the study of chaotic and regular properties of the motion in hamiltonian systems constitutes a vast area of research in the field of nonlinear dynamics . since the early 1960 s ,several methods and tools for the fast and accurate detection of the nature of orbits have been proposed and applied to this end in a great number of publications .one may refer e.g. to the pioneering paper by hnon and heiles , where the poincar surface of section ( pss ) was used to reveal the chaotic properties of a non - integrable 2 degree of freedom ( 2 dof ) hamiltonian system . of great importance in this directionwas also the algorithm proposed by benettin and co - workers regarding the computation of the full spectrum of lyapunov exponents ( les ) associated with the time evolution of deviation vectors from a reference orbit , which applies to dynamical systems of arbitrary dimension .more recently , other related methods have been proposed in the literature , like the fast lyapunov indicator " and the mean exponential growth of nearby orbits " ( megno ) , while there have also been approaches focusing on the time series constructed by the coordinates of each orbit , like the frequency map analysis " and the 0 - 1 " test .interesting accounts of these methods can be found in , as well as in a more recent review paper .a novel , very efficient method based on the evolution of initially linearly independent deviation vectors is provided by the so - called generalized alignment indices " ( gali or gali spectrum ) introduced in as a generalization of the smaller alingment index " ( sali ) .the major advantage of the gali method is that it follows the evolution of two or more deviation vectors and is thus able to extract more information about the complexity of the motion , yielding i.e. the dimensionality of the invariant torus on which a regular orbit lies and predicting faster the chaotic nature of trajectories . to date , the gali and the sali indices have been successfully applied to a wide variety of autonomous ( i.e. explicitly time - independent ) conservative flows and maps ( see e.g.  ) .a concise review of the theory and applications of both the sali and gali methods can be found in ( * ? ? ?* chapter 5 ) .the motivation of the current work is twofold : first , we wish to investigate the different dynamical properties of a non - autonomous galactic potential , whose time - dependence could mimic certain realistic general trends arising in barred n - body galaxy simulations .our second main goal is to explore the advantages of the gali method , over the more traditional les , in detecting dynamical transitions in hamiltonian systems , whose equations of motion are explicitly time - dependent .there are , of course , several studies of time - dependent ( td ) galactic and cosmological models in the literature , which use different tools to identify the chaotic vs. regular nature of orbits . defining the orbital complexity , of an orbital segment as the number of frequencies in its discrete fourier spectrum that contain a -fraction of its total power , one may compare with the short - time evolution of the les for td models . in case of a cosmological model is discussed , where orbits may experience regular and/or chaotic motion during their time evolution , while in the effects of a black hole , friction , noise and periodic driving are studied on a triaxial elliptic galaxy model , in which a type of transient chaos was found caused by a damped , oscillatory component .finally , in the so - called pattern method " was used to study a hnon - heiles potential to which an exponential function of time is added , while the dynamics of some simple td galactic models was investigated in .we recall here that in conservative systems the asymptotic nature of an orbit may be either periodic , quasiperiodic or chaotic . in the latter case , however , it may take a very long time before one can safely claim that a final " state is reached , depending on the local dynamical properties , which may be characterized by strong or weak chaos " . in dynamics in the vicinity of periodic orbits in conservative systems was studied by means of the maximum lyapunov exponent ( mle ) and the gali . herewe explore the advantages of the gali method and compare its predictions with what one finds using more traditional methods like the computation of the mle also for td systems .in particular , we focus our attention on the dynamics of a barred galaxy model containing a disc and a bulge component , which is a widely accepted model for real barred galaxies . in the spirit of a mean field approach , we consider the motion of stars ( represented by point particles ) in this potential .the richness of the dynamics of the time - independent ( ti ) version of this model has been extensively studied in terms of : ( a ) the detection of periodic orbits and the analysis of their stability ( see e.g.  ) , ( b ) the estimation of the relative fraction of chaotic vs. regular orbits , and ( c ) the statistical distributions of orbital coordinates described by -gaussian distribution functions .  here , we extend the analysis by considering a td version of this model .more specifically , we allow some mass parameters of the potential to vary linearly as functions of time . as expected , whether we study a 2 dof or 3 dof version of the model , these variations can change the stability properties of periodic orbits , `` dissolve '' islands of regular motion and alter the structure of phase space in very complicated ways .furthermore , in the td case , the vast majority of dynamical transitions of phase space orbits can not be claimed to be due to stickiness phenomena or ordinary diffusion to different regimes , as expected for ti hamiltonian systems .recently , it was found in the ti case that the relative fraction of chaotic orbits grows as the bar s strength increases .a question therefore arises , whether a similar correlation holds in the td model , in the presence of realistic " trends , which permit the mean field potential to vary in a way that is compatible with self - consistent n - body simulations regarding several components of the system .clearly , the analysis of the full n - body problem describes much better the galactic evolution and captures in great detail the different stellar structures present in the dynamics. however , there are serious difficulties and limitations when one tries to apply dynamical chaos detectors to such realistic " many - particle systems due to the lack of sufficient orbital information during the time evolution . for this reason , many researchers prefer to use mean field potentials that are frozen " in time and study the properties at specific snapshots of the simulations .  keeping in mind that a barred galaxy experiences several dynamical transitions in different epochs that can not be easily incorporated in our td mean field potential , we shall proceed by making some helpful assumptions in an attempt to understand the behavior of such widely used chaos detectors as the gali and the mle . thus , we will treat here two very general dynamical trends known to occur in barred galaxies : in the first scenario the mass of the bar component grows linearly in time ( at the expense of the disc mass ) .this increase may be caused by an exchange of angular momentum with the disc ( outer parts gain momentum from the inner parts ) , as has already been observed in n - body simulations ( see e.g.  ) .the fundamental trend in this case is that bars generally grow stronger in time , become more elongated and massive and eventually slow down .we will also consider the inverse scenario , where the bar gets weaker making the disc more massive as time evolves ( see e.g.  ) .the paper is organized as follows : in section [ model_gal ] we present the td barred galaxy model used in our study , while section [ methods ] is devoted to the description of the numerical methods employed for the computation of the mle and the galis . section [ results ] contains the main numerical results of the paper .a detailed investigation of the dynamics of particular orbits in a 2 dof version of our model is performed in section [ 2dof ] , while orbits of the full , 3 dof model , are considered in section [ sec:3dof ] .a global investigation of the dynamics of our td galactic model is given in section [ sec : global ] .finally , in section [ sec : conclusions ] the main conclusions of our work are presented .let us consider the following td 3 dof hamiltonian function which determines the motion of a star in a 3 dimensional rotating barred galaxy : the bar rotates around its ( short axis ) , while the direction is along the major axis and the along the intermediate axis of the bar .the , and are the canonically conjugate momenta , is the potential , represents the pattern speed of the bar and is the total energy of the orbit in the rotating frame of reference ( equal to the jacobi constant in the ti case ) .the corresponding equations of motion are : while the equations governing the evolution of a deviation vector needed for the calculation of the mle and the galis , are given by the variational equations :   the potential of our model consists of three components :  1 .  