continuously evolving dynamical systems universally appear as models in every branch of natural science .their usefulness is determined by their predictivity which in turn hinges on the stability of their solutions . given some uncertainty in the measured initial conditions , a physically meaningful system should offer control on the possible evolution of deviations from a given reference trajectory . global control of late - time deviations is as important as a local understanding of stability .an extensive mathematical theory exists for the global stability of solutions of dynamical systems ; the fundamental quantities in this theory are the lyapunov exponents that measure exponential deviations  .these exponents are often very hard to determine analytically , and various approaches for their calculation are used in the literature  .sometimes the stability of the same dynamical system is considered from different points of view , and the corresponding lyapunov exponents are compared ; this leads to some debate on the question of which point of view is preferable  .the local stability of solutions of dynamical systems on the other hand seems less well understood . to our knowledge, the existing mathematical theory is applicable only to second order differential systems .it was developed by kosambi , cartan and chern  , and covers as a special case the local stability analysis of geodesics on a riemannian manifold via the jacobi equation .  in this articlewe describe dynamical systems and their perturbations in a unified geometric language which is reviewed in section [ sec_language ] .this language has the advantage to specify very precisely the intrinsic geometric ingredients that define the dynamical system .it is worth emphasizing that the translation of dynamical systems into geometry is unambiguous .however , this is not true for the definition of stability .we will clarify in section [ sec_stability ] that stability in general is not an intrinsic notion but requires geometric structure not included in the definition of the dynamical system .this fact is not sufficiently appreciated in the literature .we will see that global lyapunov stability entails the choice of seminorms on the vector bundle of perturbations .we will then propose a general definition of local stability applicable to any dynamical system .this notion of local stability requires the choice of a linear connection , and reduces to standard stability criteria for second order systems .in contrast to the general case , lagrangian systems offer intrinsic geometric structures to define global and local stability , as we will discuss in section [ sec_lagrange ] . we will then consider natural lagrangian systems in section [ sec_natural ] for which independent notions of stability exist based on the translation of solutions into geodesics via the maupertuis - jacobi principle .we point out that this detour suffers from a number of limitations which are nicely resolved in the direct intrinsic approach valid for all lagrangian systems .we conclude in section [ sec_conclusion ] .in this section we review the basic geometric structure common to all dynamical systems and their perturbations . the geometric language used here will form the basis for our discussion of the stability of solutions in the following sections .  fundamentally, deterministic dynamical systems can be introduced as formal rules that describe the evolution of points in some set with respect to an external , discrete or continuous time parameter running in another set .more precisely , a dynamical system is a map which satisfies for all times . this bare definition must be enriched with additional structure if we want to model realistic dynamical systems .  in the followingwe will therefore concentrate on the important class of continuously evolving dynamical systems for which .geometrically , we focus on point sets that are differentiable manifolds , and assume that a vector field over is given : the integral curves of the vector field through a point are defined by two conditions : that their tangent vector agrees with the vector field at each point , and that . for simplicitywe assume that these integral curves are defined for all times .then a dynamical system in the sense of ( [ dynsyst ] ) is defined by the flow of the vector field , i.e. , by setting .  any integral curve of the flow vector field provides a solution trajectory of the dynamical system with initial condition .perturbations of are studied by varying this initial condition .any curve with provides a continuous family of initial conditions which evolve under the dynamical system as .all curves so obtained are continuous variations of ; as a function of the two parameters and we thus obtain a two - dimensional surface in , see figure [ figure1 ] . in consequencethe two tangent vectors and commute , =0 ] .choosing local coordinates on , the dynamical system thus becomes a system of first - order differential equations over for the integral curves of the vector field , expanding the commutator condition =0 ] .it is instructive to expand this condition in local coordinates on , which yields as expected , the perturbation equation can be read as a second order differential equation . hence we see that the simple geometric framework for dynamical systems and perturbations of solutions can be easily applied to higher order dynamics .in the previous section we have defined dynamical systems via vector fields and their associated flow on some differentiable manifold . we have also presented the geometric language to define linear perturbations of specific solutions as connecting vector fields that commute with  . in this sectionwe discuss the issue of the stability of solutions under such linear perturbations , both globally and locally .we will find that stability generally is not intrinsic to a dynamical system , but requires additional geometric structure beyond and to be well - defined .consider a solution of a dynamical system .the linear perturbations of such a solution are described by tangent vector fields along and satisfy =0 ] vanishes .so we need to require another derivative .the definition of local stability again proceeds in two steps .first , we introduce a linear connection that sends a pair of vector fields over to another vector field over . using the connection, one finds that the second covariant derivative of along the flow is a linear operator on which can be written as \cdot\xi = \tilde r(x)\cdot \xi\,,\ ] ] where  ] on then tells us that our notion for the local stability of geodesics naturally reduces to the standard definition of geodesic stability .so geodesics admit natural definitions of global as well as local stability . in the following sectionwe will show that also lagrangian systems offer intrinsic notions of global and local stability .in the previous sections we have introduced all necessary ingredients for a discussion of the stability of solutions of lagrangian systems on some configuration space manifold . in order to apply the geometric machinery we will first formulate these systems as vector flows over  .then we will demonstrate the existence of natural notions of stability that do not require further information apart from that given by the lagrangian system itself .a lagrangian system is a pair of a configuration space manifold and a lagrangian function .locally this is a function of coordinates on  and coordinates  on that give the components of a vector in the coordinate - induced basis .the trajectories of the lagrangian system are the curves with tangent vectors for which the variation of the action integral vanishes , =\delta\int_{t_1}^{t_2 }  l(\gamma(t),\dot{\gamma}(t))\,dt\,.