the program of example [ running ] satisfies some boundedness property .we also provide examples involving higher dimensional cases .then , section [ benchsafe ] focuses on checking that the set of variable values avoids an unsafe region .numerical experiments are performed on an intel core i5 cpu ( ) with yalmip being interfaced with the sdp solver mosek . for the sake of simplicity , we write instead of .[bench ]   [ ex : test ] following example [ running ] , we consider the constrained piecewise discrete - time dynamical system with \times [ 0 , 0.2 ] ] ( see the corresponding square of dots on figure [ fig : test ] ) .the other dots are obtained after successive updates of each point by the program of example [ running ] .the sets of dots in figure [ fig : test ] are obtained with and six successive iterations .  at step , program  already yields a feasible solution , from which one can extract the polynomial template and .the sos certificates extracted from this solution guarantee the boundedness property , that is .figure [ fig : test ] displays in light gray outer approximations of the set of possible values taken by the program of example [ ex : test ] as follows : ( a ) the degree six sublevel set , ( b ) the degree eight sublevel set and ( c ) the degree ten sublevel set . the outer approximation is coarse as it contains the box ^ 2 ] well - representative template basis w.r.t . to and for the program of example [ ex : test ] .we also succeeded to certify that the same property holds for higher dimensional programs , described in example [ ex : test3 ] ( ) and example [ ex : test4 ] ( ) .[ ex : test3 ] here we consider \times [ 0 , 0.2]^2 ] , , , , , and .table [ table : bench ] reports several data obtained while solving problem  at step , ( ) , either for example [ ex : test ] , example [ ex : test3 ] or example [ ex : test4 ] .each instance of problem  is recast as an sdp program , involving a total number of `` nb .vars '' sdp variables , with an sdp matrix of size `` mat .we indicate the cpu time required to compute the optimal solution of each sdp program with mosek .the symbol `` '' means that the corresponding sos program could not be solved within one day of computation .these benchmarks illustrate the computational considerations mentioned in section [ sos ] as it takes more cpu time to analyze higher dimensional programs .note that it is not possible to solve problem  at step for example [ ex : test4 ] .a possible workaround to limit this computational blow - up would be to exploit the sparsity of the system ..comparison of timing results for example [ ex : test ] , [ ex : test3 ] and [ ex : test4 ] [ cols="^,^,^,^,^,^",options="header " , ]   [ table : bench ]   here we consider the program given in example [ ex : test4 ] .one is interested in showing that the set of possible values taken by the variables of this program does not meet the ball of center and radius .[ ex : testout ] let consider the cpds with \times [ 0.5 , 0.7 ] $ ] , with , with , with and , . with ,one has and one shall prove that .note that is not a norm , by contrast with the previous examples .at steps , program  yields feasible solutions with nonnegative bounds .hence , it does not allow to certify that is empty .this is illustrated in both figure [ fig : testout ] ( a ) and figure [ fig : testout ] ( b ) , where the light grey region does not avoid the ball .however , solving the sos feasibility program at step yields a negative bound together with a certificate that avoids the ball ( see figure [ fig : testout ] ( c ) ) .finally , is a single polynomial template basis w.r.t . and with the restriction that for some for the program of example [ ex : testout ] .in this paper , we give a formal framework to relate the template generation problem to the property to prove on analyzed program : well - representative templates .we proposed a practical method to compute well - representative template bases in the case of polynomial arithmetic using sum - of - squares programming .this method is able to handle non trivial examples , as illustrated through the numerical experiments .topics of further investigation include refining the invariant bounds generated for a specific sublevel property , by applying the policy iteration algorithm .such a refinement would be of particular interest if one can not decide whether the set of variables values avoids an unsafe region when the feasible invariant bound yields a negative value for .for the case of boundedness property , it would allow to decrease the value of the bounds on the variables .finally , our method could be generalized to a larger class of programs , involving semialgebraic or transcendental assignments , by using the same polynomial reduction techniques as in  .