a triaxial ferrers bar , the density of which is given by : where is the central density , is the mass of the bar which changes in time , and , , with and being the semi - axes of the ellipsoidal bar .the corresponding potential is : where is the gravitational constant ( set equal to unity here ) , , , and is the unique positive solution of , outside of the bar ( ) , while inside the bar .the analytical expression of the corresponding forces are given in .2 .  a bulge , modeled by a plummer sphere potential is : where is the scale - length of the bulge and is its ( constant ) mass .3 .  a disc , represented by the miyamoto - nagai potential : where and are its horizontal and vertical scale - lengths , and the mass of the disc changes in time so that the total mass of the system is kept constant .the model s parameters have the following constant values : , =0.054 ( 54 ) , =6 , .5 , =0.6 , =3 , =1 , =0.08 , while the initial values of the bar and disc masses are =0.1 and =0.82 , respectively .the units used are : 1 ( length ) , 1000 ( velocity ) , 1 ( time ) , ( mass ) .the total mass is set equal to 1 and since the bulge s mass is kept constant , the disc s mass is varied as .the rate of the mass variation of the bar is chosen to be linear according to the law : where the proportionality constant is or respectively , if the mass of the bar increases or decreases in time .  in order to measure the time variation of the bar s strengthwe calculate the quantity : which estimates the relative strength of the non - axisymmetric forces . in the above expression, is the potential on the symmetry plane expressed in polar coordinates , is its axisymmetric part , while the maximum in the first term on the right hand side of ( [ qt ] ) is calculated over all values of the azimuthal angle .the maximum value of over all radii shorter than the bar extent , termed , can be used as a measure of the bar s strength .it is clear , of course , that the variation of the bar s strength modifies the values of several parameters and yields richer information about the dynamics of a self - consistent model .n - body simulations show that in general , variations of the bar s mass also change the mass ratios of the model s components , the bar s shape and the pattern speed of the galaxy .hence , if one wishes to use a mean field potential to `` mimic '' a self - consistent model as accurately as possible , one should allow for all the parameters that describe the bar ( together with all other axisymmetric components ) to depend on time , assuming that the laws of such dependence were explicitly known . in our case, however , we adopt a simpler approach and vary only the masses of the bar and the disc , as a first step towards investigating such models when time dependent parameters are taken into account .thus , we do not pretend to be able to reproduce the exact dynamical evolution of a realistic galactic simulation .rather , we wish to understand the effects of time dependence on the general features of barred galaxy models and compare the efficiency of indicators like the galis and the mle in helping us unravel the secrets of the dynamics in such problems .in order to estimate the value of the mle , , of a particular orbit we follow the evolution of the orbit and a deviation vector from it , by numerically solving the set of equations ( [ eq_motion ] ) and ( [ eq_dev_vect ] ) respectively . for this taskwe use a runge - kutta method of order 4 with a sufficiently small time step ( typically of the order of ) , which guarantees the accuracy of our computations ( i.e. the use of the half time step does not practically change our results ) .the ordinary differential equations ( odes ) ( [ eq_motion ] ) can be solved independently from equations ( [ eq_dev_vect ] ) . on the other hand ,the latter set of equations , governing the evolution of a deviation vector , has to be solved simultaneously with equations ( [ eq_motion ] ) , because the second derivatives of the potential , appearing in the right hand side of ( [ eq_dev_vect ] ) , depend explicitly on the solution of ( [ eq_motion ] ) .note that ( [ eq_motion ] ) constitutes a non - autonomous set of odes because the derivatives of depend explicitly on time .although one could transform ( [ eq_motion ] ) ( and consequently ( [ eq_dev_vect ] ) ) to an equivalent autonomous system of odes by considering time as an additional coordinate ( see e.g.  ) , this approach is not particularly helpful , and is better to be avoided .so , in order to compute the mle and the galis we numerically solve the time - dependent set of odes ( [ eq_motion ] ) and ( [ eq_dev_vect ] ) .then , according to the mle is defined as : where : is the so - called `` finite time mle '' , with and being the euclidean norm of the deviation vector at times and respectively .a detailed description of the numerical algorithm used for the evaluation of the mle can be found in .this computation can be used to distinguish between regular and chaotic orbits , since tends to zero ( following a power law ) in the former case , and converges to a positive value in the latter .but hamiltonian ( [ eq : hamilton ] ) is td , which means that its orbits could change their dynamical behavior from regular to chaotic and vice versa , over different time intervals of their evolution . in such cases ,the computation of the mle ( [ le ] ) might not be able to identify the various dynamical phases of the orbits , since by definition it characterizes the asymptotic behavior of an orbit .  in order to avoid such problems in our study, we also turn to the use of the gali method of chaos detection .the gali index of order ( gali ) is determined through the evolution of initially linearly independent deviation vectors , , with denoting the dimensionality of the phase space of our system .thus , apart from solving ( [ eq_motion ] ) , which determines the evolution of an orbit , we have to simultaneously solve ( [ eq_dev_vect ] ) for each one of the deviation vectors .then , according to , gali is defined as the volume of the -parallelogram having as edges the unit deviation vectors , .it can be shown , that this volume is equal to the norm of the wedge product ( denoted by ) of these vectors : we note that in the above equation the deviation vectors are normalized but their directions are kept intact . in practice, we apply a numerical method for calculating this norm , which is based on the singular value decomposition of an appropriate matrix .the behavior of gali for regular and chaotic orbits was theoretically studied in , where it was shown that all gali tend exponentially to zero for chaotic orbits , with exponents that depend on the first les of the orbit . however , while this relation has been verified for ti systems , in the td case studied here , the way the gali exponential rates depend on the les is less clear and certainly requires further investigation . in the case of regular orbits , gali remains practically constant and positive if is smaller or equal to the dimensionality of the torus on which the motion occurs , otherwise , it decreases to zero following a power law decay .we may , therefore , say that the galis do contain important geometric information about the tangent space of the orbits , in the sense that they identify the number of linearly independent deviation vectors in phase space .this information is especially useful near quasiperiodic orbits , where it helps us accurately determine the dimensionality of the associated torus .  in order to use gali to capture the dynamical changes of orbits in td systems we apply the following procedure : whenever gali reaches very small values ( i.e. gali ) we reinitialize its computation by taking again new random orthonormal deviation vectors , which means that we set again gali. then we let these vectors evolve under the current dynamics .an exponential decrease of gali indicates chaotic behavior .thus , the time needed for gali to become less than can be used to identify epochs where the orbit is chaotic or regular .let us first start with the simple example of an orbit whose motion is restricted in the 2 dimensional ( 2d ) space ( or 4d phase space ) , of a 2 dof reduced version of the full 3 dof model , where are set equal to zero at , and remain zero at all times .furthermore , we shall assume that the bar component gets stronger in time , according to ( [ mblinlaw ] ) , in agreement with the general trend in the dynamical evolution of barred galaxies ( e.g.  ) .in general , the fraction of the chaotic component here is expected to increase as the bar gains mass and becomes stronger . however , this does not necessarily imply that the nature of all orbits remains unchanged in time ; regular orbits can become chaotic as time goes on , and vice versa .for this system one can construct , at specific times , 2d phase portraits ( similar to the pss of the ti case ) to visualize the dynamics and follow the changes that orbits undergo .assuming that all other components except and remain constant , we vary from to , yielding a proportionality constant in ( [ mblinlaw ] ) .the initial energy value for these parameters is set to be .the strength of the bar , measured by the parameter is initially and becomes .  as a test case, we pick a specific orbit , which we call orbit a , with initial condition .this orbit is initially located inside an island of stability in the system s phase space and is thus expected to remain regular , at least for some time to come .  in the left column figure [ 2dof_orb_psss ]configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) . in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ]  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) . in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ] +  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) . in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ]  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) . note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) .in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ] +  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) .in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ]  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) . in order to get a clear picture of the structural evolution of the phase space , in each panelthe pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ] +  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) .in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ]  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) . in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ] +  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) . in order to get a clear picture of the structural evolution of the phase space , in each panel the pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ]  configuration space for successive time intervals with 2500 time units length : ( first row , interval i ) , ( second row , interval ii ) , ( third row , interval iii ) , ( fourth row , interval iv ) and ( fifth row , interval v ) .note that axis has not the same size in all panels .right column : intersection points of orbit a with the pss , for the same time intervals ( black points ) . in order to get a clear picture of the structural evolution of the phase space , in each panelthe pss corresponding to the central time of each time interval ( , , , and , from top to bottom ) is plotted in gray .the orbit is initially regular and drifts from one island of stability to another , until finally its dynamical nature can be characterized as chaotic.,title="fig:",width=185 ] +  we show the projections of orbit a on the plane , while on the right column we plot its intersection points with a pss defined by , ( black points ) for five successive time intervals , each having a duration of 2500 time units . in every panel of the right columnwe plot in gray the pss which corresponds to the central time of each time interval , i.e.  , , , and , respectively from top to bottom .orbit a is initially located inside the right island of stability of figure [ 2dof_orb_psss](b ) and oscillates symmetrically around the bar s major - axis , as shown in figure [ 2dof_orb_psss](a ) for ( interval i ) . then in figure [ 2dof_orb_psss](d ) we observe a first drift from the original island of stability to a nearby one for ( interval ii ) .the morphology of the orbit ( figure [ 2dof_orb_psss](c ) ) changes at the same time to a different shape . in figures [ 2dof_orb_psss](e),(f )this transition has fully taken place and now the motion occurs on an island different from the one it started on , but its regular nature is still preserved for ( interval iii ) . in the next time interval , however , ( interval iv ) , we see a radical change in the orbit s morphology , which indicates the transition from regularity to chaoticity ( figures [ 2dof_orb_psss](g),(h ) ) . from figures [ 2dof_orb_psss](i),(j )we deduce that for ( interval v ) , orbit a moves to regions away from the bar in the configuration space , enters the big chaotic sea on the pss , and from then on shows no regular behavior , as it remains in this chaotic region for the rest of the integration time .we stress here again that the above dynamical transitions are not related to stickiness or ordinary diffusion phenomena as one finds in ti hamiltonian systems .  in figure [ 2dof_mle_gali](a )  , and ( b ) the reinitialized gali of orbit a. the five different time intervals i , ii, iii , iv and v , that correspond to the rows of figure [ 2dof_orb_psss ] , are located between the vertical dashed gray lines.,title="fig:",width=264 ] , and ( b ) the reinitialized gali of orbit a. the five different time intervals i , ii , iii , iv and v , that correspond to the rows of figure [ 2dof_orb_psss ] , are located between the vertical dashed gray lines.,title="fig:",width=264 ]  we depict the time evolution of the finite time mle ( [ sigma_1 ] ) of orbit a. during the first parts of the motion ( intervals i and ii ) , we clearly see a decay of to zero indicating the regular nature of the orbit , in accordance with the results of figure [ 2dof_orb_psss ] . later on , at the end of interval iii and mainly during interval iv , we witness a transient behavior where stops decaying and chaos arises .then , in interval v remains positive and shows a tendency to slightly increase , which clearly suggests that orbit a becomes more chaotic as the bar s mass increases .let us now examine the behavior of gali for the same orbit . from figure [ 2dof_mle_gali](b )we see that gali oscillates around a positive value during the time intervals i and ii for which the orbit is regular and decays exponentially to zero , becoming , as soon as the orbit enters interval iv . in order to monitor the dynamical changes of the orbit we reinitialize the computation of gali as soon as gali , and plot in figure [ 2dof_gali2zerot ] the time needed for gali to decrease from gali to values smaller than along the orbit s evolution .this figure demonstrates that orbit a is initially regular and its gali becomes for the first time after within interval iv . from that point on the orbitremains chaotic as its reinitialized gali repeatedly falls to zero very fast , resulting in small values ( ) .this phase corresponds to the times that the orbit wanders in the big chaotic sea of the pss ( see figures [ 2dof_orb_psss](h ) and ( j ) ) .needed for the reinitialized gali to decrease from gali to gali , as a function of the integration time of orbit a. the fluctuations of the values ( see insert ) reflect the inhomogeneity of the dynamics in the corresponding time intervals.,width=188 ]  the important observation here is that the initial transition of orbit a from regularity to chaoticity and the subsequent variations of its dynamics are not distinctly captured by the evolution of the finite time mle . note from ( [ le ] ) and ( [ sigma_1 ] )that the mle represents a time - averaged quantity over the whole evolution of the orbit , and consequently can not reveal detailed changes of the orbit s motion as it exits an island and wanders within a large chaotic sea .on the other hand , gali _ does _ reveal such changes as it depends only on the current state of the dynamics and not on the previous history of the orbit .in fact , these advantages of the galis in capturing even brief dynamical transitions become more evident in the following sections , where we study orbits in 3 dof td models .   after describing how successful the finite time mle and the gali are in detecting changes in the chaotic vs. regular nature of orbits in the 2 dof restriction of hamiltonian ( [ eq : hamilton ] ) ,let us study some representative cases of the full 3 dof problem .we now suppose that the bar s mass decreases linearly in time from an initial value to , following the law ( [ mblinlaw ] ) with and study an orbit with quite interesting behavior , which we call orbit b. its initial condition is , and its initial energy . in this case , the bar s strength starts at and reaches the value when there is no mass left at the bar component of the model .  