\ ] ] in the variation the curve s endpoints and are held fixed .once coordinates have been chosen , solutions must satisfy the euler - lagrange equations   as any second order dynamical system , a lagrangian system can be described as a vector flow on . sincethis formulation is not so well - known , we present the relevant steps following  .first , one defines the liouville vector field on .this produces the energy function by .now recall that a vector field on is said to be vertical if it is tangent to the fiber , .moreover , a vector on , has an associated vertical lift to . combining the vertical lift and a tensor field on by ; in coordinates , . with these ingredients, we can construct the two - form on . as a final step , the vector field the lagrangian flow is the unique vector field solving this construction is easily understood in a local chart of .simple algebra shows that writing , condition ( [ elgeo ] ) is equivalent to this shows that the vector field on is a semispray , as needed for a second order dynamical system .the integral curves of indeed solve the euler - lagrange equations .   within the geometric picture of lagrangian systems as dynamical systems we are now in the position to ask for the stability of solutions . we will discuss global and local stability in turn .the definition of global stability via lyapunov exponents requires the specification of a family of seminorms on the tangent bundle .a lagrangian system provides us with an intrinsic object that can be used for this purpose , see e.g. : the generalized lagrange metric although the coefficients of depend on the fiber coordinates for generic lagrangians , the generalized lagrange metric transforms under coordinate transformations of as a -tensor on .regularity of the lagrangian corresponds to .the situation for the lagrangian system is now very similar to that of geodesics on a riemannian manifold . instead of lifting the riemannian metric to , we simply need to extend the generalized lagrange metric  to . as for lifts , there are several possibilities . as an example we consider the case , where as displayed above is simply read as a tensor over . for a reference solution with initial condition and initial perturbation , the lyapunov exponentsare then simply calculated as this is of the same form as the formula for geodesics where we used the vertical lift of the riemannian metric .so this choice of seminorms measures the exponential deviation of perturbations purely within the configuration space .  to define the local stability of solutions of we need to specify a linear connection acting on vector fields on .again the lagrangian system provides us with an intrinsic choice .we may simply use the berwald type linear connection on which is associated to any given semispray .the relevant quantities and needed to determine this connection follow by combining ( [ sem ] ) , ( [ lvec ] ) and the definition of the generalized lagrange metric : in the basis of the connection is then provided by the connection coefficients ( [ bconn ] ) .both the global and local stability of solutions of a lagrangian system can be defined intrinsically , without the introduction of further geometric structure .this makes essential use of the fact that the lagrangian system can be rewritten in the dynamical system language as a vector flow over .the solutions of natural lagrangian systems can be translated using the maupertuis - jacobi principle to geodesics on an associated riemannian manifold .we have seen that both global and local stability can be easily defined for geodesics , and this fact is often used in the literature to analyze the stability of natural lagrangian systems .the main purpose of this section is to clarify the differences between the intrinsic notions of stability discussed in the previous section and the notions of stability obtained from the translation principle .it will turn out that the intrinsic notions of stability have considerable advantages .before we translate natural lagrangian systems into geodesics , we specialize the intrinsic notions of stability developed for general lagrangian systems to this case .a natural lagrangian system is defined by a lagrangian function which is quadratic in the velocities : this corresponds to a dynamical system for which the flow vector field over is determined according to the general expression ( [ lvec ] ) , and in terms of the levi - civita connection of the kinetic energy metric , as   the intrinsic notion of global stability is based on the generalized lagrange metric of ( [ lmet ] ) which here agrees with the kinetic energy metric , . assuming a regular lagrangian and positive energy amounts to the requirement of positive definiteness of .the second order evolution equation of perturbations can be rewritten as the calculation of lyapunov exponents with respect to some extension of to is unproblematic and follows ( [ ll ] ) .the intrinsic notion of local stability is linked to the berwald type connection on .note that the natural lagrangian system has a flow vector field of the type ( [ gsol ] ) , so our general calculation above shows that local stability becomes equivalent to the kcc definition . in practice , therefore , local stability means non - positive eigenvalues of the endomorphism defined in ( [ pcomp ] ) .specializing to the case here we find where has to be evaluated for the tangent vectors along a reference solution .this is the same operator as the one we found governing the evolution of perturbations .reconsider the example of the inverted harmonic oscillator of equation ( [ ex1 ] ) .the generalized lagrange metric now gives a preferred intrinsic choice of seminorms in terms of , thus leading to the expected global lyapunov instability of the system .for local stability we use the fact that our stability criterion and the kcc criterion agree , since we consider a natural lagrangian system . from ( [ pnat ] )we find .this shows the local instability of all solutions under this intrinsic choice of connection .geodesics on a riemannian manifold with fixed affine parametrization can be viewed as a lagrangian system .the converse is also true for natural lagrangian systems .we now state a version of the jacobi - maupertuis principle , including indefinite kinetic energy metrics which occur e.g. in scalar field cosmology  ; then we will use this translation principle to investigate how the geodesic stability and the intrinsic stability of natural lagrangian systems compare .let  be a natural lagrangian system ( [ natl ] ) with non - degenerate quadratic form .then the solution trajectories of fixed energy of the lagrangian system coincide with the geodesics of the metric manifold with for constant ( the conventional choice is ) .the time parameter of the lagrangian system is related to the affine parameter of the geodesics , which we define by , as   to see why the jacobi - maupertuis principle holds , we write the euler - lagrange equations of motion of the lagrangian system as now consider geodesics on a metric manifold with the conformally rescaled metric , parametrized in terms of an arbitrary non - affine parameter also called  .these geodesics satisfy the equation now using the christoffel symbols of the rescaled metric , and may be rewritten in terms of the original kinetic energy metric and the conformal factor as this equation is equivalent to the euler - lagrange equations ( [ eqlagr ] ) provided that the contraction of this condition with is easily integrated with integration constant  , yielding the hamiltonian constraint ( [ hamilton ] ) .we solve this for and substitute back into the above condition for the equivalence of the geodesic and euler - lagrange equations . rearranging the terms we obtain this is satisfied for all possible trajectories only if for positive constant  , which demonstrates the first part of the claim .finally , defines an affine parameter along the geodesics of .we may now use the hamiltonian constraint and our result for to show that relates the affine parameter to the lagrangian time .   