Abstract: template abstract domains allow to express more interesting properties than classical abstract domains . however , template generation is a challenging problem when one uses template abstract domains for program analysis . in this paper , we relate template generation with the program properties that we want to prove . we focus on one - loop programs with nested conditional branches . we formally define the notion of well - representative template basis with respect to such programs and a given property . the definition relies on the fact that template abstract domains produce inductive invariants . we show that these invariants can be obtained by solving certain systems of functional inequalities . then , such systems can be strengthened using a hierarchy of sum - of - squares ( sos ) problems when we consider programs written in polynomial arithmetic . each step of the sos hierarchy can possibly provide a solution which in turn yields an invariant together with a certificate that the desired property holds . the interest of this approach is illustrated on nontrivial program examples in polynomial arithmetic .   # 1  # 1#2#1,#2


the oberbeck - boussinesq approximation , so named after the pioneering works by oberbeck and boussinesq , is the basis of most of the contemporary studies on natural or mixed convection flows .very interesting historical surveys on the origins of this approximation are available in the recent papers by zeytounian and bois .although the nature of this approximation is very clear and unambiguous with reference to the mass and momentum balance equations for the fluid , the formulation of the approximated energy balance equation is not so definite and univocal .the questions concerning the energy balance equations are the following :  )   which is the specific heat involved in the energy balance ?  is there a pressure work term in the energy balance , proportional to the convective derivative of the pressure field ?the textbooks on fluid dynamics and heat transfer generally give clear answers to these questions .the problem , as it will be discussed in section [ enbal ] , is that the answers are different .  in a recent technical note , the present author carried out a first analysis of the existing formulations of the local energy balance adopted in the oberbeck - boussinesq approximation of buoyant flows .  the purpose of this short paper is to extend the analysis performed in ref . in order to point out the manifold nature of the energy balance formulations , within the framework of the oberbeck - boussinesq approximation , available in the literature .then , a thermodynamic argument is proposed in order to give answers to questions a ) and b ) . in particular, it will be concluded that the answer to question a ) depends on the fluid being a liquid or a gas . for a perfect gasthe answer to question a ) is definitely : `` the specific heat at constant volume '' . for a liquidthe answer to question a ) is less definite , but sufficiently reliable : `` the specific heat at constant pressure '' .the answer to question b ) is : `` no pressure work term appears in the energy balance '' . in a final section , the analysis of the energy balance is applied to the topic of buoyant flows in fluid saturated porous media .as is well known , the oberbeck - boussinesq approximation implies that the local mass and momentum balance equations be written as  where the summation over repeated indices is assumed . in eq .( [ 2 ] ) , is the substantial or convective derivative . in eqs .( [ 1 ] ) and ( [ 2 ] ) , is the velocity field , is the position vector , is the time , is the temperature , is the gravitational acceleration , is the kinematic viscosity , is the isobaric coefficient of thermal expansion , and are the reference density and the reference temperature respectively , while is the substantial or convective derivative . in eqs .( [ 1 ] ) and ( [ 2 ] ) , the properties and are also referred to the temperature . in eqs .( [ 1 ] ) and ( [ 2 ] ) , the properties and are referred to the temperature . the implicit assumptions behind eqs .( [ 1 ] ) and ( [ 2 ] ) are that : is the difference between the pressure and the hydrostatic pressure , and that one considers the density as coincident with the reference value except for the gravitational body force term . for that term , the density is assumed to be a function of the temperature only , thus considering the dependence on the pressure as negligible . the linear equation of state   \label{3}\ ] ]is implicitly invoked in eq .( [ 2 ] ) . in eq . ( [ 3 ] ) , the dependence on is assumed to be sufficiently weak to be approximated linearly in the surroundings of the reference value .for the energy balance , the formulation of the oberbeck - boussinesq approximation is not so definite in the literature .in fact , one may have chandrasekhar s and white s formulation where is the thermal conductivity , is the dynamic viscosity and is the strain tensor the source term in eq .