Behavior Of Lyapunov Exponent Research Articles

Dynamics of coupled quasiperiodic generator and Rossler system

The interaction of a system with quasi-periodic autonomous dynamics and a chaotic system (Rossler system) is considered. The behavior of Lyapunov exponents is studied to identify possible types of system dynamics: chaos with additional zero Lyapunov exponents, three-frequency and two-frequency quasi-periodic regimes, periodic oscillations and the mode of oscillation death. Keywords: quasi-periodic oscillations, dynamical chaos, Rossler system, Lyapunov exponents.

Динамика связанных квазипериодического генератора и системы Ресслера

The interaction of a system with quasi-periodic autonomous dynamics and a chaotic system (Rössler system) is considered. The behavior of Lyapunov exponents is studied to identify possible types of system dynamics: chaos with additional zero Lyapunov exponents, three-frequency and two-frequency quasi-periodic modes, periodic oscillations and the mode of oscillation death.

Lyapunov Spectrum Properties and Continuity of the Lower Joint Spectral Radius

In this paper we study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles [in the sense of (Bonatti and Viana in Ergod Theory Dyn Syst 24(5):1295–1330, 2004)] over mixing subshifts of finite type. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the entropy spectrum at boundary of Lyapunov spectrum in the sense that h_{top}(E(alpha _{t})) rightarrow h_{top}(E(beta ({mathcal {A}})), where E(alpha )={xin X: lim _{nrightarrow infty }frac{1}{n}log Vert {mathcal {A}}^{n}(x)Vert =alpha }, for such cocycles. We prove the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.

Superstable cycles and magnetization plateaus for antiferromagnetic spin-1 Ising and Ising–Heisenberg models on diamond chains

We consider the appearance of superstable cycles in the dynamical approach to the antiferromagnetic and ferromagnetic spin-1 Ising and Ising–Heisenberg models on diamond chains, and their connection with magnetization plateaus. The rational mappings, which provide the statistical properties of the model, are derived by using recurrence relations technique. We consider stability properties of the mapping, providing evidences of a connection between magnetization plateaus and dynamical properties, as the behavior of Lyapunov exponents. The newfound correspondence between superstable point and zero magnetic plateau in spin-1 models on a diamond chain has been tested for a range of parameters.

A new problem of adiabatic invariance related to the rigid body dynamics

We study a new problem of adiabatic invariance, namely a nonlinearoscillator with slowly moving center of oscillation; the frequency ofsmall oscillations vanishes when the center of oscillation passesthrough the origin (the fast motion is no longer fast), and this canproduce nontrivial motions. Similar systems naturally appear in thestudy of the perturbed Euler rigid body, in the vicinity of properrotations and in connection with the 1:1 resonance, as models for thenormal form. In this paper we provide, on the one hand, a rigorousupper bound on the possible size of chaotic motions; on the other handwe work out, heuristically, a lower bound for the same quantity, andthe two bounds do coincide up to a logarithmic correction. We alsoillustrate the theory by quite accurate numerical results, including,besides the size of the chaotic motions, the behavior of LyapunovExponents. As far as the system at hand is a model problem for therigid body dynamics, our results fill the gap existing in theliterature between thetheoretically proved stability properties of proper rotations and thenumerically observed ones, which in the case of the 1:1 resonance didnot completely agree, so indicating a not yet optimal theory.