using the translation principle ,the stability of solutions of natural lagrangian systems can now be discussed in terms of the stability of the corresponding geodesics .it turns out that this approach has a number of limitations that are not always appreciated in the literature :  * the boundaries of the riemannian manifold ; * the absence of perturbations that modify energy and the change to affine time ; * the problem of one - dimensional systems .these points equally impact the discussion of global and local stability .  [[ boundaries - of - the - riemannian - manifold . ] ] boundaries of the riemannian manifold .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the maupertuis - jacobi principle translates the solutions of a natural lagrangian system with given energy into geodesics of an associated -dimensional metric manifold with metric , see ( [ mjmetric ] ) .this riemannian metric has a singular boundary at all points where the conformal factor vanishes , i.e. , for , even assuming regularity of the kinetic energy metric .the ricci scalar generically diverges for a vanishing .this means that the boundary singularity is a curvature singularity , not an artifact due to a possibly poor choice of coordinates .the fact that the riemannian manifold used to model the lagrangian system has a singular boundary can give rise to difficulties in the discussion of the equivalent dynamical system , see e.g.  .all solutions for which the velocity vanishes at some time parameter will hit this boundary ; therefore their evolution can not be determined in the geodesic picture , even though it is perfectly well - defined on the pure dynamical side . consequently ,neither the global nor the local stability of solutions hitting the boundary can be determined : the geodesics can not be extended outside a limited range of their proper parameter , so the late time limit that needs to be taken for the lyapunov exponents is ill - defined ; also local stability can not be checked along a complete trajectory .moreover note that fixed point solutions , with vanishing velocity everywhere , are not tractable in the geodesic picture at all .this strikingly contrasts the dynamical system analysis , in which stability of fixed points is well understood .the relevance of these issues is easily illustrated already in simple radially symmetric systems with .first , consider the harmonic oscillator potential and solutions of energy . in this case, the riemannian manifold will be restricted to with metric .the ricci scalar is given by , so that the boundary is indeed singular .one may check that all radial geodesics , characterized in polar coordinates by , reach the boundary in finite proper time .therefore the manifold is also geodesically incomplete .these conclusions hold in full generality for similar potentials , too . to illustrate the consequences for the stability analysis we consider the potentials where is the heaviside step function , see figure [ figure2 ] .solutions with and initial position close to are mapped to the same riemannian manifold for both potentials .so the differences of the two dynamical systems can not be resolved in their geometrical image .in particular , the stability of radial solutions under small perturbations is ( under any reasonable intuitive idea of stability ) dramatically different in the two systems , which the geodesic picture is unable to reveal .  on the positive side, we remark that in complex systems solutions reaching the boundary of the manifold are the exception more than the rule , at least if the kinetic energy metric is positive definite .the set of initial conditions for solutions reaching the boundary will be very restricted .solutions with generic initial conditions are expected to be confined to the interior of the manifold . in the examples above ,only radial geodesics reach the boundary since an arbitrarily small angular momentum already prevents points of vanishing velocity along the trajectory .  [[ perturbations - that - modify - energy - and - the - change - to - affine - time . ] ] perturbations that modify energy and the change to affine time .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the energy of the solutions of a lagrangian system directly enters the definition of the metric of the corresponding riemannian manifold . hence a generic lagrangian system can not be mapped completely into a single metric manifold .note that two metric manifolds and corresponding to different choices of fixed energy do not possess isometric regions , which makes it impossible to use part of a given manifold to discuss dynamics at different energy .for the discussion of both global or local stability this means that only perturbations leaving the energy fixed can be considered . from the lagrangian point of view an initial perturbationis represented by a vector in the -dimensional space , where provides initial conditions for the reference solution .a perturbation modifies the energy of the trajectory unless , where is the energy function defined in section [ lageo ] .we can assume , because the system otherwise would be in a fixed point , so restricts to a -dimensional subspace of . in the geodesic translationwe therefore expect to have control only over this subspace .  another point to noteis that the preferred role of the lagrangian time is lost in translation from the lagrangian picture to the geodesic picture . in the latterthe only geometrically natural parameter is the affine parameter along the geodesics .that this has impact on the discussion of stability becomes clear in simple examples , where time rescalings may render unstable solutions stable and vice versa .the non - existence of energy - changing perturbations and the loss of lagrangian time reduce the control of stability of natural lagrangian systems when discussed in the geodesic picture .this fact is nicely reflected in the reduction of the number of relevant exponents in the lyapunov spectrum . to see this note that the evolution equation for the perturbation vector in ( [ 2perturbation2 ] ) can be recast as the jacobi equation ( [ jacobi ] ) using the levi - civita connection of . in a parallely transported orthonormal basis with the jacobi equation becomes choosing as initial perturbation , , gives the solution , , and the corresponding lyapunov exponent clearly vanishes .the perturbation corresponding to this exponent is an infinitesimal shift along the geodesic .the choice of initial conditions , , gives the solution , , and hence a second vanishing lyapunov exponent . in this case , the initial perturbation corresponds to an infinitesimal shift of the initial velocity along the geodesic . as a consequence ,only lyapunov exponents in the spectrum remain significant .[ [ one - dimensional - systems . ] ] one - dimensional systems .