in figure [ 3doforbitb ]   ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ]  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ] +  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ]  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ] +  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ]  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ] +  we show the projections on the ( left column ) and the plane ( right column ) of orbit b for three different time intervals . even by mere inspection one can observe the complexity of the evolution of orbit b. although it is not safe to make accurate predictions for the nature of an orbit based on its form in the configuration space , we can say that orbit b looks regular in intervals i and iii ( upper and bottom rows of figure [ 3doforbitb ] respectively ) , while it appears more complicated in interval ii ( middle row of figure [ 3doforbitb ] ) .these observations suggest that the orbit is initially regular and after a chaotic phase becomes regular again .exactly because orbit projections ( especially for systems of more than 2 dof ) are so difficult to interpret , we compute the and the gali of orbit b , in order to analyze the stages through which the orbit passes . in figure [ 3dof2_mle_gali](a )  , and ( b ) the reinitialized gali of orbit b. the three different time intervals i , ii , and iii , that correspond to the rows of figure [ 3doforbitb ] , are located between the vertical dashed gray lines.,title="fig:",width=264 ] , and ( b ) the reinitialized gali of orbit b. the three different time intervals i , ii , and iii , that correspond to the rows of figure [ 3doforbitb ] , are located between the vertical dashed gray lines.,title="fig:",width=264 ]  we see that decays for , implying that the orbit is regular , then increases to higher values , indicating the possibility of chaotic motion , and finally for decays again to zero , suggesting a return to regularity .it is remarkable how clearly gali captures the different dynamical phases of the orbit s evolution ( recall that that whenever gali , we set gali and repeat its computation using three orthonormal deviation vectors ) . gali oscillates around a high non - zero value , asserting that the motion is regular until about when it starts to decrease rapidly to zero indicating that the orbit is chaotic .this phase lasts until , when gali jumps and remains practically constant until the end of the integration period , indicating that the orbit has become regular again .  by comparing the panels of figure [ 3dof2_mle_gali ]we see that does not convincingly identify the transitions from regularity to chaoticity and vice versa , for reasons that were already discussed .furthermore , we see that , even in the last interval of the orbit s evolution ( ) , where begins to fall towards zero , it decreases so slowly that the nature of the orbit is far from clear . on the other hand , gali can successfully detect the local properties of the dynamics and provide us with a clear knowledge of the chaotic vs. regular nature of an orbit , even for small time windows , where the orbit s nature often changes rapidly .the distinction between the regular and chaotic intervals of orbit b is well depicted in figure [ 3dof_gali3zerot ] ,   that the reinitialized gali needs to decrease from gali to gali , as a function of the integration time of orbit b. note that for gali no longer falls to zero , indicating that the motion has entered a regular domain.,width=377 ]  where the time needed for the reinitialized gali to become is plotted as a function of the integration time .small values for correspond to chaotic epochs , where gali goes to zero exponentially fast , while larger values correspond to intervals where gali takes longer to decay to zero .observe also in figure [ 3dof_gali3zerot ] the remarkable fact ( shown by an upwardly pointing arrow ) that , after , gali no longer falls to zero ( see figure [ 3dof2_mle_gali](b ) ) until the end of the integration time !this is certainly not expected in ti systems .it does occur , however , for orbit b of the 3 dof td model , as well as orbit c of a similar system ( see below ) .let us now study the case of a linear increase of the bar s mass from the initial value to , as we did in the case of the 2 dof model .we take again in ( [ mblinlaw ] ) . as an example , we consider the orbit c with initial condition and initial energy , which undergoes an interesting sequence of dynamical transitions . in this case , starts from and reaches the value at the end of the orbit s evolution .  in figure [ 3doforbita ]   ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ]  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ] +  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ]  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ] +  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ]  ( left column ) and the plane ( right column ) for different time intervals : ( upper row , interval i ) , ( middle row , interval ii ) and ( lower row , interval iii).,title="fig:",width=188 ] +  we plot the projections of orbit c on the and the planes ( left and right column respectively ) for three different time intervals .the orbit looks more or less regular in intervals i and iii , although its shape is quite different in the two intervals . in intervalii it looks a bit more complicated and seems to represent a transition between the two different configurations of intervals i and iii .however , as has already been mentioned , the mere inspection of the orbit is not enough to accurately inform us about its chaotic or regular nature .  , and ( b ) the reinitialized gali of orbit c. the three different time intervals i , ii , and iii , that correspond to the rows of figure [ 3doforbita ] , are located between the vertical dashed gray lines.,title="fig:",width=264 ] , and ( b )the reinitialized gali of orbit c. the three different time intervals i , ii , and iii , that correspond to the rows of figure [ 3doforbita ] , are located between the vertical dashed gray lines.,title="fig:",width=264 ]  in figure [ 3dof1_mle_gali](a ) we plot the time evolution of for orbit c. from this figure we see that initially decays to zero , suggesting the regular character of the orbit .then , at we observe a small increase of , which indicates a dynamical change in the orbit s behavior .this is soon followed by a monotonic decrease of , which might indicate that orbit c becomes regular again .all this , however , is highly speculative .by contrast , the time evolution of the reinitialized gali ( see figure [ 3dof1_mle_gali](b ) ) shows a lot more clearly the transition epoch between the two different regular states . initially gali remains constant and different from zero , providing clear evidence that the orbit is regular .then , for it decreases to over a relatively long time interval , indicating a fundamental change in the character of the orbit . finally , after reinitializing the index s computation at , gali begins to converge to a positive constant , demonstrating the remarkable fact that the orbit has again become regular !thus , here also , as in the case of orbit b ( see figure [ 3dof2_mle_gali](b ) ) , an interlude of chaotic behavior is followed by a transition to regularity , which lasts until the end of our integration time .   after establishing the efficiency of the gali method in identifying time intervals where an orbit of a td model is regular or chaotic , let us use it to study in a more global way the dynamics of hamiltonian ( [ eq : hamilton ] ) . in particular , we will investigate the case considered in section [ sec : mass_increase ] , where the mass of the bar component increases linearly from to , corresponding to in ( [ mblinlaw ] ) .  