fractal dimension of higher dimensional chaotic repellors abstract using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in typical dynamical systems in terms of the lyapunov exponents and the characteristic escape time from the repellor our main example a three dimensional chaotic scattering billiard yields a new structure for its invariant manifolds this system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

wada basin boundaries and basin cells in dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is nonempty we consider a two dimensional diffeomorphism f that is f is an invertible map and both f and its inverse are differentiable with continuous derivatives which has at least three basins fractal basin boundaries contain infinitely many periodic points generally only finitely many of these periodic points are outermost on the basin boundary that is accessible from a basin for many systems all accessible points lie on stable manifolds of periodic points a point x on the basic boundary is a wada point if every open neighborhood of x has a nonempty intersection with at least three different basins we call the boundary of a basin a wada basin boundary if all its points are wada points our main goal is to have definitions and hypotheses for wada basin boundaries that can be verified by computer the basic notion basin cell will play a fundamental role in our results for numerical verifications assuming each accessible point on the boundary of a basin b is on the stable manifold of some periodic orbit we show that partial derivative b over bar is a wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins in addition we find conditions for basins b 1 b 2 b n n greater than or equal to 3 to satisfy partial derivative b over bar 1 partial derivative b over bar 2 partial derivative b over bar n our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a wada basin boundary our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of wada basin boundaries

a new class of efficient and robust energy stable schemes for gradient flows we propose a new numerical technique to deal with nonlinear terms in gradient flows by introducing a scalar auxiliary variable sav we construct efficient and robust energy stable schemes for a

complexity in hamiltonian driven dissipative chaotic dynamical systems the existence of symmetry in chaotic dynamical systems often leads to one or several low dimensional invariant subspaces in the phase space we demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is hamiltonian but the full system is dissipative in particular an infinite number of distinct attractors can coexist these attractors can be quasiperiodic strange nonchaotic and chaotic with different positive lyapunov exponents finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors however arbitrarily small perturbations can lead to dynamically distinct chaotic attractors this work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom

pressure and fractal indices for the gibbs measures of hyperbolic julia sets the authors give a simple algorithm to compute the topological pressure for the julia sets j of hyperbolic rational endomorphism this allows one to compute the spectra of the generalised dimensions and entropies with respect to the whole class of gibbs measures on j

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

cohomology of restricted lie rinehart algebras and the brauer group abstract we give an interpretation of the brauer group of a purely inseparable extension of exponent 1 in terms of restricted lie rinehart cohomology in particular we define and study the category p lr a of restricted lie rinehart algebras over a commutative algebra a we define cotriple cohomology groups h p l r l m for l p lr a and m a beck l module we classify restricted lie rinehart extensions thus we obtain a classification theorem for regular extensions considered by hochschild

indeterminate sub critical bifurcations in parametric resonance in this paper we show that an indeterminate sub critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems this occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub critical bifurcation here in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co existing attractor the system will settle

fractal boundaries in open hydrodynamical flows signatures of chaotic saddles we introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently it is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors the fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set we point out that these boundaries coincide with streak lines passing through appropriately chosen points as an illustrative numerical experiment we consider a model of the von k arm an vortex street a time periodic two dimensional flow of a viscous fluid around a cylinder

superdiffusion in a honeycomb billiard we investigate particle transport in the honeycomb billiard which consists of connected channels placed on the edges of a honeycomb structure the spreading of particles is superdiffusive due to the existence of ballistic trajectories which we term perfect paths simulations give a time exponent of 1 72 for the mean square displacement and a starlike i e anisotropic particle distribution we present an analytical treatment based on the formalism of continuous time random walks and explain the anisotropic distribution under the assumption that the perfect paths follow the directions of the six lattice axes furthermore we derive a relation between the time exponent and the exponent of the distribution function for trajectories close to a perfect path in billiards with randomly distributed channels conventional diffusion is always observed in the long time limit although for small disorder transient superdiffusional behavior exists our simulation results are again supported by an analytical analysis