+ + + + + + + + + + + + + + + + + + + + + + + +  while it is often argued that the geodesic translation is useful for complex systems  , one must emphasize that it completely breaks down for simple one - dimensional systems .we already noted above that one obtains two vanishing lyapunov exponents for any translated system ; for this means that all lyapunov exponents vanish .so global stability can not distinguish arbitrary one - dimensional systems .the situation is similarly bad for local stability .all one - dimensional riemannian manifolds are flat , so the eigenvalues of the riemann tensor are zero .but then also local stability can not distinguish arbitrary dynamical systems .both concepts become meaningless for one - dimensional systems .  after analyzing the different aspects of the geometric picture, we conclude that the riemannian way to define stability suffers from a number of limitations .general dynamical systems can not be mapped into a unique riemannian manifold , which leads to the presence of singular boundaries that may spoil the method of any predictive power .perturbations that modify the energy can not be discussed at all .systems with a small number of degrees of freedom are affected more severely ; the method is not applicable to the one - dimensional case .none of these limitations apply to the intrinsic definitions of stability .the generalized lagrange metric is just without any conformal factor , so that no singular boundaries can occur .full control on perturbations that change energy is expected ; indeed , the additional contribution from the potential in ( [ soform ] ) in contrast to ( [ geomsplit ] ) removes possible degeneracies that could be responsible for a reduced number of lyapunov exponents .one - dimensional systems do not pose a problem for the intrinsic formalism , as we have already seen in the example of the inverted harmonic oscillator .the use of geometric language clearly specifies the intrinsic ingredients of a given physical system , and at the same time clarifies which mathematical structures may be meaningfully used . in this articlewe have analyzed the stability of solutions of dynamical systems in geometric language .we have restricted ourselves to the important class of continuously evolving systems which can be formulated as vector flows over some smooth manifold .although the solutions simply are integral curves of the vector field , this is not a restriction to first order systems ; the order rather depends on the choice of .linear perturbations of reference solutions are characterized by vector fields  over that commute with . within this basic frameworkwe have reconsidered the question of the stability of solutions under linear perturbations .for the analysis of global stability exists the extensive mathematical theory of lyapunov exponents . herethe use of the geometric formulation nicely clarifies that global stability in general is not an intrinsic notion .it requires outside information that has to be added to the dynamical system  .more precisely , a family of seminorms has to be given on the tangent bundle .this arbitrariness of choice is not sufficiently appreciated in the literature that applies the global stability concept .  in order to analyze the local stability of solutions of have proposed a very intuitive new criterion that requires the additional choice of a linear connection on to measure local accelerations .hence also local stability in general is not intrinsic to the dynamical system .the connection criterion for local stability is applicable to any dynamical system . for the case of second order systemswe have shown how it reduces to the stability theory of kosambi , cartan and chern .in fact , the new criterion is precisely equivalent to kcc theory for natural lagrangian systems .some dynamical systems contain sufficient geometrical structure to allow for completely intrinsic definitions of global and local stability .it is well - known that this is the case for geodesics on a riemannian manifold .but notably , we have shown that this is also the case for general lagrangian systems .writing them as dynamical systems shows that the definition of global lyapunov stability proceeds from the generalized lagrange metric which simply is extended to the bundle  .local stability according to the newly proposed connection criterion uses the berwald type connection on associated to the lagrangian semispray .no previous notion of local stability seems to cover general lagrangian systems .  for natural lagrangian systems given by lagrangians quadratic in the velocities ,the jacobi - maupertuis principle is commonly applied to translate solutions of fixed energy into geodesics of an associated riemannian manifold .we have compared the resulting notions of geodesic stability with the intrinsic notions derived from the lagrangian system .we have pointed out that geodesic stability suffers from various illnesses : boundaries of the riemannian manifold that are curvature singular ; the non - ability to resolve perturbations that modify energy ; and the complete breakdown of the analysis for one - dimensional systems . none of these issues troubles the intrinsic definitions of stability in the lagrangian picture .the main achievement of this article lies in clarifying in a geometric framework the structures that underlie any discussion of the global or local stability of solutions of dynamical systems .it has emerged that the geometry of dynamical systems forces us to define a new criterion for local stability in terms of a choice of linear connection .our comparison of this criterion to the geodesic stability of natural lagrangian systems already shows a number of advantages of the intrinsic geometric approach .we expect that the practical relevance of our construction will be proven in future applications .the authors wish to thank christian gross and paul smyth for stimulating discussions .they gratefully acknowledge full financial support from the german research foundation dfg through the emmy noether grant wo 1447/1 - 1 .v. i. oseledec , trans .* 19 * ( 1968 ) 197 .y. b. pesin , russian math . surveys * 32 * ( 1977 ) 55 .g. benettin , l. galgani and j. m. strelcyn , phys .rev . a * 14 * ( 1976 ) 2338 .g. benettin , l. galgani , a. giorgilli and j. m. strelcyn , meccanica * 15 * ( 1980 ) 9 .g. benettin , l. galgani , a. giorgilli and j. m. strelcyn , meccanica * 15 * ( 1980 ) 21 .e. gozzi and m. reuter , chaos solitons fractals * 4 * ( 1994 ) 1117 .l. caiani , l. casetti , c. clementi and m. pettini , phys .* 79 * ( 1997 ) 4361 [ arxiv : chao - dyn/9702011 ] .l. casetti , m. pettini and e. g. d. cohen , phys .* 337 * ( 2000 ) 238 .b. terzic and h. e. kandrup , phys .a * 311 * ( 2003 ) 165 [ arxiv : astro - ph/0211248 ] .m. cerruti - sola and m. pettini , phys .e * 53 * ( 1996 ) 179 .m. di bari , d. boccaletti , p. cipriani and g. pucacco , phys .e * 55 * ( 1997 ) 6448 .p. cipriani and m. di bari , phys . rev .* 81 * ( 1998 ) 5532 .e. cuervo - reyes , arxiv:0807.1156 [ math - ph ] . m. cerruti - sola , g. ciraolo , r. franzosi and m. pettini , phys .e * 78 * ( 2008 ) 046205 .d. d. kosambi , math .z. * 37 * ( 1933 ) 608 .e. cartan , math .z. * 37 * ( 1933 ) 619 .s. s. chern , bull .sci . math .* 63 * ( 1939 ) 206 .i. bucataru and r. miron , arxiv:0705.3689 [ math.dg ] . t. harko and v. s. sabau , phys .d * 77 * ( 2008 ) 104009 [ arxiv:0803.4369 [ gr - qc ] ] .i. bucataru , novi sad j. math .* 29 * ( 1999 ) 69 .k. yano and s. ishihara , _ tangent and cotangent bundles _ , marcel dekker , new york 1973 .m. crampin , j. phys .a * 16 * ( 1983 ) 3755 .i. bucataru , diff .* 25 * ( 2007 ) 335 .k. townsend and m. n. r. wohlfarth , class .quant . grav .* 21 * ( 2004 ) 5375 [ arxiv : hep - th/0404241 ] .l. a. elias and a. saa , phys .d * 75 * ( 2007 ) 107301 [ arxiv : gr - qc/0702058 ] .o. hrycyna and m. szydlowski , chaos solitons fractals * 28 * ( 2006 ) 1252 [ arxiv : gr - qc/0505155 ] .

