Algorithmic information for interval maps with an indifferent fixed point and infinite invariant measure

Measuring the average information that is necessary to describe the behavior of a dynamical system leads to a generalization of the Kolmogorov-Sinai entropy. This is particularly interesting when the system has null entropy and the information increases less than linearly with respect to time. We consider a class of maps of the interval with an indifferent fixed point at the origin and an infinite natural invariant measure. We show that the average information that is necessary to describe the behavior of the orbits increases with time n approximately as nalpha, where alpha < 1 depends only on the asymptotic behavior of the map near the origin.     

On decomposing mixed-mode oscillations and their return maps

Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological, and engineering systems. These mixed-mode oscillations (MMOs) are often found in systems with multiple time scales. Previous differential equation modeling and analysis of MMOs have mainly focused on local mechanisms to explain the small oscillations. Numerical continuation studies reported different MMO patterns based on parameter variation. This paper aims at improving the link between local analysis and numerical simulation. Our starting point is a numerical study of a singular return map for the Koper model which is a prototypical example for MMOs, which also relates to local normal form theory. We demonstrate that many MMO patterns can be understood geometrically by approximating the singular maps with affine and quadratic maps. Motivated by our numerical analysis we use abstract affine and quadratic return map models in combination with two local normal forms that generate small oscillations. Using this decomposition approach we can reproduce many classical MMO patterns and effectively decouple bifurcation parameters for local and global parts of the flow. The overall strategy we employ provides an alternative technique for understanding MMOs.     

Large Deviation Principle for Benedicks-Carleson Quadratic Maps

Since the pioneering works of Jakobson and Benedicks &; Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks &; Carleson to build induced Markov maps of arbitrarily small scale and associated towers, to which the absolutely continuous measures can be lifted. These various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle, for the absolutely continuous measures. This result encompasses dynamics far from equilibrium, and thus significantly extends presently known local large deviations results for quadratic maps.     

Resonances and asymptotic behavior of Birkhoff series

For hamiltonian systems with two degrees of freedom a mechanism accounting for the divergence of perturbation series and the asymptotic relation between true and formal dynamics is proposed. In the special case of conservative quadratic maps numerical and analytical support is given for a piecewise geometric structure of the Birkhoff series, that is a sequence of pseudoconvergence radii is found which decreases to zero and is associated with the resonances approaching the rotation angle of the linear map.     

Induced maps,Markov extensions and invariant measures in one-dimensional dynamics

A way to study ergodic and measure theoretic aspects of interval maps is by means of the Markov extension. This tool, which ties interval maps to the theory of Markov chains, was introduced by Hofbauer and Keller. More generally known are induced maps, i.e. maps that, restricted to an element of an interval partition, coincide with an iterate of the original map.We will discuss the relation between the Markov extension and induced maps. The main idea is that an induced map of an interval map often appears as a first return map in the Markov extension. For S-unimodal maps, we derive a necessary condition for the existence of invariant probability measures which are absolutely continuous with respect to Lebesgue measure. Two corollaries are given.     

Turning point properties as a method for the characterization of the ergodic dynamics of one-dimensional iterative maps

Dynamical as well as statistical properties of the ergodic and fully developed chaotic dynamics of iterative maps are investigated by means of a turning point analysis. The turning points of a trajectory are hereby defined as the local maxima and minima of the trajectory. An examination of the turning point density directly provides us with the information of the position of the fixed point for the corresponding dynamical system. Dividing the ergodic dynamics into phases consisting of turning points and nonturning points, respectively, elucidates the understanding of the organization of the chaotic dynamics for maps. The turning point map contains information on any iteration of the dynamical law and is shown to possess an asymptotic scaling behaviour which is responsible for the assignment of dynamical structures to the environment of the two fixed points of the map. Universal statistical turning point properties are derived for doubly symmetric maps. Possible applications of the observed turning point properties for the analysis of time series are discussed in some detail. (c) 1997 American Institute of Physics.     

Torus maps and the problem of a one-dimensional optical resonator with a quasiperiodically moving wall

We study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically (with d≥2 incommensurate frequencies). We show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d-dimensional torus that preserves a foliation by irrational straight lines. In particular, the Jacobian of this map has (d−1) eigenvalues equal to 1. We present rigorous and numerical results about several dynamical features of such maps. We also show how these dynamical features translate into properties for the field in the cavity. In particular, we show that when the torus map satisfies a KAM theorem—which happens for a Cantor set of positive measure of parameters—the energy of the electromagnetic field remains bounded. When the torus map is in a resonant region—which happens in open sets of parameters inside the gaps of the previous Cantor set—the energy grows exponentially.     

Influence of dynamical noise on time series generated by nonlinear maps

We consider periodic and chaotic dynamics of discrete nonlinear maps in the presence of dynamical noise. We show that dynamical noise corrupting dynamics of a nonlinear map may be considered as a measurement “pseudonoise” with the distribution determined by the Jacobian of the map. The formula for the distribution of the measurement “pseudonoise” for one-dimensional quadratic maps has also been obtained in an explicit form. We expect that our results apply to an arbitrary distribution of low-level dynamical noise and hope that these results could help to find a universal method of discriminating dynamical from measurement noise.     

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green–Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.     