in the ti version of model ( [ eq : hamilton ] ) was considered for fixed values and respectively . in that work, these two cases were referred as models ` s ' and ` m ' respectively .ensembles of 50000 different initial conditions were integrated up to time units , and the gali method was used to accurately determine the percentages of chaotic orbits . the analysis performed in showed that chaotic behavior is more dominant for the ` m ' model ( i.e. the one with the more massive bar component ) .our td model ( [ eq : hamilton ] ) coincides with model ` s ' of at and becomes model ` m ' at .thus , it is of interest to check for this model if the same sets of initial conditions considered in show a tendency to increase their chaoticity as time grows from to , in agreement with the general trend found in .for this reason , we evolve the same three classes of initial condition distributions considered in :  * distribution : 5000 orbits equally spaced in the space with  ] ,  ] ,  ] and , * distribution : 5000 orbits whose spatial coordinates are chosen randomly over the mass density distribution of model ` s ' ( according to the so - called rejection method ) within the rectangular box , , , with , and obtained from ( [ eq : hamilton ] ) with taking for each initial condition a fixed random value in the interval $ ] ( see for more details ) .the only difference between the distributions of our study and those of is that we use only 5000 initial conditions , instead of 50000 used in , in order to facilitate our computations .since our model is td the percentage of chaotic orbits of the three orbital distributions is expected to change in time . in order to monitor these changeswe adopt the following strategy : we divide the total integration time of 20000 time units in eight successive time windows of length time units . at the beginning of each time window , we reinitialize gali to unity and follow a new set of three orthonormal deviation vectors for each orbit . then for each time window we calculate the `` current percentage of chaotic orbits '' as the fraction of orbits whose gali becomes in the duration of the time window . the results of this procedure are shown in figure [ 3dof_galipercics ] ,   ( dotted line ) , ( dashed line ) , ( solid line ) of model ( [ eq : hamilton ] ) , when is linearly increased from to ( see text for more details ) . the percentages are obtained for successive time windows of length time units , and are attributed to the starting times of these intervals . for comparison ,the percentages of the ti models ` s ' ( with ) and ` m ' ( with ) obtained in , are also plotted at and respectively , by filled squares ( distribution ) , filled circles ( distribution ) , and filled triangles ( distribution ).,width=377 ]  where the percentages obtained for each time window are attributed to its starting time .the percentages found in for models ` s ' and ` m ' are also plotted at and respectively , by filled squares ( distribution ) , filled circles ( distribution ) , and filled triangles ( distribution ) .  in accordance with the results of we see an increase of the fraction of chaotic orbits for all three ensembles of orbits .distribution contains more chaotic orbits among the three ensembles , while the percentages of chaotic orbits is smallest for distribution , in agreement with what was observed in .obviously , one should not expect to find the same numerical values with the percentages reported in .for example , our td model is identical to model ` s ' of only momentarily for , and thus the percentages of the td model reported for do not correspond exactly to model ` s ' . apart from the fact that the percentages reported in were obtained for a ti model , another difference is that in each orbit was integrated in a fixed hamiltonian system for 10000 time units , while in our study each orbit is integrated for only 2500 time units in a td potential. nevertheless , the dynamical trends obtained in our study are in good agreement with the ones presented in , clearly showing the efficiency of the gali method in analyzing td models .autonomous hamiltonian systems are conveniently studied for fixed values of the total energy , where the location and extent of their regular and chaotic regions are time independent and can be accurately identified by a variety of methods especially in the low degree of freedom case . even in such ti systemsthe dynamics can exhibit remarkable complexity , as there exist regimes of `` strong '' and `` weak '' chaos , as well as varying degrees of regularity , as the motion can occur on invariant tori of different dimensions and exhibit surprising localization properties in configuration and frequency space .naturally , therefore , hamiltonian systems which are explicitly td are expected to be a lot more complicated , since , in the absence of ti integrals , all the above attributes evolve in time .for example , in the ti case , orbits do not change their nature : if they are initially regular they will always remain so , while if chaotic they can get trapped for long times on the boundary of regular regimes ( exhibiting a `` weak '' form of chaos ) , but will never entirely relinquish their chaoticity .this is not so in the td case , where individual trajectories may indeed display sudden transitions from regular to chaotic behavior and vice versa during their time evolution .  in the present paper , we have sought to shed some new light on these fascinating phenomena by studying the dynamics of a mean field model of a barred galaxy , whose mass parameters are allowed to vary linearly in time .our primary goal was to show that transitions from order to chaos and vice versa do occur in this model and can be monitored much more accurately by local methods such as the gali spectrum , rather than the more traditional approach of les .in addition , we wanted to investigate some astronomical properties of this td system as it does incorporate some of the features appearing in n - body simulations and ti analytic potentials .  in this regard ,we have chosen for simplicity to vary only two parameters in time , and , keeping the size of the bar and the pattern speed fixed . since the total mass is constant , whatever mass the bar loses is gained by the disc component and vice - versa . moreover , to investigate more thoroughly the observed dynamical transitions , we have extended the maximal integration time of the orbits to ( 20 billion years ) , which corresponds to a total interval of nearly 2 hubble times . however , our results demonstrate that rich behavior and fundamental changes can also be observed within a single hubble time of .most importantly , the gali method turns to be again very efficient and accurate in the detection of chaotic motion in a td system , as in the case of ti models .furthermore , our work reveals that the method is especially suited for detecting intervals where an orbit changes its state fundamentally . by following the times that the galis require to fall to zero ,one can describe in detail the orbit s successive passages from order to chaos and vice versa .finally , we focus on a more global astronomical study , where one would like to estimate qualitatively and quantitatively the relative fraction of regular and chaotic motion in such galaxy models . to this end , we choose different sets of initial conditions , launch them in phase space and classify regular and chaotic orbits , depending on whether the gali fluctuates around a non - zero value or falls exponentially to zero . we have thus been able to verify that the conclusion of an earlier publication on the ti model , that the percentage of chaos grows as the mass of the bar increases , holds true in the td case as well .it would be highly interesting to investigate these questions in more realistic models , where besides the mass parameters the rotation frequency of the galaxy is also allowed to vary accordingly .  in closing, it is important to point out the advantages of the gali method over the computation of the finite time mle , as the galis do succeed in clearly capturing the transitions between different regular states and identifying the intermediate chaotic phases .by contrast , the manifestation of these different dynamical behaviors is much less pronounced in the time evolution of the mle described by in this paper .evidently , the practice of averaging lyapunov exponents over an orbit s history smoothens out their fluctuations over short - lived events and gives them meaning only in the sense of the long time limit .this work was partially supported by a grant from the gsrt , greek ministry of development , for the project `` complex matter '' , awarded under the auspices of the era complexity network .it is also supported by the european union ( european social fund ) and greek national funds through the operational program `` education and lifelong learning '' of the national strategic reference framework ( nsrf ) - research funding program : thales .investing in knowledge society through the european social fund .t. m. would like to thank dr .i. martinez - valpuesta and dr .r. machado for their fruitful comments and discussions on this work .s. is grateful for the hospitality of the max planck institute for the physics of complex systems in dresden , germany , during his visit in july - august 2012 , when part of this work was carried out .