( [ 5 ] ) , , is the thermal power generated by the viscous dissipation .one may have the enthalpy formulation  where the last term on the right hand side is an additional source term : the pressure work acting on the fluid element .finally , one may have landau - lifshitz s , bejan s and kundu - cohen s formulation in order to decide on the most convenient expression of the energy balance , let us write the general non - approximated form of this balance , _i.e. _ the local version of the first law of thermodynamics where is the internal energy per unit mass , is the fluid stress tensor and is the heat flux density .the meaning of the terms on the right hand side of eq .( [ 8 ] ) is straightforward : the first term is the incoming heat flux contribution to the energy change , while the second term is the mechanical work input due to the forces acting on the boundary of the fluid element .the latter term depends only on mechanical quantities , _i.e. _ on the velocity and pressure fields within the fluid domain . on the other hand ,the evaluation of the energy change on the left hand side of eq .( [ 8 ] ) implies the use of thermodynamics .thermodynamics ensures that for every single - phase or two - phase stable equilibrium states . in the special case of a perfect gas, it is well known that  , so that in the case of either a liquid or a real gas , one must rely on the oberbeck - boussinesq approximation by assuming that an approximate equation of state can be applied .this implies that the pressure of the fluid does not change appreciably . since andsince the pair yields a unique stable equilibrium state , then one concludes that all the thermodynamic properties may be considered as functions of .this conclusion holds for the internal energy per unit mass , so that a relationship can be established .the thermodynamic coefficient , in general , does not coincide either with or with .in fact , is the total derivative of the function with respect to , and not the partial derivative of with kept constant .as is well known , the latter is the correct thermodynamic definition of  .the equation of state is one regarding a set of stable equilibrium states of the fluid with approximately the same pressure .then , one has eq .( [ 11 ] ) is not the definition of the specific heat at constant pressure .as is well known , the latter is defined as where is the enthalpy per unit mass .then , one can easily write the following relationship : where the definition of the coefficient of isobaric expansion has been used . then , is smaller than and differs from , except for the limiting case of a perfect gas . indeed , in the latter case , one can easily show that the equation of state of the perfect gas and eq .( [ 13 ] ) ensure that , so that eqs .( [ 9 ] ) and ( [ 10 ] ) are perfectly consistent .with reference to a newtonian fluid and on account of eq .( [ 1 ] ) , one may express .then , by employing eq .( [ 10 ] ) , by replacing with and by assuming , eq .( [ 8 ] ) can be approximated as in eq .( [ 15 ] ) , the value of the thermodynamic coefficient , given by eq .( [ 13 ] ) , is referred to the temperature like any other fluid property involved in the oberbeck - boussinesq system of governing equations .( [ 15 ] ) coincides with eq .( [ 4 ] ) only for the limiting case of a perfect gas and differs from eq .( [ 7 ] ) as .   with and for water at atmospheric pressure , scaledwidth=40.0% ]  [ fig1 ]   with for water at atmospheric pressure , title="fig:",scaledwidth=40.0% ]   [ fig2 ]  one can question the extent to which and differ from in the case of liquids . for water at atmospheric pressure ,a precise evaluation of the discrepancies , and can be done by means of the data reported in appendix c of bejan s textbook .in fact , on account of eq .( [ 13 ] ) , one has figure [ fig1 ] evidently suggests that the discrepancy is rather close to and that is much smaller .figure [ fig2 ] shows that is , on average , of order . as a consequence , for water at atmospheric pressure ,the approximation is a definitely reliable one . in general , it is not easy to find data for the specific heat at constant volume of a liquid . usually , thermodynamic tables report the values of , while is evaluated from the mayer relationship  where is the coefficient of isothermal compressibility , on account of eqs .( [ 16 ] ) and ( [ 18 ] ) and of the data reported in refs . , one may evaluate the discrepancies , and for some organic liquids at and atmospheric pressure .these data are reported in table [ tab : tab1 ] .this table suggests again that eq .( [ 17 ] ) is definitely reliable with an error smaller than , while the approximation would lead to an error higher than ..specific heat ratios for some organic liquids at and atmospheric pressure [ cols="<,^,^,^",options="header " , ]   [ tab : tab1 ]in the preceding section , a thermodynamic strategy has been established to determine the most appropriate formulation of the local energy balance equation within the oberbeck - boussinesq approximation .the basis of this approach is twofold .  * the convective derivative of the internal energy per unit mass , ,is evaluated by considering the thermodynamic process undergone by the fluid element .this process can be reliably modeled as an isobaric process . *the mechanical work input is evaluated according to the stress - strain relationship for a newtonian fluid , as well as to the constraint satisfied by the velocity field .  