Renormalization Group for Strongly Coupled Maps

Systems of strongly coupled chaotic maps generically exhibit collective behavior emerging out of extensive chaos. We show how the well-known renormalization group (RG) of unimodal maps can be extended to the coupled systems, and in particular to coupled map lattices (CMLs) with local diffusive coupling. The RG relation derived for CMLs is nonperturbative, i.e., not restricted to a particular class of configurations nor to some vanishingly small region of parameter space. After defining the strong-coupling limit in which the RG applies to almost all asymptotic solutions, we first present the simple case of coupled tent maps. We then turn to the general case of unimodal maps coupled by diffusive coupling operators satisfying basic properties, extending the formal approach developed by Collet and Eckmann for single maps. We finally discuss and illustrate the general consequences of the RG: CMLs are shown to share universal properties in the space-continuous limit which emerges naturally as the group is iterated. We prove that the scaling properly ties of the local map carry to the coupled systems, with an additional scaling factor of length scales implied by the synchronous updating of these dynamical systems. This explains various scaling laws and self-similar features previously observed numerically.     

Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps

The numerical approximation of Perron-Frobenius operators allows efficient determination of the physical invariant measure of chaotic dynamical systems as a fixed point of the operator. Eigenfunctions of the Perron-Frobenius operator corresponding to large subunit eigenvalues have been shown to describe “almost-invariant” dynamics in one-dimensional expanding maps. We extend these ideas to hyperbolic maps in higher dimensions. While the eigendistributions of the operator are relatively uninformative, applying a new procedure called “unwrapping” to regularised versions of the eigendistributions clearly reveals the geometric structures associated with almost-invariant dynamics. This unwrapping procedure is applied to a uniformly hyperbolic map of the unit square to discover this map’s dominant underlying dynamical structure, and to the standard map to pinpoint clusters of period 6 orbits.     

Dynamic phase transition from localized to spatiotemporal chaos in coupled circle map with feedback

We investigate coupled circle maps in the presence of feedback and explore various dynamical phases observed in this system of coupled high dimensional maps. We observe an interesting transition from localized chaos to spatiotemporal chaos. We study this transition as a dynamic phase transition. We observe that persistence acts as an excellent quantifier to describe this transition. Taking the location of the fixed point of circle map (which does not change with feedback) as a reference point, we compute a number of sites which have been greater than (less than) the fixed point until time t. Though local dynamics is high dimensional in this case, this definition of persistence which tracks a single variable is an excellent quantifier for this transition. In most cases, we also obtain a well defined persistence exponent at the critical point and observe conventional scaling as seen in second order phase transitions. This indicates that persistence could work as a good order parameter for transitions from fully or partially arrested phase. We also give an explanation of gaps in eigenvalue spectrum of the Jacobian of localized state.     

Statistics of Poincaré recurrences for maps with integrable and ergodic components

Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to zero the distribution converges to the exponential e(-t) for almost any point x, if the system is mixing and the set A is a ball or a cylinder. We consider instead a system, a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points and we generalize around other points with numerical computations. The result could be extended to quasi-integrable area preserving maps such as the standard map for small coupling. We then analyze the distribution of return times in a region which is composed by two invariants subdomains: one with a mixing dynamics and the other with an integrable dynamics given by our shear flow. We show that the statistics of first return in this mixed region is asymptotically given by the exponential law, but this limit is attained by an intermediate regime where exponential and polynomial laws are linearly superposed and weighted by some factors which are proportional to the relative sizes of the chaotic and regular regions. The result on the statistics of first return times for mixed regions in the phase space can provide a basis to analyze such a property for area preserving maps in mixed regions even when a rigorous result is not available. To this end we present numerical investigations on the standard map which confirm the results of the model.     

Asymptotic chaos

We provide in table I a list of normal forms of ordinary differential equations describing the dynamics of physical systems in conditions near to the simultaneous onset of up three instabilities. The first (quadratic) terms in the Taylor series for the nonlinear terms in these amplitude equations (as they are called in fluid dynamics) are given in each case. We focus on a particular example involving three instabilities and derive an asymptotic version of the corresponding normal form as a limit of small dissipation is approached. The numerical investigation of this asymptotic normal form strongly suggests that chaotic behavior occurs as close as one wants to the onset of the triple instability. This chaotic behavior is also exhibited by a return map constructed by direct numerical experiments on the amplitude equation. We also derive by analytic methods a model return map that qualitatively reproduces much of the dynamics observed numerically in the solutions of the asymptotic normal form in nearly homoclinic conditions. In the limit of strong contraction, this model map of the plane reduces to a unidimensional map that is valuable in understanding the dynamics of the original system.     

Noise-induced asymptotic periodicity in a piecewise linear map

We examine asymptotically periodic density evolution in one-dimensional maps perturbed by noise, associating the macroscopic state of these dynamical systems with a phase space density. For asymptotically periodic systems density evolution becomes periodic in time, as do some macroscopic properties calculated from them. The general formalism of asymptotic periodicity is examined and used to calculate time correlations along trajectories of these maps as well as their limiting conditional entropy. The time correlation is shown to naturally decouple into periodic and stochastic components. Finally, asymptotic periodicity is studied in a noise-perturbed piecewise linear map, focusing on how the variation of noise amplitude can cause a transition from asymptotic periodicity to asymptotic stability in the density evolution of this system.     

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.     

Classical decays in decoherent quantum maps

The linear entropy and the Loschmidt echo have proved to be of interest recently in the context of quantum information and of the quantum to classical transitions. We study the asymptotic long-time behavior of these quantities for open quantum maps and relate the decays to the eigenvalues of a coarse-grained superoperator. In specific ranges of coarse graining, and for chaotic maps, these decay rates are given by the Ruelle-Pollicott resonances of the classical map.