Abstract: we study the distinction and quantification of chaotic and regular motion in a time - dependent hamiltonian barred galaxy model . recently , a strong correlation was found between the strength of the bar and the presence of chaotic motion in this system , as models with relatively strong bars were shown to exhibit stronger chaotic behavior compared to those having a weaker bar component . here , we attempt to further explore this connection by studying the interplay between chaotic and regular behavior of star orbits when the parameters of the model evolve in time . this happens for example when one introduces linear time dependence in the mass parameters of the model to mimic , in some general sense , the effect of self - consistent interactions of the actual n - body problem . we thus observe , in this simple time - dependent model also , that the increase of the bar s mass leads to an increase of the system s chaoticity . we propose a new way of using the generalized alignment index ( gali ) method as a reliable criterion to estimate the relative fraction of chaotic vs. regular orbits in such time - dependent potentials , which proves to be much more efficient than the computation of lyapunov exponents . in particular , gali is able to capture subtle changes in the nature of an orbit ( or ensemble of orbits ) even for relatively small time intervals , which makes it ideal for detecting dynamical transitions in time - dependent systems .


most scientific papers start with an _ abstract _ that focuses on the main ideas of the work but leaves out many of the details . from an information theoretic point of viewthis allows for very efficient processing which is crucial if information processing capabilities are limited . in general ,abstractions are formed by reducing the information content of an entity until it contains only information that is relevant for a particular purpose .this partial neglect of information can lead to different entities being treated as equal or , phrased differently , the separation of structure from noise . consider the abstract concept of a `` chair '' , where many aspects such as the size , color , material or particular shape are considered as noise that is irrelevant to the purpose of `` sitting down '' .the ability to form abstractions is thought of as a hallmark of intelligence , both in cognitive tasks and in basic sensorimotor behaviors traditionally it is conceptualized as being computationally costly because particular entities have to be grouped together by neglecting irrelevant information .here we argue that abstractions arise as a consequence of _ limited _ computational capacity .the inability to distinguish different entities leads to the formation of abstractions .note that this information processing limitation can be induced through limited computational capacity , but also through limited sample sizes or low signal - to - noise ratios . in this paperwe study abstractions in the process of decision - making , where `` similar '' situations elicit the same behavior when partially ignoring the current situational context .  followingthe work of decision - making with limited information - processing resources has been studied extensively in psychology , economics , political science , industrial organization , computer science and artificial intelligence research . in this paperwe use a information - theoretic model of decision - making under resource constraints . in particular , present a framework in which gain in expected utility is traded off against the adaptation cost of changing from an initial behavior to a posterior behavior .the variational problem that arises due to this trade - off has the same mathematical form as the minimization of a _ free energy _difference functional in thermodynamics . here, we discuss the close connection between the thermodynamic decision - making framework and rate - distortion theory which is an information theoretic framework for lossy compression .the problem in lossy compression is essentially the problem of separating structure from noise and is thus highly related to finding abstractions . in the context of decision - making the rate - distortion frameworkcan be applied by conceptualizing the decision - maker as a channel from observations to actions _ with limited capacity _ , which is known in economics as the framework of `` rational inattention '' .  in the next sectionwe discuss how the rate - distortion framework can be obtained for bounded - rational decision - makers that face a number of tasks . in section3 we demonstrate two simple applications to explore the type of abstractions that emerge from limited information processing capabilities .in section 4 we summarize the findings and discuss the presented approach .in , a bounded - rational actor that initially follows a policy changes its behavior to in a way that optimally trades off the expected gain in utility against the transformation costs for adapting from to . this trade - offis formalized by the following variational principle = \underset{q(x)}{\operatorname{argmax}}~\underbrace{\sum_x q(x)u(x)}_{\mathbf{e}_{q(x)}[u ] } - \frac{1}{\beta } \underbrace{\sum_x q(x ) \log \frac{q(x)}{p_0(x)}}_{d_{\mathrm{kl}}(q||p_0 ) } , \label{eq : freeenergy}\ ] ] where is known as the _ inverse temperature _ and is known as the difference in _free energy_negative free energy in physics which is composed of the expected utility w.r.t . and the kullback - leibler ( kl ) divergence between and . acts as a conversion - factor between transformation cost ( usually in nats or bits ) and the expected utility .the distribution that maximizes the variational principle is given by with the _ partition sum _ .the influence of the transformation cost and thus the boundedness of the actor is governed by the parameter which determines `` how far ''the final behavior can deviate from the initial behavior measured in terms of kl - divergence .the perfectly rational actor that maximizes his utility can be recovered as the limit case where transformation cost is ignored , whereas corresponds to an actor that has infinite transformation cost or no computational resources and , thus , sticks with his prior policy .  note that in the notation shown here , is conceptualized as a function over gains . in case corresponds to a loss - function , the same variational principle allows to find the distribution that optimally trades off minimum expected loss against transformation cost . in this casethe argmin over has to be taken and the sign of is inverted . in case a continuous random variable , sums have to be replaced by the corresponding integrals .consider an actor that is embedded into an environment and receives ( potentially partial and noisy ) information about the current state of the environment , that is the actor observes the value of a random variable .this observation allows the actor to reduce uncertainty about the current state of the environment and adapt its behavior correspondingly .formally this is expressed with the conditional distribution over the action . the thermodynamic framework for decision - making introduced in the previous sectioncan straightforwardly be harnessed for describing such a bounded - rational agent that receives information by plugging in the conditional distribution into equation [ eq : freeenergy ] - \frac{1}{\beta } d_{\mathrm{kl}}(p(x|y)||p_0(x ) ) , \label{eq : freeenergyconditional}\ ] ] with the solution   notice that the utility function in general depends on the observation , leading to , but to indicate the conditioning on a specific value of we write .the initial distribution can be interpreted as a default- or prior - behavior in the absence of an observation , thus we will refer to as `` the prior '' .the information processing cost is then given as the kl divergence between and the prior with the conversion factor that relates the units of transformation cost and the units of utility .in the free energy principle ( equation [ eq : freeenergyconditional ] ) , the prior is assumed to be given .a very interesting question is which prior distribution maximizes the free energy difference for all observations _ on average_. to formalize this question , we extend the variational principle in equation [ eq : freeenergyconditional ] by taking the expectation over and the argmax over   - \frac{1}{\beta } d_{\mathrm{kl}}(p(x|y)||p_0(x))\right].