Zero Lyapunov Exponents Research Articles

Three-Dimensional Torus Breakdown and Chaos With Two Zero Lyapunov Exponents in Coupled Radio-Physical Generators

Abstract Using an example a system of two coupled generators of quasi-periodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero, and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involves saddle tori occurring at their doublings. This transition is associated with typical structure of parameter plane, like cross-road area and shrimp-shaped structures, based on the two-frequency quasi-periodic dynamics. Using double Poincaré section, we have shown destruction of three-frequency torus.

A Remark on Stochastic Flows in a Hilbert Space

This paper is an extension of known results of Pesin’s entropy formula and SRB measures for random compositions of infinite-dimensional mappings to the continuous-time setting of stochastic flows. Consider a stochastic flow ϕ on a separable infinite dimensional Hilbert space preserving a probability measure μ, which is supported on a random compact set K. We show that if ϕ is C2 (on K) and satisfies some mild integrable conditions of the differentials, then Pesin’s entropy formula holds if μ has absolutely continuous conditional measures along the unstable manifolds. The converse is also true under an additional condition on K when the system has no zero Lyapunov exponent.

Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds—a contribution to the Gallavotti–Cohen chaotic hypothesis

This note presents a non-rigorous study of the linear response for an SRB (or ‘natural physical’) measure ρ of a diffeomorphism f in the presence of tangencies of the stable and unstable manifolds of ρ. We propose that generically, if ρ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of with respect to f along X is convergent:This suggests that an SRB measure may exist for small perturbations of f, with weak differentiability.

Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle

We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also describe the real Lie groups which can appear in the monodromy of the Kontsevich-Zorich cocycle. The number of zero exponents is then as small as possible, given its monodromy.

Affine Manifolds and Zero Lyapunov Exponents in Genus 3

In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichm\"uller curve in genus three has two zero Lyapunov exponents in the Kontsevich-Zorich cocycle, then it lies in the principal stratum and has at most quadratic trace field. Moreover, there can be at most finitely many such Teichm\"uller curves.

The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula

Consider a random cocycle $\Phi$ on a separableinfinite-dimensional Hilbert space preserving a probability measure$\mu$, which is supported on a random compact set $K$. We show thatif $\Phi$ is $C^2$ (over $K$) and satisfies some mild integrableconditions of the differentials, then Pesin's entropy formula holdsif $\mu$ has absolutely continuous conditionalmeasures on the unstable manifolds. The converse is also true under an additional condition on $K$ when the system has no zero Lyapunov exponent.

A Volume Preserving Diffeomorphism with Essential Coexistence of Zero and Nonzero Lyapunov Exponents

We show that there exists a C∞ volume preserving topologically transitive diffeomorphism of a compact smooth Riemannian manifold which is ergodic (indeed is Bernoulli) on an open and dense subset \({\mathcal{G}}\) of not full volume and has zero Lyapunov exponent on the complement of \({\mathcal{G}}\) .

Some results on perturbations of Lyapunov exponents

In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.

Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms

We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the C1 topology: Either there is at least one zero Lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure, then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independent interest that hold Cr-generically for any r ≥ 1; namely, the continuity of the ergodic decomposition, the persistence of invariant sets, and the L1-continuity of Lyapunov exponents.

Characterizing the phase synchronization transition of chaotic oscillators

The chaotic phase synchronization transition is studied in connection with the zero Lyapunov exponent. We propose a hypothesis that it is associated with a switching of the maximal finite-time zero Lyapunov exponent, which is introduced in the framework of a large deviation analysis. A noisy sine circle map is investigated to introduce this hypothesis and it is tested in an unidirectionally coupled Rössler system by using the covariant Lyapunov vector associated with the zero Lyapunov exponent.

Influence of noise on the behavior of oscillators near the synchronization boundary

Nonautonomous behavior of oscillators in the presence of noise is considered. The influence of noise on the dynamics of local zero Lyapunov exponents for nonautonomous dynamic systems that are near the synchronization boundary is considered. It is shown that the action of noise on a nonautonomous dynamic system that is near the synchronization boundary produces domains of synchronous motion in the series realization, which alternate with asynchronous domains. In accordance with this, the distribution of local zero Lyapunov exponents corresponding to laminar phases shift toward negative values. This effect is demonstrated with a discrete-time system (map of a circle onto itself) that is a reference model to describe the synchronization phenomenon and also with a reference system exhibiting chaotic dynamics (Ressler system).