Abstract: we reconsider both the global and local stability of solutions of continuously evolving dynamical systems from a geometric perspective . we clarify that an unambiguous definition of stability generally requires the choice of additional geometric structure that is not intrinsic to the dynamical system itself . while global lyapunov stability is based on the choice of seminorms on the vector bundle of perturbations , we propose a definition of local stability based on the choice of a linear connection . we show how this definition reproduces known stability criteria for second order dynamical systems . in contrast to the general case , the special geometry of lagrangian systems provides completely intrinsic notions of global and local stability . we demonstrate that these do not suffer from the limitations occurring in the analysis of the maupertuis - jacobi geodesics associated to natural lagrangian systems .


the study of complex networks has attracted a great deal of interest since the discoveries of the small - world and scale - free ba:1999 properties in many natural and man - made networks ab:2002,newman : siam , blmch:2006 . while in initial studies the nodes and links of a networkare treated as identical , recent studies have extended to the heterogeneous networks of scaled nodes and weighted links where many new features and properties are discovered .meanwhile , in many practical systems the scalars , which usually reflect the different characteristics among the nodes , and the weights , which usually characterize the information transport capacities on the links , are closely correlated .typical examples include the co - authorship networks , where the scalar can be regarded as the number of papers published by one researcher and the weight represents the collaboration times between two researchers , and the world - wide airline network ( wwan ) , where the scalar can be the airport capacity and the weight represents the amount of transportation between two airports . in both casesthe links which connect nodes of larger scalar often assume larger weights .however , the latter is different to the former in that in wwan the transportation is directed and , in most cases , the mass flows in two directions are not balanced , i.e. , there exists _ gradient _ on the links .the gradient networks , which are defined as _ the directed graphs formed by local gradients of a scalar field distributed on the nodes _ , are ubiquitous in nature and play an important role in many biological and technical systems .for example , it has been shown that the congestion tendency of traffic networks can be drastically reduced when gradient is considered .it is well known that the collective behavior of complex systems is strongly influenced by their underling coupling structures .one typical example is chaos synchronization in complex networks .compare to the regular networks , the synchronizability of both the small - world and scale - free networks ( sfn ) are drastically enhanced due to the decreased average distance . however , as shown by the recent studies , the heterogeneous distribution of both the node degree and the link weight could suppress the synchronizability .the contradiction between the fact that many empirical networks have the property of heterogeneity and the finding that synchronizability are suppressed in heterogeneous networks has formed a paradox in the study of network synchronization .this paradox has stimulated the searching of optimal network configuration in sfn .specifically , for a sfn of given topology and total coupling cost , people want to know how to distribute the couplings could efficiently promote the synchronizability . in ref . the authors proposed to distribute the incoming coupling strengths according to the local information of node degree ( hereafter we mark it as m - scheme ) .it is found that the synchronizability is solely determined by the average degree , independent of the degree distribution and the system size . in particular , under the condition of uniform coupling capacity distribution, sfn achieves its maximum synchronizability which is superior to network of homogeneous degree distribution . in ref . the incoming coupling strengths are proposed to be distributed according to the betweenness centrality of links ( hereafter we mark it as c - scheme ) , it is found that synchronizability reaches its maximum only when the distributions of these two quantities match . in both cases ,the couplings are directed and in general they are not balanced , i.e. , one direction weights over another direction .as more and more evidences point to the important roles of gradient couplings , an interesting question is : _ which kind of gradient will be more efficient in improving the network synchronizability ? _ the answer relies on two parallel investigations : how to set the gradient direction and how to distribute the gradient weight . in setting the gradient direction ,an intuitive method is to let the gradient start from the larger degree node and point to the smaller one . butanalysis of non - diagonalizable networks suggest that this setting can be arbitrary , given no loops in the gradient network . in distributing the gradient weight ,the answer is even diverse and confusing , the proposed methods range from the uniform weight distribution hcab:2005 ( hereafter we marked it as h - scheme ) to the distributions based on local information of node degree and on global information of link betweenness . therefore the global picture of the function of gradient / asymmetric coupling is still not clear and further study is needed .  in this work, we will investigate the problem of network synchronization from the gradient network point of view and give a generic understanding to the functions of gradient / asymmetric couplings in improving network synchronizability . under the framework of gradient network , not only the previous results about asymmetric coupling can be well unified , but also the picture of how to distribute the gradient direction and weight becomes clear and simple .for example , now it is straightforward to figure out the necessary conditions for heterogeneous network to have a stronger synchronizability than homogeneous network and , more importantly , to what extend the gradient could benefits the network synchronizability .the rest of the paper will be organized as follows . in sectionii we will introduce the idea of gradient network and some of its basic properties .based on the previous experiences and the new understanding of gradient network , in section iii we will suggest the basic criteria for constructing optimal networks and propose a new coupling scheme as application . in section iv we will analysis the properties of the gradient network emerges in the new coupling scheme and , in section v , show its efficiency in improving network synchronizability by the method of eigenvalue analysis . in section vi we will discuss the multiple effects of gradient and show how the optimal gradient changes its value according to the network parameters . in sectionvii we present the simulation results on coupled nonidentical chaotic oscillators .finally we give the discussion and conclusion in section viii .for any pair of connected nodes , there are actually two directed couplings .when the weights in two directions are equal , we say the couplings are symmetric , otherwise , the couplings are asymmetric . to highlight the role of asymmetric couplings ,a natural way is to separate the mutual couplings into two parts : the symmetric part and the asymmetric part .while the symmetric part reflects the common strength that two nodes affect each other , the asymmetric part represents the dominant role that one node put to another node .extend this kind of separation to the whole network we will find that the original asymmetrically coupled network can be separated into two subgraphs : _ one undirected symmetric network and one directed gradient network_. both two subgraphs have the same topology as the original network , but the coupling direction and weight have been changed , as schematically shown in fig .1 . specially, the gradient network consists of only directed links .while synchronization of symmetric network has been well explored in previous studies , the separation