Nonautonomous Maps Research Articles

BREATHERS IN INHOMOGENEOUS NONLINEAR LATTICES: AN ANALYSIS VIA CENTER MANIFOLD REDUCTION

We consider an infinite chain of particles linearly coupled to their nearest neighbors and subject to an anharmonic local potential. The chain is assumed weakly inhomogeneous, i.e. coupling constants, particle masses and on-site potentials can have small variations along the chain. We look for small amplitude and time-periodic solutions, and, in particular, spatially localized ones (discrete breathers). The problem is reformulated as a nonautonomous recurrence in a space of time-periodic functions, where the dynamics is considered along the discrete spatial coordinate. Generalizing to nonautonomous maps a center manifold theorem previously obtained for infinite-dimensional autonomous maps [44], we show that small amplitude oscillations are determined by finite-dimensional nonautonomous mappings, whose dimension depends on the solutions frequency. We consider the case of two-dimensional reduced mappings, which occur for frequencies close to the edges of the phonon band (computed for the unperturbed homogeneous chain). For an homogeneous chain, the reduced map is autonomous and reversible, and bifurcations of reversible homoclinic orbits or heteroclinic solutions are found for appropriate parameter values. These orbits correspond respectively to discrete breathers for the infinite chain, or "dark" breathers superposed on a spatially extended standing wave. Breather existence is shown in some cases for any value of the coupling constant, which generalizes (for small amplitude solutions) an existence result obtained by MacKay and Aubry at small coupling [57]. For an inhomogeneous chain, the study of the nonautonomous reduced map is in general far more involved. Here, the problem is considered when the chain presents a finite number of defects. For the principal part of the reduced recurrence, using the assumption of weak inhomogeneity, we show that homoclinics to 0 exist when the image of the unstable manifold under a linear transformation (depending on the defect sequence) intersects the stable manifold. This provides a geometrical understanding of tangent bifurcations of discrete breathers commonly observed in classes of systems with impurities as defect strengths are varied. The case of a mass impurity is studied in detail, and our geometrical analysis is successfully compared with direct numerical simulations. In addition, a class of homoclinic orbits is shown to persist for the full reduced mapping and yields a family of discrete breathers with maximal amplitude at the impurity site.

OLD AND NEW RESULTS ON STRANGE NONCHAOTIC ATTRACTORS

Classical and new results concerning the topological structure of skew-products semiflows, coming from nonautonomous maps and differential equations, are combined in order to establish rigorous conditions giving rise to the occurrence of strange nonchaotic attractors on 𝕋d × ℝ. A special attention is paid to the relation of these sets with the almost automorphic extensions of the base flow. The scope of the results is clarified by applying them to the Harper map, although they are valid in a much wider context.

Positive entropy on nonautonomous interval maps and the topology of the inverse limit space

Entropy on nonautonomous maps { f i } ı = 0 ∞ of the interval is defined 2 ways. Under one definition, called forward entropy, it is shown that positive entropy implies that the inverse limit space of ( { f i } ı = 0 ∞ , I ) contains an indecomposable subcontinuum. Under the second definition, called backwards entropy, it is shown that the inverse limit space of ( { f i } ı = 0 ∞ , I ) is not locally connected.

Homoclinic orbits of non-autonomous maps and their approximation†

We consider homoclinic orbits in non-autonomous discrete time dynamical systems of the form where it is assumed that an n independent fixed point exists. A numerical method for computing finite approximations of transversal homoclinic orbits is introduced and a detailed error analysis is presented. The non-autonomous setup requires special tools. We prove that the analytic condition of transversality of the orbit corresponds to a transversal intersection of the corresponding invariant fiber bundles. The approximation method and the validity of the error estimate is illustrated by an example.

Integrability conditions for nonautonomous quad-graph equations

This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations.

An alternating integrable map whose square is the QRT map

We present a novel alternating integrable map (i.e., a non-autonomous integrable map whose sole time-dependence is a term of the form (−1) n ). We show that the square (i.e., second iterate) of this map is the QRT map.

The discrete Painlevé I equations: transcendental integrability and asymptotic solutions

The integrability of the discrete Painleve I equation (dP-I) is reviewed and its integrability studied. We establish the existence of a conserved quantity which is algebraic in the case of the autonomous dP-I equation and it is argued that the non-autonomous dP-I map has a non-algebraic invariant. Our analysis leads to, among other results the construction of asymptotic solutions with interesting structures.

A Perturbation Method for Nonlinear Two-Dimensional Maps

A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.

Special function solutions for asymmetric discrete Painlevé equations

We study the solutions of a family of discrete Painleve equations. The equations that we examine are given as a system of two first-order non-autonomous mappings. The solutions we are interested in are the ones obtained whenever the Painleve equation can be reduced to a discrete Riccati equation, which can be linearized through a Cole-Hopf transformation. The special solutions thus obtained involve generalizations or reductions of the hypergeometric (and q-hypergeometric) function.