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

indeterminate bifurcational phenomena in hardening systems indeterminate bifurcations are emerging as an important ingredient of nonlinear dissipating dynamical systems in this paper we show that indeterminate bifurcations with an unpredictable outcome are a typical ingredient of nonlinear hardening systems such phenomena are clearly important concepts in the theory of nonlinear resonance and their discovery complements the earlier work carried out on softening systems as an illustrative example we examine the dynamics of a parametrically excited hardening system and show that when the trivial solution is located on a highly intertwined basin boundary its loss of stability gives a dynamic jump whose outcome is indeterminate in the sense that we cannot predict on to which coexisting attractor the system will settle

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

fast and stable schemes for phase fields models we propose and analyse new stabilized time marching schemes for phase fields model such as allen cahn and cahn hillard equations when discretized in space with high order finite differences compact schemes the stabilization applies to semi implicit schemes for which the linear part is simplified using sparse pre conditioners the new methods allow to significant obtain a gain of cpu time the numerical illustrations we give concern applications on pattern dynamics and on image processing inpainting segmentation in two and three dimension cases

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

origin of chaotic transients in excitatory pulse coupled networks we develop an approach to understanding long chaotic transients in networks of excitatory pulse coupled oscillators our idea is to identify a class of attractors sequentially active firing saf attractors in terms of the temporal event structure of firing and receipt of pulses then all attractors can be classified into two groups saf attractors and non saf attractors we establish that long transients typically arise in the transitional region of the parameter space where the saf attractors are collectively destabilized bifurcation behavior of the saf attractors is analyzed to provide a detailed understanding of the long irregular transients although demonstrated using pulse coupled oscillator networks our general methodology may be useful in understanding the origin of transient chaos in other types of networked systems an extremely challenging problem in nonlinear dynamics and complex systems

intermingled basins in coupled lorenz systems we consider a system of two identical linearly coupled lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength we verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa we investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system

intertwined basins of attraction generated by the stable manifold of a saddle point this article deals with the intertwined basins of attraction we give a necessary and sufficient condition for the existence of intertwined basins of attraction

noise induced escape through a fractal basin boundary we study noise induced escape within a discrete dynamical system that has two co existing chaotic attractors in phase space separated by a locally disconnected fractal basin boundary it is shown that escape occurs via a unique accessible point on the fractal boundary the structure of escape paths is determined by the original saddles forming the homoclinic structure of the system and by their hierarchical interrelations

a dynamical meaning of fractal dimension when two atlractors of a dynamical system have a common basin boundary b small changes in initial conditions which lie near b can result in radically different long term behavior of the trajectory a quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary b

q deformed statistical mechanical property in the dynamics of trajectories en route to the feigenbaum attractor we demonstrate that the dynamics toward and within the feigenbaum attractor combine to form a q deformed statistical mechanical construction the rate at which ensemble trajectories converge to the attractor and to the repellor is described by a q entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor the values of the q indices involved are given by the unimodal map universal constants while the thermodynamic structure is closely related to that formerly developed for multifractals as an essential component in our demonstration we expose in great detail the features of the dynamics of trajectories that either evolve toward the feigenbaum attractor or are captured by its matching repellor the dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period doubling transition to chaos elements in our analysis are the following i the preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops ii the limiting form of this rank structure results in an infinite number of families of well defined phase space gaps in the positions of the feigenbaum attractor or of its repellor iii the gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions iv the power law with log periodic modulation associated with the rate of approach of trajectories toward the attractor and to the repellor is explained in terms of the progression of gap formation v the relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated vi a mean field evaluation of the atypical partition function a thermodynamic interpretation of the time evolution process and a crossover to ordinary exponential statistics are given we make clear the dynamical origin of the anomalous thermodynamic framework existing at the feigenbaum attractor

on basin boundaries in this paper we make a study of a class of basin boundaries we show that an efficient characterization of a given element of this class can be constructed with segments of differentiable curves

indeterminate trans critical bifurcations in parametrically excited systems this paper may be regarded as a companion paper to a recent article m s soliman j m t thompson proc r soc lond a 438 511 518 1992 that addressed the phenomena of indeterminate sub critical bifurcations in typical parametrically excited systems this earlier work focused on the principal subharmonic resonance which occurs when the excitation frequency is almost twice the linear natural frequency in contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same we show that under this condition indeterminate trans critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation the outcome is then unpredictable even under infinitesimal perturbations from the trivial solution

extreme phase sensitivity in systems with fractal isochrons abstract sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors however even systems with quasi periodic dynamics can exhibit it in this context we report on the fractal properties of the isochrons of some continuous time asymptotically periodic systems we define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons similar results are also obtained with discrete time systems as an illustration of the framework we compute the phase sensitivity coefficient for popular models of bursting neurons suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive unreliable phase response