\ ] ]  the inner argmax - operator over and the expectation over can be swapped because the variation is not over . with the kl - term expanded this leads to   the solution to the argmax over is given by .( see 2.1.1 in or ) . plugging in for yields the following variational principle for bounded - rational decision - making with a minimum average relative entropy prior } - \frac{1}{\beta } \underbrace{\sum_y p(y ) d_{\mathrm{kl}}(p(x|y)||p(x))}_{i(x;y ) } , \label{eq : ratedistortionvariational}\ ] ] where is the _ mutual information _ between and .the variational problem can be interpreted as maximizing expected utility with an upper bound on the mutual information or in the dual point of view , as minimizing the mutual information between actions and observations with a lower bound on the expected utility .the problem in equation [ eq : ratedistortionvariational ] is equivalent to the problem formulation in rate - distortion theory ( ) , where is usually conceptualized as a distortion function which leads to a flip in the sign of and an argmin instead of an argmax .the solution that extremizes the variational problem is given by the self - consistent equations ( see )   note that the solution for the conditional distribution in the rate - distortion problem ( equation [ eq : freeenergysolutionii ] ) is the same as the solution in the free energy case of the previous section ( equation [ freeenergysolution ] ) , except that the prior is now defined as the marginal distribution ( see equation [ eq : optimalprior ] ) .this particular prior distribution minimizes the the average relative entropy between and which is the mutual information between actions and observations .  in the limit - case where transformation costs are ignored, is equal to the perfectly rational policy for each value of _ independent _ of any of the other policies and becomes a mixture of these solutions .note that if there is a subset of perfectly rational solutions that is shared among tasks , then only this subset will be assigned probability mass since it reduces mutual information ( see section 3.3 ) importantly , high values of the mutual information term in equation [ eq : ratedistortionvariational ] will not lead to a penalization , which means that actions can be very informative about the observation .the behavior of an actor with infinite computational resources will thus be very observation - specific .in the case the mutual information between actions and observations is minimized to , leading to , the maximal abstraction where all elicit the same response .the actor s behavior becomes independent of the observation due to the lack in computational resources to change its behavior . within this limitation the actorwill , however , still emit actions that maximize the expected utility .  for values of the rationality parameter in between these limit - cases , that is , the bounded - rational actor trades off _ observation - specific _ actions that lead to a higher expected utility for particular observations at the cost of high mutual information between observations and actions , against _ abstract _ actions that yield a `` good '' expected utility for many observations and lead to a lower mutual information term .an alternative interpretation , closer to the rate - distortion framework , is that the perceptual channel through which is transmitted to the actor has a limited _ capacity _ given by .for large values of , the transmission of is not severely influenced and the actor can choose the best action for this particular observation .for lower values of however , the actor becomes very uncertain about the true value of and has to choose abstract actions that are `` good '' under all observations which are compatible with the actor s belief over .the self - consistent solutions that maximize the variational principle in equation [ eq : ratedistortionvariational ] can be computed by starting with an initial distribution and then iterating equation [ eq : freeenergysolutionii ] and equation [ eq : optimalprior ] in an alternating fashion .this procedure is well known in the rate - distortion framework as a blahut - arimoto - type algorithm .the iteration is guaranteed to converge to a unique maximum ( see 2.1.1 in and .note that has to have the same support as .implemented in a straightforward manner , the blahut - arimoto iterations can become computationally costly since the iterations involve evaluating the utility function for every action - observation - pair and computing the normalization constant . in case of continuous - valued random variables , closed - form analytic solutions exist only for special cases .in the following we present the application of the rate - distortion framework for decision - making introduced in the previous section to multi - task decision problems .we assume that we are given a number of tasks within the same environment and that the observations from the environment are fully informative about the current task , that is we observe the value of a discrete random variable corresponding to a unique task .note that this assumption can easily be relaxed .more formally we make the following assumptions : we are given a set of tasks which define the set of observations with if and only if .each task is defined through the utility function , where is an action .the action - space is the same for all tasks .we assume that the probability over tasks is known and given by .the goal of the decision maker is to find task - specific distributions that maximize the expected utility _ given _ its computational constraints .this problem is formalized in the variational principle in equation [ eq : ratedistortionvariational ] with the self - consistent solutions in equations [ eq : freeenergysolutionii ] , [ eq : optimalprior ] . in this principle for bounded - rational decision - making ,information processing costs arise from changing the prior - behavior to the task - specific behavior and are measured in terms of kl - cost in accordance with the thermodynamic framework for decision - making .we designed the following two - task problem , to demonstrate the role of the rationality parameter that governs the trade - off between expected utility and mutual information . in both tasks ,the action  ] that leads to the second - best utility in both environments .the simulation results summarized in table [ tab : exp1 ] show that for a high value of the inverse temperature the decision - maker picks the maximum - utility action in each task with probability . at a low value of actor uses the same action distribution for both tasks due to its boundedness , resulting in .this leads to a maximal _ abstraction _ over both tasks which is solved optimally by putting all the probability mass on the suboptimal action .note that the limit shown here is in general still far from the fully bounded limit  in this particular example however lowering further has no effect ..two - task decision problem .possible actions and their utilities for both tasks are given in the first three columns of the table .the results of the blahut - arimoto iterations for a large value of are shown in the middle three columns . in this casethe maximum - utility action for each task is picked with full certainty .the results for a small value of are shown in the last three columns .the decision maker does not have computational resources to change its behavior according to the task and thus always picks the suboptimal action that leads to a high utility in both tasks . [cols="^,^,^,^,^,^,^,^,^ " , ]   figure [ fig : exp1 ] shows the transition from perfect rationality to full boundedness . starting at the entropy of the conditionals is zero , since for a given task the actor picks the maximum - utility action with certainty . by lowering the inverse temperature , both the mutual information and the expected utility  ] * * rate - utility function showing the information processing rate as a function of the expected utility .the rate specifies the minimal average number of bits of the observation that need to be processed in order to achieve a certain expected utility . for the limit the decision maker picks the maximum utility action for each environment deterministically thus following the maximum expected utility ( meu ) principle . ]abstractions are formed by reducing the information content of an entity until it only contains relevant information . for a discrete random variable translates into forming a partitioning over the space where `` similar '' elements are grouped into the same subset of and become indistinguishable within the subset . in physicschanging the granularity of a partitioning to a coarser level is known as _ coarse - graining _ which reduces the resolution of the space in a nonuniform manner . in the rate - distortion frameworkthe partitioning emerges in the shared prior as a _ soft - partitioning _ ( see ) , where actions with the same average utility get the same probability mass and become essentially indistinguishable .  