Power spectra for deterministic chaotic dynamical systems

We generate new hierarchy of many-parameter family of maps of the interval [0, with an invariant measure, by composition of the chaotic maps of reference . Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent, of these maps analytically, where the results thus obtained have been approved with numerical simulation. In contrary to the usual one-dimensional maps and similar to the maps of reference [1], these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain region of parameters values, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at these values of parameter whose Lyapunov characteristic exponent begins to be positive.

1998

We describe a one-dimensional chaotic map where the Liapunov exponent is a smooth function of a control parameter.

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.

The European Physical Journal B, 2009

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the (z 1 , z 2)-logarithmic map, corresponds to a generalization of the z-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the z-logistic map is numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy S q. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to q-Gaussian attractor distributions. We also study the generalized q-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The q-sensitivity indices are obtained as well. Our results are, like those for the z-logistic maps, numerically compatible with the q-generalization of a Pesin-like identity for ensemble averages.

International Symposium on Physical Design, 1996

The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In particular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the unstable (and stable) tangent spaces are not constant over the chaotic set; we call this unstable dimension variability. A simple two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed.

Open Systems & Information Dynamics (OSID), 2001

A hierarchy of universalities in families of 1-D maps is discussed. Breakdown of universalities in families of 3-D maps is shown on selected examples of such families.

Pure and Applied Mathematics Journal, 2015

In this work, I studied a new class of topological λ-type chaos maps, λ-exact chaos and weakly λ-mixing chaos. Relationships with some other type of chaotic maps are given. I will list some relevant properties of λ-type chaotic map. The existence of chaotic behavior in deterministic systems has attracted researchers for many years. In engineering applications such as biological engineering, and chaos control, chaoticity of a topological system is an important subject for investigation. The definitions of λ-type chaos, λ-type exact chaos, λ-type mixing chaos, and weak λ-type mixing chaos are extended to topological spaces. This paper proves that these chaotic properties are all preserved under λr-conjugation. We have the following relationships: λ-type exact chaos⇒ λ-type mixing chaos ⇒ weak λ-type mixing chaos ⇒λ-type chaos.

Journal of Nonlinear Mathematical Physics, 2002

We give a hierarchy of many-parameter families of maps of the interval [0, 1] with an invariant measure and using the measure, we calculate Kolmogorov-Sinai entropy of these maps analytically. In contrary to the usual one-dimensional maps these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain region of parameters space, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at certain values of the parameters.

2000

In this work, an asymptotic measure is introduced in order to characterise chaotic dynamics. This is the asymptotic distance between trajectories d, , which can actually help either as a complementary measure to Lyapunov exponents, or as an alternative parameter characterising chaos when Lyapunov exponents are either very difficult, or impossible to work out. Some analytical relationships between the leading Lyapunov exponent and the values of d, both for discrete maps and continuous systems are here reported, together with experimental comparisons drawn from the simulation of Chua's circuit in different operational conditions.

Pure and Applied Mathematics Journal, 2014

In this paper, we will study a new class of chaotic maps on locally compact Hausdorff spaces called Lambda-type chaotic maps and θ-type chaotic maps. The Lambda-type chaotic map implies chaotic map which implies θ-type chaotic map. Further, the definition of topological Lambda-type chaos implies John Tylar definition which implies topological θ-type chaos definition. Relationships with some other types of chaotic maps defined on topological spaces are given.