Stochastic averaging and asymptotic behavior of the stochastic Duffing–van der Pol equation

Consider the stochastic Duffing–van der Pol equation x ̈ =−ω 2x−Ax 3−Bx 2 x ̇ +ε 2β x ̇ +εσx W ̇ t with A⩾0 and B>0. If β/2+ σ 2/8 ω 2>0 then for small enough ε>0 the system (x, x ̇ ) is positive recurrent in R 2⧹{0}. Let λ ̃ ε denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that λ ̃ ε∼ε 2 λ ̃ as ε→0, where λ ̃ is the top Lyapunov exponent along trajectories for a stochastic differential equation obtained from the stochastic Duffing–van der Pol equation by stochastic averaging. In the course of proving this result, we develop results on stochastic averaging for stochastic flows, and on the behavior of Lyapunov exponents and invariant measures under stochastic averaging. Using the rotational symmetry of the stochastically averaged system, we develop theoretical and numerical methods for the evaluation of λ ̃ . We see that the sign of λ ̃ , and hence the asymptotic behavior of the stochastic Duffing–van der Pol equation, depends strongly on ωB/ A. This dimensionless quantity measures the relative strengths of the nonlinear dissipation Bx 2 x ̇ and the nonlinear restoring force Ax 3.

Observer-Dependence of Chaos Under Lorentz and Rindler Transformations

The behavior of Lyapunov exponents λ and dynamical entropies h, whose positivity characterizes chaotic motion, under Lorentz and Rindler transformations is studied. Under Lorentz transformations, λ and h are changed, but their positivity is preserved for chaotic systems. Under Rindler transformations, λ and h are changed in such a way that systems, which are chaotic for an accelerated Rindler observer, can be nonchaotic for an inertial Minkowski observer. Therefore, the concept of chaos is observer-dependent.

Critical exponents in the transition to chaos in one-dimensional discrete systems

We report the numerically evaluated critical exponents associated with the scaling of generalized fractal dimensions during the transition from order to chaos. The analysis is carried out in detail in the context of unimodal and bimodal maps representing typical one-dimensional discrete dynamical systems. The behavior of Lyapunov exponents (LE) in the cross over region is also studied for a complete characterization.

Scaling behavior of Lyapunov exponents in Abelian-Higgs theories

Scaling behavior of Lyapunov exponents in Abelian-Higgs theories

Stochastic modeling of a billiard in a gravitational field: Power law behavior of Lyapunov exponents

We consider the motion of a point particle (billiard) in a uniform gravitational field constrained to move in a symmetric wedge-shaped region. The billiard is reflected at the wedge boundary. The phase space of the system naturally divides itself into two regions in which the tangent maps are respectively parabolic and hyperbolic. It is known that the system is integrable for two values of the wedge half-angleθ1 andθ2 and chaotic forθ1<θ<θ2. We study the system at three levels of approximation: first, where the deterministic dynamics is replaced by a random evolution; second, where, in addition, the tangent map in each region is, replaced by its average; and third, where the tangent map is replaced by a single global average. We show that at all three levels the Lyapunov exponent exhibits power law behavior nearθ1 andθ2 with exponents 1/2 and 1, respectively. We indicate the origin of the exponent 1, which has not been observed in unaccelerated billiards.

Behaviour of Lyapunov exponents near crisis points in the dissipative standard map

We numerically study the behaviour of the largest Lyapunov characteristic exponent λ1 in dependence on a control parameter in the 2D standard map with dissipation. In order to investigate the system's motion in parameter intervals slightly above crisis points we introduce "partial" Lyapunov exponents which characterize the average exponential divergence of nearby orbits on a semi-attractor at a boundary crisis and on distinct parts of a "large" chaotic attractor near an interior crisis. In the former case we find no significant difference between λ1 in the pre-crisis regime and the partial Lyapunov exponent describing transient chaotic motions slightly above the crisis. For the latter case we give a quantitative description of the drastic increase of λ1. Moreover, a formula which connects the critical exponent of a chaotic transient above a boundary crisis with a pointwise dimension is derived.

Adiabatic invariants and asymptotic behavior of Lyapunov exponents of the Schr�dinger equation

We give an upper bound for the high-energy behavior of the Lyapunov exponent of the one-dimensional Schrodinger equation. We relate this behavior to the diffrentiability properties of the potential. As an application, this result provides an upper bound for the asymptotic length of the gaps of the Schrodinger equation.