Dynamics associated with a quasiperiodically forced Morse oscillator: Application to molecular dissociation.

The dynamics associated with a quasiperiodically forced Morse oscillator is studied as a classical model for molecular dissociation under external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase-space barriers, allowing escape from bounded to unbounded motion. In contrast to the ubiquitous Poincar\'e map reduction of a periodically forced system, we derive a sequence of nonautonomous maps from the quasiperiodically forced system. We obtain a global picture of the dynamics, i.e., of transport in phase space, using a sequence of time-dependent two-dimensional lobe structures derived from the invariant homoclinic tangle of a persisting invariant saddle-type torus in a Poincar\'e section of an associated autonomous system phase space. Transport is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, and this provides the framework for studying basic features of molecular dissociation in the context of classical phase-space trajectories. We obtain a precise criterion for discerning between bounded and unbounded motion in the context of the forced problem. We identify and measure analytically the flux associated with the transition between bounded and unbounded motion, and study dissociation rates for a variety of initial phase-space ensembles, such as an even or weighted distribution of points in phase space, or a distribution on a particular level set of the unperturbed Hamiltonian (corresponding to a quantum state). A double-phase-slice sampling method allows exact numerical computation of dissociation rates. We compare single- and two-frequency forcing. Infinite-time average flux is maximal in a particular single-frequency limit; however, lobe penetration of the level sets of the unperturbed Hamiltonian can be maximal in the two-frequency case. The variation of lobe areas in the two-frequency problem gives one added freedom to enhance or diminish aspects of phase-space transport on finite time scales for a fixed infinite-time average flux, and for both types of forcing the geometry of lobes is relevant. The chaotic nature of the dynamics is understood in terms of a traveling horseshoe map sequence.

Slow sweep through a period-doubling cascade: Delayed bifurcations and renormalisation

We investigate analytically the effect on a period-doubling cascade of slowly sweeping the bifurcation parameter, by means of asymptotic calculations. First we analyse the behaviour of the orbits when sweeping through one period-doubling bifurcation. There is still an effective bifurcation called a dynamic bifurcation, but which is delayed. We compute the delay. Sweeping in forward and backward directions yields qualitatively different results. The presence of noise in the system, even of small intensity, turns out to play a key role. We analyse the various behaviours as a function of the relative sizes of the sweep rate and the noise intensity. In order to tackle the sweep through a whole cascade we extend the renormalisation theory for period doubling of one-dimensional maps to non-autonomous maps. The particular we show that a slow linear of the parameter results in a new unstable eigenvalue which provides a scaling law for the sweep rate. Taking also into account the effect of noise, we then derive an asymptotic scaling for the successive period-doubling bifurcations. As a result of the sweep (and the noise), only a finite number of doublings can be observed: we estimate this maximum number. We also show how to estimate the location of the real bifurcation of the real points from results of experiments with non-zero sweep velocity. When the functions involved to be analytic we show that the delay of a period-doubling bifurcation due to sweep depends crucially on their degree of smoothness. In both the analytic and non-analytic cases, we give insight into the geometry of some invariant curves reminiscent of an accordion. Our analysis explains numerical and experimental results on the problem.

Do integrable mappings have the Painlevé property?

We present an integrability criterion for discrete-time systems that is the equivalent of the Painlev\'e property for systems of a continuous variable. It is based on the observation that for integrable mappings the singularities that may appear are confined, i.e., they do not propagate indefinitely when one iterates the mapping. Using this novel criterion we show that there exists a family of nonautonomous integrable mappings which includes the discrete Painlev\'e equations by ${\mathit{P}}_{\mathrm{I}}$, recently derived in a model of two-dimensional quantum gravity, and ${\mathit{P}}_{\mathrm{II}}$, obtained as a similarity reduction of the integrable modified Korteweg--de Vries lattice. These systems possess Lax pairs, a well-known integrability feature.

Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems

The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles.

Relaxation, postponement, and features of the attractor in a driven varactor oscillator

The driven varactor oscillator is investigated by numerical integration of its ODEs using the standard model of circuit theory. Attention is given to some properties of the basic relaxation mechanism. For time dependent amplitudes of the sinusoidal driving voltage the post-ponement of the bifurcations is characterized by transient Lyapunov numbers. The postponement of the first bifurcation shows the same dependence on the sweep velocity as in the case of the nonautonomous quadratic map. The shapes of the attractors are displayed in extended phase space. Generalized Renyi-dimensionsD 0 andD 1 have been determined in the chaotic region. A corresponding twodimensional Pioncare map indicates selfsimilarity.

Transversal homoclinic points and hyperbolic sets for non-autonomous maps I

A concept of generalized hyperbolic sets for non-autonomous maps is developed. Starting from transversal homoclinic orbits such generalized hyperbolic sets are constructed. The Shadowing Lemma is proven for maps admitting a generalized hyperbolic set. Time dependent symbolic dynamics is introduced and related to non-autonomous maps.