of gradient network from the asymmetric network could simplify the problem and make the analysis easier .now our attention solely turns to the study of gradient network and investigate its effects on network synchronization .please note that the so formed gradient network is different to those conventional ones .conventionally gradient network is defined as the collection of directed links pointing to each node from whichever of its near neighbors has the highest / lowest scalar .for network of nodes and average degree , according to the conventional definition there will be only directed links in the gradient network . while for the mutually coupled network , every link may have a asymmetric part and will contributes one gradient link .thus the gradient network generated from coupled system actually has totally directed links . by noting that the gradients pointing to one node may have different weights and the behavior of this nodeis mainly determined by the largest one ( a specific parameter will be introduced later to balance the weight distribution ) , it is reasonable to reduce the gradient network from links to links by only keeping the largest gradient pointing to each node . in other words , for gradient network of heterogeneous weight distribution, its main feature can be qualitatively captured by the reduced gradient network consisting of only the important gradients . with this simplification ,the gradient network generated from coupled system will consist with those conventional ones , therefore many results in previous studies can be directed used here .hereafter we will only analyze the properties and functions of this kind of reduced gradient network , while the correctness of this simplification will be verified later by eigenvalue analysis and direct simulations .to concrete the idea further , we describe in following the conventional way of constructing gradient network .consider complex network which has nodes and links .the set of edges is specified by a adjacency matrix , if and are connected , otherwise , while . for a given node of degree , the set of its neighbors is denoted by  .consider also a scalar field defined on the set of nodes , therefore each node has a scalar value associated to it .for node of degree , the gradient is defined as the directed link pointing to from whichever of its neighbors has the highest scalar .if the neighbors have the same scalar or if the node has the same scalar as all its neighbors , the selection will be random .gradient network is just constructed as the collection of all these directed links . by this method ,the generated gradient network consists of nodes and links . except the node of the largest scalar , which receives gradient from a node of smaller scalar than itself, all other nodes only receive gradient from nodes of higher scalar than themselves .therefore , except the -node loop formed by the largest scalar node and one of its neighbor node , all the other links formed a tree structure with nodes and at the root .it has been shown that , for homogeneous network of random scalar distribution , the degree distribution of gradient network follows a power law scaling , , with  .  comparing to those conventional gradient networks , the gradient network generated from coupled system possesses some new features .firstly , the node scalar usually are not of random distribution , it has a close relation to the properties of the node .secondly , the gradient are weighted .gradient between nodes of larger scalar difference usually assumes a different weight to that of smaller scalar difference .finally , for sparsely connected networks we need to consider the problem of network breaking , i.e. the degeneration problem .all these new features extend the conventional concept on gradient network and arise new problems for investigation .our mission is just to characterize this kind of new gradient network and explore its functions in steering the collective behavior of coupled networks .gradient , also known as bias , has been employed in nonlinear studies for many years and proven to be a powerful technique in many fields such as chaos control and chaos synchronization .for example , for coupled oscillators on lattice it has been shown that the increase of gradient could greatly enhance the synchronizability of the system .the asymmetric coupling schemes proposed in refs .mzk:2005,chahb:2005,hcab:2005 suggest that gradient can be used to complex networks as well .gradient has been also employed in turbulence control , where the introduction of gradient could significantly improve the control efficiency .while different study proposes different scheme in setting the gradient , these schemes are nonetheless not unified .in particular , for complex networks , people are still not clear of the proper way of gradient configuration , i.e. how to set the gradient direction and weight .now , from the view point of gradient network , these studies can be well unified and the principles for constructing optimal network can be vividly portrayed .  by reviewing the previous studies, we can get some important clues in constructing the matrix .( 1 ) based on the recent findings that hubs are firstly synchronized than nonhubs and the firstly synchronized hubs act as the `` core '' in pattern formation of complex networks , it seems plausible to build mutual links between hubs in the first place . in other words , the gradient between the hubs ,if they are directly connected , should be small . by this settingwe wish to build an efficient channel between the hubs and to form the synchronous core quickly .( 2 ) to guarantee that the synchronous manifold of the `` core '' can be efficiently propagated to the nonhubs while keeping the stability of the core itself , it is reasonable to set the gradient start from hubs and point to nonhubs. according to this requirement we can set the gradient direction .( 3 ) in practical systems the gradients pointing to one node generally have different weight , neighbors of higher scalar may generate larger gradient than those of lower scalar . to distinguish this difference , it is reasonable to set the gradient weight being proportional to the scalar potential . setting in this way, the behavior of each node will more likely to follow its larger degree neighbors .therefore , in determining the final synchronous state of the whole network , the high degree nodes have a larger contribution than the smaller degree nodes .( 4 ) for most practical systems , each node has only limited information of the whole network , e.g. the degree information of itself or its neighbors . which means that , in constructing the coupling matrix , the convenient method should employ only the local network information .based on the above understandings , we propose a new coupling scheme for network synchronization . in particular ,we consider networks of coupled chaotic oscillators following equations,\text { } i=1, ... n,\]]where governs the local dynamics of uncoupled node ,  ] .the network is synchronizable iff all the eigenvalues except the one , which corresponding to the synchronous manifold , can be contained within this range after a linear scaling , i.e. , with the largest and the smallest positive eigenvalues , respectively .in other words , the quantity of synchronizability can be described by the eigenratio , with a smaller represents a stronger synchronizability . meanwhile ,when network is synchronized , larger usually means smaller coupling cost since  .the eigenvalues of asymmetric matrix are usually complex values yhx:1998,hcab:2005 , but for the coupling matrix constructed in eq .matrix , they are reals . noticing that the coupling matrix can be written as , with the diagonal matrix of degrees and the normalization factors for rows of . from the following identity find that the eigenvalues of the asymmetric matrix are the same as that obtained from the symmetric matrix  , which are real and nonnegative values .  