Characterization of chaos in random maps

Physical Review E, 2013

We investigate the effects of random perturbations on fully chaotic open systems. Perturbations can be applied to each trajectory independently (white noise) or simultaneously to all trajectories (random map). We compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time-dependent conditionally-map-invariant measure. For the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map. In random maps we show that the escape rate κ and dimensions D of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation. We discuss the accuracy (bias) and precision (variance) of finite-size estimators of κ and D, and show that the improvement of the precision of the estimations with the number of trajectories N is extremely slow (∝1/ ln N ). We also argue that the finite-size D estimators are typically biased. General theoretical results are combined with analytical calculations and numerical simulations in area-preserving baker maps.

Journal of Statistical Physics, 1984

We consider single-humped symmetric one-dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one. To calculate the probability distribution function, and in turn the Lyapunov exponent and the correlation function, a perturbation expansion is developed for the invariant measure. Besides deriving some general results, we treat several examples in detail and compare our theoretical results with recent numerical ones. Furthermore, a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes.

Some results on the stochastic control of two-dimensional chaotic map, i.e., baker map are presented. The approach is based on probabilistic coupling of the controlled dynamics with a controlling system and subsequent lift of the coupled dynamics in a suitable functional space. The lifted dynamics is described in terms of probability densities and is governed by linear Perron{Frobenius and Koopman operators. Su cient condition of controllability and estimation for time to achieve control for a given accuracy in terms of spectral decomposition of Perron-Frobenius operator are obtained.

Journal of Applied Mathematics and Stochastic Analysis, 2004

Letρ(x,t)denote a family of probability density functions parameterized by timet. We show the existence of a family{τ1:t>0}of deterministic nonlinear (chaotic) point transformations whose invariant probability density functions are preciselyρ(x,t). In particular, we are interested in the densities that arise from the diffusions. We derive a partial differential equation whose solution yields the family of chaotic maps whose density functions are precisely those of the diffusion.

Nonlinearity

We consider discrete-time dynamical systems with a linear relaxation dynamics that are driven by deterministic chaotic forces. By perturbative expansion in a small time scale parameter, we derive from the Perron-Frobenius equation the corrections to ordinary Fokker-Planck equations in leading order of the time scale separation parameter. We present analytic solutions to the equations for the example of driving forces generated by N -th order Chebychev maps. The leading order corrections are universal for N ≥ 4 but different for N = 2 and N = 3. We also study diffusively coupled Chebychev maps as driving forces, where strong correlations may prevent convergence to Gaussian limit behavior.

Preprint ICTP IC/94/347, Trieste, …, 1994

The problem of a qualitative change in dynamics of n-dimenskmal chaotic maps under the influence of parametric perturbations is considered. We prove that for certain maps, •• the quadratic maps family, a piecewise linear maps family, and a two-dimensional map having a hyperbolic attractor,-there are perturbations which lead to suppression of chaos. Arguments that for such maps the set of parameter values corresponding to the ordered behaviour has the positive Lebesgue measure, are given.

Il Nuovo Cimento D, 1995

Local and global statistical properties of a class of one dimensional dissipative chaotic maps and a class of 2 dimensional conservative hyperbolic maps are investigated. This is achieved by considering the spectral properties of the Perron-Frobenius operator (the evolution operator for probability densities) acting on two different types of function space. In the first case, the function space is piecewise analytic, and includes functions having support over local regions of phase space. In the second case, the function space essentially consists of functions which are "globally" analytic, i.e analytic over the given systems entire phase space. Each function space defines a space of measurable functions or observables, whose statistical moments and corresponding characteristic times can be exactly determined.

Chaos, Solitons & Fractals, 2009

Journal of Mathematical Analysis and Applications, 2002

Sensitive dependence on initial conditions is widely understood as being the central idea of chaos. We rst give su cient conditions (both topological and ergodic) on an endomorphism to ensure the sensitivity property. Then, a strong sensitivity concept is introduced. Su cient conditions on a transformation implying strong sensitivity are given. We also provide bounds for the strong sensitivity constant.

In this paper, we present results of numerical experiments on chaotic transients in families of the logistic and Hénon maps. The duration of chaotic transients (the rambling time) for logistic maps estimated according to a rigorous criterion shows monotonic regularities with respect to both the period and the number of periodic window in a series of a given period. Due to inapplicability of this criterion to multidimensional maps, a more universal, though approximate, criterion is systematically studied on the family of logistic maps to optimize a choice of the free parameter value. The same approximate criterion is used to estimate rambling time for a number of periodic windows for the family of Hénon maps. The dependence of the rambling time on the width of periodic windows is tested.

Communications in Mathematical Physics

To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large-but non-invariant-subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth. Two signatures of chaotic behavior in dynamical systems are the positivity of Lyapunov exponents and fast decay of time correlations. For deterministic maps that are not uniformly expanding or hyperbolic, these properties can be difficult to prove even when the underlying geometry suggests a strong likelihood of chaotic behavior. Our main message is that the situation for random maps is different, and one of the aims of this paper is to propose a systematic way to establish the positivity of Lyapunov exponents for such maps. For a prototypical class of 2D maps with certain requisite geometry, we recover, under small perturbations, Lyapunov exponents that correctly reflect that geometry. We prove also that the time correlations of such maps decay exponentially fast. We begin with a discussion of the underlying issues before proceeding to a more detailed discussion of our results.

Europhysics Letters (EPL), 2004

PACS. 05.45.-a -Nonlinear dynamics and nonlinear dynamical systems. PACS. 42.65.Sf -Dynamics of nonlinear optical systems; optical instabilities, optical chaos and complexity, and optical spatio-temporal dynamics. PACS. 05.45.Vx -Communication using chaos.

When a probabilistic description of deterministic chaos is feasible, it can describe the dynamical evolution of a given system beyond the Lyapunov horizon where a point-like evolutionary description fails. In the one dimensional examples that we have studied a probabilistic description is very informative about the spectral properties of a given system. One example is an intermittent map whose behaviour is similar to that of a bistable stochastic system. When weakly perturbed by an oscillating kick the system responds sensitivelyat forcing periods predicted by the system's spectra.

Journal of Applied Mathematics and Stochastic Analysis, 1997

A random map is a discrete time dynamical system in which one of a number of transformations is selected randomly and implemented. Random maps have been used recently to model interference effects in quantum physics. The main results of this paper deal with the Lyapunov exponents for higher dimensional random maps, where the individual maps are Jabloński maps on the n-dimensional cube.

Journal of Applied Mathematics and Physics, 2019

In this article, we have discussed basic concepts of one-dimensional maps like Cubic map, Sine map and analyzed their chaotic behaviors in several senses in the unit interval. We have mainly focused on Orbit Analysis, Time Series Analysis, Lyapunov Exponent Analysis, Sensitivity to Initial Conditions, Bifurcation Diagram, Cobweb Diagram, Histogram, Mathematical Analysis by Newton's Iteration, Trajectories and Sensitivity to Numerical Inaccuracies of the said maps. We have tried to make decision about these mentioned maps whether chaotic or not on a unique interval of parameter value. We have performed numerical calculations and graphical representations for all parameter values on that interval and have tried to find if there is any single value of parameter for which those maps are chaotic. In our calculations we have found there are many values for which those maps are chaotic. We have showed numerical calculations and graphical representations for single value of the parameter only in this paper which gives a clear visualization of chaotic dynamics. We performed all graphical activities by using Mathematica and MATLAB.