to demonstrate this , we use a binary grid of size , where each cell of the grid can be white or colored in black .actions are particular patterns on this grid thus the actionspace becomes .the utility function defines the following three tasks :  1 .the utility equals the number of colored pixels , but one row and one column has to be all - white , otherwise the utility is zero . 2 .any pattern with exactly four colored pixels scores a utility of , all other patterns have utility zero .any pattern with an even number of colored pixels scores a utility equal to the total number of colored pixels ; all other patterns have a utility of zero .figure [ fig : exp2 ] shows samples each from the conditionals for each task and the prior for . since the inverse temperature is high , all the samples with nonzero probability are actions that yield maximum utility in their particular task .note that the patterns that lead to a maximum utility in task ( 1 ) are a subset of the patterns that lead to maximum utility in task ( 2 ) but also lead to a nonzero utility in task ( 3 ) .since transformation costs are mostly ignored in this case , the patterns appearing for task ( 3 ) are very different from the patterns in task ( 1 ) .note however that the additional patterns in task ( 2 ) that would also lead to maximum utility are assigned a probability of zero .the subset of patterns which are also optimal in task ( 1 ) is sufficient to achieve maximum expected utility and by not including the additional `` specialized '' patterns for task ( 2 ) the mutual information can be reduced significantly .the prior consists essentially of two kinds of patterns : the ones that are optimal in task ( 1 ) and ( 2 ) simultaneously and the patterns that are optimal in task ( 3 ) .the first two tasks have essentially become indistinguishable because the actor will respond with exactly the same action - distribution .  by lowering the inverse temperature to ( see figure [ fig : exp3 ] ) , the mutual information constraint gets more weight and suboptimal patterns are picked for task ( 3 ) , similar to the simulation in the previous section .the behavior of the actor has now become indistinguishable for all three tasks at the expense of a lower expected utility .importantly , the effective resolution of the prior has reduced from two distinct sets of patterns to a single set of indistinguishable patterns ( in terms of their expected utility ) .the level of granularity of the prior has been reduced even further .  .the number above each pattern indicates the probability of the pattern in the corresponding distribution .* * samples for task ( 1 ) .all shown patterns yield maximum utility in the task . ** samples for task ( 2 ) .all shown patterns yield maximum utility in this task , however task ( 2 ) has more patterns that would potentially lead to maximum utility only the subset that coincides with the maximum utility patterns in ( 1 ) has nonzero probability though .this is a consequence of sharing the same prior and mutual information minimization . ** samples for task ( 3 ) .the patterns in task ( 1 ) and ( 2 ) would also have a nonzero probability in task ( 3 ) , but the sampled patterns shown here yield twice the utility and have thus all the probability mass . * * samples from the shared prior .the prior is a mixture over the patterns shown in the conditional distributions . ]  .the number above each pattern indicates the probability of the pattern in the corresponding distribution . ** samples for task ( 1 ) . ** samples for task ( 2 ) . ** samples for task ( 3 ) . compared to the case in figure [fig : exp2 ] , the increased weight of the mutual information term has led to the selection of suboptimal actions in task ( 3 ) , similar to the previous simulation . ** samples from the shared prior . in the fully abstract regimeall conditional distributions are exactly equal to the prior , leading to . ]in this work , we discussed the connection between the thermodynamic framework for decision - making with information processing costs and rate - distortion theory .this connection implies a novel interpretation of the rate - distortion framework for multi - task bounded - rational decision - making .importantly , abstractions emerge naturally in this framework due to _ limited _ information processing capabilities .the authors in find a very similar emergence of `` natural abstractions '' and `` ritualized behavior '' when studying goal - directed behavior in the mpd case using the _ relevant information _ method , which is a particular application of rate - distortion theory .although not shown here , the approach presented in this paper straightforwardly carries over to an inference case by treating as observations and as the belief - state . in the inference case ,limited information processing capacities make it impossible to detect certain patterns which in turn renders different entities indistinguishable , leading to the formation of abstractions .this idea has been explored previously in .both , the work just mentioned and our work are inspired by the information bottleneck method , which is mathematically very similar to the rate - distortion problem ( with a particular choice of distortion function ) and thus also to the approach presented here .note that limited information processing capabilities can arise for various reasons .the most obvious reason , perhaps , is the lack of computational power which is in many cases equivalent to certain time - constraints ( such as reaction times ) or memory constraints .other reasons for information processing limits are small sample sizes or low signal - to - noise ratios that put an upper limit on the mutual information independent of available computational power .  in the approach presented here , we assume that the decision - maker draws samples from .responding to a certain task with a sample from could then be implemented for instance with a rejection sampling procedure .the prior will then be the proposal - distribution that has the highest average acceptance rate over all tasks .the computational cost of finding is not part of the current framework .these implications have to be explored in further work .

Abstract: a distinctive property of human and animal intelligence is the ability to form abstractions by neglecting irrelevant information which allows to separate structure from noise . from an information theoretic point of view abstractions are desirable because they allow for very efficient information processing . in artificial systems abstractions are often implemented through computationally costly formations of groups or clusters . in this work we establish the relation between the free - energy framework for decision making and rate - distortion theory and demonstrate how the application of rate - distortion for decision - making leads to the emergence of abstractions . we argue that abstractions are induced due to a _ limit _ in information processing capacity .