TWO DIMENSIONAL CHAOS: THE BAKER MAP UNDER CONTROL

Computers & Mathematics With Applications, 1997

we have proposed a new probabilistic method for the control of chaotic systems [l]. In this paper, we apply our method to characteristic cases of chaotic maps (one and two-dimensional examples). As these chaotic maps are structurally stable, they cannot be controlled using conventional control methods without significant change of the dynamics. Our method consists in the probabilistic coupling of the original system with a controlling system. This coupling can be understood ss a feedback control of probabilistic nature. The chosen periodic orbit of the original system is a global attractor for the probability densities.

Physica A: Statistical Mechanics and its Applications, 1996

We discuss the characterization of chaotic behaviours in random maps both in terms of the Lyapunov exponent and of the spectral properties of the Perron-Frobenius operator. In particular, we study a logistic map where the control parameter is extracted at random at each time step by considering finite dimensional approximation of the Perron-Frobenius operator.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 1993

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.

International Journal of Bifurcation and Chaos, 2016

Digital stabilization of unstable equilibria of linear systems may lead to small amplitude stochastic-like oscillations. We show that these vibrations can be related to a deterministic chaotic dynamics induced by sampling and quantization. A detailed analytical proof of chaos is presented for the case of a PD controlled oscillator: it is shown that there exists a finite attracting domain in the phase-space, the largest Lyapunov exponent is positive and the existence of a Smale horseshoe is also pointed out. The corresponding two-dimensional micro-chaos map is a multi-baker map, i.e. it consists of a finite series of baker’s maps.

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.

Journal of Difference Equations and Applications

The paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value.

Chaos, Solitons & Fractals, 2009

Techniques for stabilizing unstable state in nonlinear dynamical systems using small perturbations fall into three general categories: feedback, non-feedback schemes, and a combination of feedback and non-feedback. However, the general problem of finding conditions for creation or suppression of chaos still remains open. We describe a method for dynamical control of chaos. This method is based on a definition of the hierarchy of solvable chaotic maps with dynamical parameter as a control parameter. In order to study the new mechanism of control of chaotic process, Kolmogorov-Sinai entropy of the chaotic map with dynamical parameter based on discussion the properties of invariant measure have been calculated and confirmed by calculation of Lyapunov exponents. The introduced chaotic maps can be used as dynamical control.

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.

29th IEEE Conference on Decision and Control, 1990

In this paper, we stud7 a family of two-dimensional nonlinear feedback systems which do ;lot satisfy the Lipschitz continuity condition and exhibit chaotic beh:,vior. The geometric Poincar6 map is determined analytically and a bifurcation study in terms of two canonical parameters, and the associated asymptotic behavior of the systems are presented. Ergodic theory of one-dimensional dynamic systems is used to derive a probabilistic description of the Zhaotic motions.