from the viewpoint of gradient network , the previous coupling schemes mzk:2005,chahb:2005,hcab:2005 can be well unified and the role of gradient can be easily identified . in m- scheme the gradient is generated according to the degree difference and adjusted via parameter , i.e. and .a negative value of represents that the gradient flows from the smaller degree node to the larger degree node , while for positive the gradient flows in the opposite direction . as reported in ref . and repeated in fig .3(a ) , the maximum synchronizability happens at .since the gradient increases its weight as increases from , it is of certain surprise to find that , instead of enhancement , larger gradient will suppress synchronizability when . another intriguing observation is the sharp change of eigenratio as varies : at while at .while the optimization at can be understood by the heterogeneous distribution of the coupling capacity among the nodes ( at the coupling capacity is the same for all nodes ) and the decreased total coupling cost as increases from the optimal value ( as increases from the total coupling of the network will be decreased ) , the sharp change of demonstrates the nontrivial effect of gradient played in network synchronization . in fig .3(a ) we also plotted the variations of the eigenratio as a function of the gradient for c - scheme , where gradient is generated by the betweenness difference between the connected nodes , and for h - scheme , where gradient is generated by the aging difference . for c - scheme ,the gradient weight is adjusted by parameter , with negative values represent that the gradient flows from node of smaller betweenness to node of larger betweenness , while the opposite happens for positive . again, the absolute value of represents the weight of gradient .the behavior of is quite similar to that of m - scheme , i.e. optimal configuration exists at around while larger gradients suppress synchronizability , except that the variation of eigenratio is much slow than that of m - scheme . for h- scheme the gradient weight is adjusted via the parameter , with negative values represent that the gradient flows from the older ( larger degree ) node to the younger ( smaller degree ) node , while opposite happens for positive . again, the absolute value of represents the weight of gradient .different to the former two schemes , it is found in fig .3(a ) that for h - scheme the eigenratio monotonically decreases as decreases from to , or , similarly , as the gradient from older to younger increases its weight .noticing that equals the situation in m - scheme , it seems that for h - scheme the increase of gradient will always enhance synchronizability .however , from the view point of gradient network , the increase of gradient in h - scheme may induce the breaking problem which will suppress synchronization .meanwhile , a younger node receives gradients _ equally _ from all its older neighbors despite their detail difference , this may confuse the target synchronous state to which the younger node should follow .therefore the gradient functions are not fully explored in h - scheme .  as a comparison , we also plot in fig .3(a ) the result of the new scheme described by eq .[ matrix ] . for ,the gradient flows from smaller degree node to larger degree node and the opposite happens when .the weight of the gradient is adjusted by the absolute value of .it can be found that , as increases , the eigenratio _ monotonically _ decreases from large values to small values . as we will show later , the smallest value that can reach is only determined by the largest eigenvalue , which is almost constant for different coupling schemes .( in the extreme situation of , the coupling matrix takes the form of the reduced gradient network .there will be only two eigenvalues , and , while equals . ) when , we recover to the situations of and used in the previous schemes , respectively .significantly , the maximum synchronizability at can be achieved at around , while can be further decreased as increases in the new scheme . in this sense, we say that the new scheme is more efficient in employing gradient than the other schemes .noticing the fact that increasing degree heterogeneity could suppress synchronizability , we go on to compare the synchronizabilities of the different schemes as a function of heterogeneity .the variation of as a function of is plotted in fig .2(b ) together with four reference configurations : sfn and rn without gradient , rn of m - scheme , and rn of the new coupling scheme . to make the comparison fair, we adopt and , where the maximum synchronizabilities are reached for the corresponding schemes . to make a fair comparison between the new scheme and the h - scheme , we adopt and , since under this setting the total gradient is equal for these two schemes .it can be found that the new scheme has a clear advantage over all the other schemes .it is also found that , under the new scheme , networks of higher heterogeneity shows a much clear advantage over the homogeneous ones , while for the other schemes the advantage is relatively weak .based on this observation , we say that the new scheme not only efficiently enhances the synchronizability of sfns , as compared to their original configurations , but also makes sfns prominently superior to rn .this finding provides a more stronger explanation to the paradox of network synchronization .another quantity used to characterize synchronizability is , the second smallest eigenvalue . since a larger value of usually represents a smaller coupling cost in reaching the global synchronization , i.e.  .the variation of as a function of heterogeneity is plotted in fig .3(c ) , it is found that the new scheme has a larger value of in comparison with the other schemes .to explore the underlying mechanisms behind the new scheme , and also to manifest the basic principles we proposed for network construction in sec .iii , we go on to characterize the gradient network by other two quantities : the distribution of gradient weight and the distribution of eigenvalues . the gradient weight , as described by eq .[ gradient ] , is some value between , where the connected nodes have the similar scalar , and , where the link connect the largest degree and the smallest nodes . in fig .4(a ) we plot the weight distributions of the gradient networks for all the coupling schemes .a clear difference is that the weight distribution of the new scheme has a long tail .according to eq .[ matrix ] , a larger represents a larger degree difference between the connected nodes .this is in accordance with the second principle in network construction , i.e. the gradient weight should be proportional to the the scalar potential . as a comparison , in other schemes the node receives gradients from it neighbors in a relatively mean fashion .for example , in h - scheme , the younger node receives equal gradient from all the older neighbors , while disregards the degree difference among them . the advantage of the new scheme is also reflected in the eigenspectrum . in fig .4(b ) we plot the eigenvalue distributions for different coupling schemes .it can be found that , while the largest eigenvalue is similar for all the schemes , the new scheme is distinct with a larger value of and an absolutely higher probability around .this property makes the eigenvalues to be restricted within a tight region around , and make the value of eigenratio be quickly decreased as increases .the extreme situation will be all the eigenvalues , except and , equal , where the maximum synchronizability is achieved and .  