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

prey predator systems with intertwined basins of attraction in this article we show the existence of intertwined basins of attraction for a class of prey predator systems which improves the extant results by deleting some crucial and hard testing conditions

the global dynamics of discrete semilinear parabolic equations a class of scalar semilinear parabolic equations possessing absorbing sets a lyapunov functional and a global attractor are considered the gradient structure of the problem implies that provided all steady states are isolated solutions approach a steady state as t to infty the dynamical properties of various finite difference and finite element schemes for the equations are analysed the existence of absorbing sets bounded independently of the mesh size is proved for the numerical methods discrete lyapunov functions are constructed to show that under appropriate conditions on the mesh parameters numerical orbits approach steady state solutions as discrete time increases however it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented furthermore it is also shown that the explicit euler scheme introduces spurious solutions with period 2 in the timestep as a result the absorbing set is destroyed and there is initial data leading to blow up of the scheme however small the mesh parameters are taken to obtain stabilization to a steady state for this scheme it is necessary to restrict the timestep in terms of the initial data and the space step implicit schemes are constructed for which absorbing sets and lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step both one step and multistep bdf methods are studied

time marching multilevel techniques for evolutionary dissipative problems in this article we use a pseudospectral fourier discretization in conjunction with a multilevel splitting of high and low modes to solve dissipative partial differential equations we develop unconditionally stable explicit techniques for the temporal integration of the linear terms and apply them to the high modes equation improving the overall temporal stability of the multilevel method and resulting in a competitive fully explicit numerical scheme for nonlinear problems in the cases where the linear term determines the time step restriction numerical experiments with the burgers equation in one and two dimensions showed substantial cpu cost reduction when comparing the resulting method with the standard spectral collocation associated with regular temporal integration schemes

an unconditionally stable hybrid numerical method for solving the allen cahn equation we present an unconditionally stable second order hybrid numerical method for solving the allen cahn equation representing a model for antiphase domain coarsening in a binary mixture the proposed method is based on operator splitting techniques the allen cahn equation was divided into a linear and a nonlinear equation first the linear equation was discretized using a crank nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method the nonlinear equation was then solved analytically due to the availability of a closed form solution various numerical experiments are presented to confirm the accuracy efficiency and stability of the proposed method in particular we show that the scheme is unconditionally stable and second order accurate in both time and space

triple cohomology of lie rinehart algebras and the canonical class of associative algebras abstract we introduce a bicomplex which computes the triple cohomology of lie rinehart algebras we prove that the triple cohomology is isomorphic to the rinehart cohomology provided the lie rinehart algebra is projective over the corresponding commutative algebra as an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra and extend sridharan s result on almost commutative algebras

analysis of chaotic saddles in low dimensional dynamical systems the derivative nonlinear schrodinger equation abstract in this paper we present a computational study of nonattracting chaotic sets known as chaotic saddles in a low dimensional dynamical system describing stationary solutions of the derivative nonlinear schrodinger equation a driven dissipative model for alfven waves these chaotic saddles have gaps which are filled at chaotic transitions such as a saddle node bifurcation and an interior crisis we give a detailed explanation of how to numerically determine the chaotic saddles and describe how a chaotic attractor after an interior crisis point can be decomposed into two chaotic saddles dynamically connected by a set of coupling unstable periodic orbits created by a gap filling explosion after the crisis this coupling between two chaotic saddles is responsible for the intermittent dynamics displayed by the chaotic system after the interior crisis