in the light of these arguments ,three possible pitfalls that can be encountered in the determination of the local energy balance equation are described in the following .one could say that the convective derivative can be evaluated by assuming that the fluid element is undergoing an isochoric thermodynamic process .therefore the validity of eq .( [ 9 ] ) is extended , not only to the perfect gases , but to every fluid system .then one would be led to eq .( [ 4 ] ) instead of eq .( [ 15 ] ) .this reasoning is erroneous as the nature of the thermodynamic process undergone by the fluid element is implicitly defined by the equation of state assumed , _i.e. _ eq . ( [ 3 ] ) .this equation of state is based on the hypothesis that the density may change only as a consequence of temperature variations . in other words, one is assuming that the set of stable equilibrium states available to the fluid element are , with a very good approximation , at constant pressure .   a possible misleading argument in the deduction of eq .( [ 15 ] ) is connected to the mechanical work term .this term , as it is clearly explained in chandrasekhar , must be simplified according to the constraint satisfied by the velocity field , _ i.e. _ eq .( [ 1 ] ) . on the other hand , in some textbooks ( see , for instance , kundu and cohen ) , a part of the mechanical work term , namely the expansion - contraction work contribution , , is rewritten by forgetting eq .( [ 1 ] ) and by using the exact local mass balance instead , if one brings this term to left hand side of the local energy balance eq .( [ 8 ] ) , then one has to evaluate instead of .therefore , by following the thermodynamic argument described in section [ enbal ] , one would have where eqs .( [ 10 ] ) , ( [ 13 ] ) and ( [ 14 ] ) have been used . on account of eqs .( [ 20c])-([22c ] ) , one would be led to the formulation of the local energy balance expressed by eq .( [ 7 ] ) , instead of eq .( [ 15 ] ) .however , this is a tricky procedure , since the velocity field evaluated through the oberbeck - boussinesq model is constrained to be solenoidal by eq .( [ 1 ] ) . then , even if it is based on an exact mass balance , eq .( [ 20c ] ) can not be coherently invoked within the oberbeck - boussinesq approximation .one could evaluate the differential of the internal energy per unit mass , , by using the definition of enthalpy per unit mass , .then , one has   if one assumes that the thermodynamic process undergone by the fluid element is isobaric , then one has then , eq .( [ 20b ] ) yields where eq .( [ 13 ] ) has been used .( [ 22b ] ) leads directly to eq .( [ 15 ] ) .if one assumes that the thermodynamic process undergone by the fluid element is isochoric , then one has the thermodynamic identity can be easily proved on the basis of the elementary thermodynamic differential relationships .the complete proof can be found , for instance , in ref . . therefore , by substituting eqs .( [ 23b ] ) and ( [ 24b ] ) into eq .( [ 20b ] ) , one obtains from eq .( [ 25b ] ) , one justifies the relationship obviously , eq . ( [ 26b ] ) leads to eq .( [ 6 ] ) .  while the procedure based on eqs .( [ 20b])-([22b ] ) and leading to eq .( [ 15 ] ) is correct , the procedure based on eqs .( [ 20b ] ) , ( [ 23b])-([26b ] ) and leading to eq .( [ 6 ] ) is completely unjustified for the following two reasons .* the process undergone by the fluid element can not be considered as isochoric as already pointed out in subsection [ isocho ] .* if one has to suppose that the process is isochoric , the straightforward reasoning is to assume the general validity of eq .( [ 9 ] ) , so that one is led to eq .( [ 4 ] ) , as described in subsection [ isocho ] .in other words , the misleading thermodynamic analysis behind eq .( [ 6 ] ) adds an unjustified assumption ( an isochoric process ) to a tricky procedure that replaces the simple conclusion with the more complicated one expressed by eq . ( [ 26b ] ) .this circumstance induces some reflections on how the erroneous assumption of isochoric process can lead to ambiguous results , depending on the procedure followed . on the contrary, the assumption of isobaric process leads exactly to the same result , _i.e. _ eq . ( [ 15 ] ) , either if the reasoning stems from the evaluation of or , as in the present subsection , if one works out the differential of the enthalpy per unit mass , .the remarks on the appropriate form of the local energy balance in the framework of the oberbeck - boussinesq approximation can be easily reformulated with reference to the theory of fluid saturated porous media .in fact , following nield and bejan , the volume - averaged energy equations for the solid and fluid phases can be written as  where is the seepage velocity and the subscripts and denote the solid and the fluid phase , respectively . in eq .( [ 21 ] ) , is the porosity , while is the viscous dissipation term obtained through a volume average of the term appearing in eq .