from figs .4(a ) and ( b ) , it seems that increasing gradient will monotonically increases the synchronizability .however , as shown in fig .2(b ) , too large gradients may suppress synchronizability , since gradient could induce the risk of network breaking .now we show this phenomenon via eigenvalue analysis . in fig .4(c ) we plot eigenratio versus gradient for networks of different heterogeneities .it is found that , as the heterogeneity decreases ( i.e. increase the value of ) , in the region of , the gradient gradually changes its role from enhancing synchronizability to suppressing synchronizability .that is , for networks of lower heterogeneity and smaller mean degree , there exist an optimal gradient at .when , increasing gradient will enhance synchronizability , but when , the opposite happens .the position where this transition appears depends on the values of and , for densely connected sfn the value will be significantly delayed .the breaking effect can be further understood from fig .4(d ) , where we plot the eigenratio for a number of network realizations for fixed and .it is found that , for smaller gradient , keeps on small values , representing that the network is still well connected . in this region ,increase will enhance the synchronizability ; but for larger gradient , intermittently jumps to very large values , reflecting that the network is broken in some realizations . therefore in this region increase suppress the synchronizability .we now provide the results of direct simulations . it was shown that , although the msf method was proposed for complete synchronization of coupled identical systems , the eigenratio could still provide a qualitative description for the collective behaviors , e.g. phase synchronization , of coupled nonidentical systems chahb:2005,mzk:2005,zk:2006 .we employ sfn of nonidentical chaotic r oscillators as the model .the dynamics of a singular oscillator reads  ] .the coupling form is .the degree of synchronization in this model can be characterized by monitoring the amplitude of the mean field  . for small coupling strength , oscillates irregularly and is approximately zero , reflecting a smaller degree of synchronization ; while oscillates regularly and increases sharply as coupling strength exceeds a critical value , reflecting a larger degree of synchronization . in fig .5(a ) we plot the behavior of as a function of the coupling strength for all the coupling schemes , it is found that the new scheme has a clear advantage over the other schemes even for smaller , especially in the small the region of small couplings .to demonstrate the positive effect that gradient plays in the new scheme , we plot in fig .5(b ) the behavior of as a function of the gradient degree . again , as predicted by the eigenvalue analysis , increases monotonically as increases .  to show the breaking effect ,we have carried the same simulations but for a homogeneous sparse network , i.e. and . as shown by the inset in fig .5(b ) , there is indeed a maximum around . before this value, the increase of gradient will monotonically enhance the synchronizability , while after this value , increase gradient will suppress synchronizability .as we have analyzed , too large gradient induce the breaking effect and could suppress synchronization .however , in the region of , nodes belong to the same tree are still synchronized and the network breaks into synchronized clusters , therefore it is expected that the decrease of after will be slow . since this time although the network is not globally synchronized , strong coherence still exists within the clusters .the simulation results of fig .5(b ) also testified our analysis that the network breaking effect is closely related to the mean degree and the degree heterogeneity , i.e. eq . [ break ] . since in fig .5(b ) , where the substrate is densely connected sfn , we can not find the transition phenomenon , while for the sparsely connected rn network for the inset plot , we find the transition .now we are able to answer the paradox described at the beginning of this paper : under what conditions that heterogeneous networks will be superior to homogeneous ones in synchronizability .the key point is gradient . by modifying the strength of the directed couplings, we can adjust the gradient direction and weight . to make the sfn outstand ,the proper way to set the direction is making the gradient flow from the larger degree node and point to the smaller degree node ; and a convenient method to set the gradient weight can be making it be proportional to the degree difference .if we set the direction and weight in the opposite ways , sfn will be much difficult to synchronize than rn .this finding thus indicates that , with proper gradient , both two kinds of heterogeneities , the heterogeneous degree distribution and heterogeneous weight distribution , can be used to improve network synchronizability instead of suppression .the advantage that sfn superior to rn becomes even clear when considering the breaking effect . according to our finding , in achieving a stronger synchronizability ,heterogeneous network can bear larger gradient than homogeneous network .the optimal gradient , where the maximum synchronizability is achieved , is closely related to the heterogeneity exponent and the mean degree . the finding that sparse network can not bear too much gradient may have important implications to the function of many natural networks which have sparse links while gradient exists , e.g. , the metabolic network where and the protein network where  .remarkably , because most technological ( e.g. the internet and www ) and biological networks ( e.g. the protein and neural networks ) have the property of disassortativity , i.e. larger degree nodes tend to repulse from each other , the breaking effect will be more relevant in these systems .another advantage enjoyed by this new scheme is that , in constructing the coupling matrix , it only employs the local network information .more accurately , the degree information of the node itself and its neighbors . while schemes based on global network information , e.g. the betweenness centrality employed in c - scheme and the oriented tree employed in ref .nm:2006 , are possible for small size networks , the constructions based on local information , e.g. the node degree employed in m - scheme and h - scheme , could be more efficient and practical in practical .we note that in the new scheme the setting of the gradient is determined by the information of the neighbors , instead of the node itself .this is one of the key points making the new scheme prominent .we also note that the proposed new scheme only provide one choice in employing the gradient , other methods which may involve complex construction rules and global network information , i.e. to design the gradient case by case depending on the specific topology structure , may further promote the synchronizability .but these methods may not as convenient as this new scheme .in conclusion , we have employed gradient network to the problem of network synchronization . under this framework, previous studies on asymmetric networks can be well unified and the principles for synchronizability enhancement become clear and systematic .our studies not only indicates that , comparing to homogeneous networks , scale - free networks are the natural choice for synchronization , but also predict that , for practical networks of low heterogeneity and small mean degree , there should exist an optimal gradient where the synchronizability is maximized .we expect this prediction to be verified by empirical findings in future .x.g . wang acknowledges the great hospitality of arizona state university , where part of the work was done during a visit .lai , k. park , and l. huang were supported by nsf under grant no .itr-0312131 and by afosr under grant no .fa9550 - 06 - 1 - 0024 and no .f49620 - 01 - 01 - 0317 .

Abstract: the contradiction between the fact that many empirical networks possess power - law degree distribution and the finding that network of heterogeneous degree distribution is difficult to synchronize has formed a paradox in the study of network synchronization . surprisingly , we find that this paradox can be well solved when proper gradients are introduced to the network links , i.e. heterogeneous degree distribution is in favor of synchronization in gradient networks . we analyze the general properties of gradient networks and explore their functions in enhancing network sychronizability . based on these understandings , we suppose the basic principles for constructing efficient gradient networks and propose a specific coupling scheme as verification . comparing to the previous asymmetric coupling schemes , the new scheme not only possesses a much stronger synchronizability but also uses few network information . moreover , under the framework of gradient network , the factors which had been employed in former studies in improving network synchronizability can be well unified and identified . the validity of our findings is verified by analytical estimates on the behavior of eigenvalues as well as directed simulations on coupled nonidentical oscillators . our study therefore suggests that , in addition to the topology advantage , scale - free networks also manifest their dynamical advantage given proper gradients are considered .