Zero Lyapunov exponent in the vicinity of the saddle-node bifurcation point in the presence of noise

We consider a behavior of the zero Lyapunov exponent in the vicinity of the bifurcation point that occurs as the result of the interplay between dynamical mechanisms and random dynamics. We analytically deduce the laws for the dependence of this Lyapunov exponent on the control parameter both above and below the bifurcation point. The developed theory is applicable both to the systems with the random force and to the deterministic chaotic oscillators. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. We also discuss how the revealed regularities are expected to take place in other relevant physical circumstances.

Localization-delocalization transition of photons in one-dimensional randomn-mer dielectric systems

In this work, we have both theoretically and experimentally investigated the photonic transmission in a one-dimensional random $n$-mer (RN) dielectric system. Due to the positional correlation in the RN structure, the localization-delocalization transitions of photons happen at expected frequencies of photons. Multiple resonant transmissions are found in the photonic band gap. At each resonant mode, zero-Lyapunov exponent and undecayed field distribution of electromagnetic waves have been found through the whole system. Furthermore, the channel is opened for photonic transport at the resonant frequency, and the density of states of photons increases step by step as frequency increases. The theoretical results are experimentally demonstrated in RN dielectric multilayer films of $\mathrm{Si}{\mathrm{O}}_{2}∕\mathrm{Ti}{\mathrm{O}}_{2}$ for visible and near infrared light.

Delocalization of phonons and quantized thermal conductance in a randomn-mer system

In this paper we report the phononic transport and its influence on thermal conductance in a one-dimensional random $n$-mer system. Based on an elastic wave equation, we studied frequency spectrum, phononic transmission, and thermal conductivity of the system. Multiple resonant transmissions are observed, which originate from delocalization of phonons in the correlated disorder system. At each resonant mode, zero-Lyapunov exponent and undecayed vibration of atoms have been found through the whole chain. Meanwhile channels are opened for phonon transmission and heat transport. As a result, thermal conductance is significantly enhanced at each resonant mode, forming a quantized feature as frequency increases. These properties demonstrate the possibility of manipulating phonon propagation and thermal conductance in phononic bandgap materials and may have potential applications in designing filter and waveguide for acoustic waves.

Time-dependent mode structure for Lyapunov vectors as a collective movement in quasi-one-dimensional systems

The time-dependent mode structure of the Lyapunov vectors associated with the stepwise structure of the Lyapunov spectra and its relation to the momentum autocorrelation function are discussed in quasi-one-dimensional many-hard-disk systems. We obtain the complete mode structures (Lyapunov modes) for all components of the Lyapunov vectors, including the longitudinal and transverse components of both the spatial and momentum parts, and their phase relations. These mode structures are suggested by the form of the Lyapunov vectors for the zero-Lyapunov exponents. The spatial node structures of these modes are explained by the reflection properties of the hard walls used in the models. Our main result is that the largest time-oscillating period of the Lyapunov modes is twice as long as the time-oscillating period of the longitudinal momentum autocorrelation function. This relation is satisfied irrespective of the number of particles and the boundary conditions. A simple explanation for this relation is given based on the form of the time-dependent Lyapunov mode.

Space-time complexity in Hamiltonian dynamics.

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with "flights," trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially close trajectories. The complexity function is similar to the propagator p(t(0),x(0);t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space-time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t-->eta=ln t, s-->xi=ln s makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(epsilon;xi,eta) possesses invariants (alpha,beta) that describe the fractal dimensions of the space-time structures of trajectories. The invariants (alpha,beta) can be linked to the transport properties of the system, from one side, and to the Riemann invariants for simple waves, from the other side. This analog provides a new meaning for the transport exponent mu that can be considered as the speed of a Riemann wave in the log-phase space of the log-space-time variables. Some other applications of new notions are considered and numerical examples are presented.

Determining Zero Lyapunov Exponents in Continuous Time Dynamical Systems

Using the tangent bundle conjecture for the Lyapunov exponents, which considers these quantities as a generalization of eigenvalues to nonlinear systems, this work tries to identify zero exponents of a given system, if it exhibits one, via algebraic techniques. The existence of zero Lyapunov exponents for the Toda, Hénon-Heiles and two decoupled simple harmonic oscillator systems are shown.