( [ 15 ] ) .as is well known , the specific form of the term depends on the momentum balance model adopted for the fluid saturated porous medium .the inter - phase heat transfer coefficient in eqs .( [ 20 ] ) and ( [ 21 ] ) describes the thermal energy flow between the solid and the fluid phase . since reference is made to the oberbeck - boussinesq approximation , all the solid and fluid properties in eqs .( [ 20 ] ) and ( [ 21 ] ) are evaluated at the reference temperature . with respect to nield and bejan , eq .( [ 21 ] ) has been adapted on the basis of eq .( [ 15 ] ) in order to include the thermodynamic coefficient defined through eq .( [ 13 ] ) . in cases of local thermal equilibrium between the solid phase and the fluid phase , , one can add eqs .( [ 20 ] ) and ( [ 21 ] ) , so that one has where nature of the oberbeck - boussinesq approximation has been revisited in order to clarify some thermodynamic aspects connected to the formulation of the energy balance .the present analysis , motivated by the manifold formulation of this balance in the existing literature , leads to the following conclusions :  1 .the energy balance for a fluid , within the oberbeck - boussinesq approximation , is given by the equation where ; 2 .the thermodynamic coefficient coincides with the specific heat at constant volume , , for a perfect gas ; 3 .the thermodynamic coefficient is definitely well approximated by the specific heat at constant pressure , , for a liquid ; 4 .no pressure work term of the type must be introduced on the right hand side of the energy balance equation .  finally , the most appropriate form of the energy balance for buoyant flows in a fluid saturated porous medium has been discussed .i am deeply indebted with prof. don nield for his encouragement throughout the course of this work , as well as for his very helpful and insightful comments .99 a. oberbeck , ber die wrmeleitung der flssigkeiten bei bercksichtigung der strmungen infolge von temperaturdifferenzen , ann . phys .chem . , neue folge * 7 * , 271 - 292 ( 1879 ) .j. boussinesq , _ thorie analytique de la chaleur _ , vol .ii , gauthier - villars , paris , 1903 . r. kh .zeytounian , joseph boussinesq and his approximation : a contemporary view , c. r. mcanique , * 331 * , 575 - 586 ( 2003 ) .bois , joseph boussinesq ( 1842 - 1929 ) : a pioneer of mechanical modelling at the end of the 19th century , c. r. mcanique , * 335 * , 479 - 495 ( 2007 ) .a. barletta , comments on a paradox of viscous dissipation and its relation to the oberbeck - boussinesq approach , int .j. heat mass transfer , * 51 * , 6312 - 6316 ( 2008 ) .s. chandrasekhar , _ hydrodynamic and hydromagnetic stability _ , oxford university press , oxford , 1961 , pp . 10 - 18 .f. m. white , _ fluid mechanics _ , 4th ed .mcgraw - hill , new york , 2002 , pp .231 - 233 .o. g. martynenko , p. p. khramtsov , _ free - convective heat transfer _ , springer , berlin , 2005 , pp . 1 - 5 .d. l. turcotte , a. t. hsui , k. e. torrance , g. schubert , influence of viscous dissipation on bnard convection , j. fluid mech . , * 64 * , 369 - 374 ( 1974 ). l. d. landau , e. m. lifshitz , _ fluid mechanics _ , 2nd ed ., pergamon press , oxford , 1987 , chapter 5 .a. bejan , _ heat transfer _ , wiley , new york , 1993 , pp .227 - 231 .k. kundu , i. m. cohen , _ fluid mechanics _ , 2nd ed . ,elsevier , new york , 2002 , pp .117 - 120 .v. s. arpaci , p. s. larsen , _ convection heat transfer _ , prentice - hall , new york , 1984 , pp .e. p. gyftopoulos , g. p. beretta , _ thermodynamics : foundations and applications _ , macmillan , new york , 1991 , chapter 19 .d. r. lide ( editor - in - chief ) , _ crc handbook of chemistry and physics _ , 87th ed . ,taylor and francis , boca raton , fl , 2007 .d. a. nield , a. bejan , _ convection in porous media _ , 3rd ed ., springer , berlin , 2006 , chapter 2 .w. p. breugem , d. a. s. rees , a derivation of the volume - averaged boussinesq equations for flow in porous media with viscous dissipation , transp .porous media , * 63 * , 1 - 12 ( 2006 ) .d. a. nield , the modeling of viscous dissipation in a saturated porous medium , asme j. heat transfer * 129 * , 1459 - 1463 ( 2007 ) .

Abstract: a thermodynamic argument is proposed in order to discuss the most appropriate form of the local energy balance equation within the oberbeck - boussinesq approximation . the study is devoted to establish the correct thermodynamic property to be used in order to express the relationship between the change of internal energy and the temperature change . it is noted that , if the fluid is a perfect gas , this property must be identified with the specific heat at constant volume . if the fluid is a liquid , a definitely reliable approximation identifies this thermodynamic property with the specific heat at constant pressure . no explicit pressure work term must be present in the energy balance . the reasoning is extended to the case of fluid saturated porous media .  * keywords * oberbeck - boussinesq approximation ; energy balance ; specific heat ; porous media +  paper submitted to the + international journal of heat and mass transfer