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

the transition to chaotic attractors with riddled basins abstract recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby the basin is riddled with holes here we consider the dynamics near the transition to this situation as a parameter is varied using a simple analyzable model we obtain the characteristic behaviors near this transition numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered

inpainting of binary images using the cahn hilliard equation image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas we outline here the use of a model for binary inpainting based on the cahn hilliard equation which allows for fast efficient inpainting of degraded text as well as super resolution of high contrast images

basins of attraction in piecewise smooth hamiltonian systems piecewise smooth hamiltonian systems arise in physical and engineering applications for such a system typically an infinite number of quasiperiodic attractors coexist here we use the term attractors to indicate invariant sets to which typically initial conditions approach as a result of the piecewise smoothness of the underlying system these attractors are therefore characteristically different from the attractors in dissipative dynamical systems we find that the basins of attraction of different attractors exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales this practically prevents prediction of attractors from specific initial conditions and parameters the mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems we demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation

three dimensional optical billiard chaotic scattering we report on the first experimental measurements taken from an easily studied visually remarkable chaotic scattering system that we call an optical billiard in particular we measure the fractal dimension of the basin boundaries generated by the reflection of light from various configurations of polished reflecting spheres we find that our experimentally determined fractal dimension values agree well with the values we find from simulations we also find that depending on the configuration of the spheres the boundary can be either a nowhere differentiable surface or else can possess the topological wada property whereby any basin boundary point is simultaneously on the boundary of four distinct basins other configurations yielding other topologies of the chaotic scattering set are suggested for future study

unpredictability of the asymptotic attractors in phase coupled oscillators an array of phase coupled oscillators may exhibit multiple coexisting chaotic and nonchaotic attractors the system of coupled circle maps is such an example we demonstrate that it is common for this type of system to exhibit an extreme type of final state sensitivity in both parameter and phase space numerical computations reveal that there exist substantial regions of the parameter space where arbitrarily small perturbations in parameters or initial conditions can alter the asymptotic attractor of the system completely consequently asymptotic attractors of the system cannot be predicted reliably for specific parameter values and initial conditions

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

on chaos and fractal behavior in a generalized duffing s system abstract in this paper three duffing s equations their attractors and their initial condition maps are examined for elements of universality in the literature all three equations have been examined briefly before but their behavior has only been incompletely described this paper more completely examines these system and answers important questions concerning the reason for chaotic system behavior

repellers semi attractors and long lived chaotic transients abstract we study the chaotic transients observed in many deterministic systems in general they are related to strange repellers or semi attractors if they are repelling in some and attracting in other directions we propose formulas relating the average life time of the transient to dimensions of the repeller and to lyapunov exponents of the flow on it the formulas are tested numerically in a number of cases

fractal control boundaries of driven oscillators and their relevance to safe engineering design when a metastable damped oscillator is driven by strong periodic forcing the catchment basin of constrained motions in the space of the starting conditions x 0 0 develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds the four dimensional basin in the phase control space spanned by x 0 0 f where f is the magnitude and the frequency of the excitation will likewise acquire a fractal boundary and we here explore the engineering significance of the control cross section corresponding for example to x 0 0 0 the fractal boundary in this section is a failure locus for a mechanical or electrical system subjected while resting in its ambient equilibrium state to a sudden pulse of excitation we assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well absolute and transient basins are examined giving control space maps analogous to familiar pictures of the mandelbrot set generalizing from this prototype study it is argued that in engineering design against boat capsize or earthquake damage for example a study of safe basins should augment and perhaps entirely replace conventional analysis of the steady state attracting solutions

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices

dynamic bifurcations in non stationary systems transitions with an unpredictable outcome in nonlinear non stationary systems dynamic bifurcations result in a transition to a qualitatively new state in this paper we examine how the dynamics of transition of such systems may be assessed using the concept of transient basins of attraction we delineate the phenomenon of indeterminate dynamic bifurcations where it is shown that the response after the system passes through critical parameter values may be extremely sensitive to the choice of initial conditions or parameter states this new form of unpredictability in systems whose parameters vary with time is clearly an important concept to be assimilated in the theory of non stationary dynamics