Orbits Of Chaotic Dynamical Systems Research Articles

Accurate determination of heteroclinic orbits in chaotic dynamical systems

Accurate calculation of heteroclinic and homoclinic orbits can be of significant importance in some classes of dynamical system problems. Yet for very strongly chaotic systems initial deviations from a true orbit will be magnified by a large exponential rate making direct computational methods fail quickly. In this paper, a method is developed that avoids direct calculation of the orbit by making use of the well-known stability property of the invariant unstable and stable manifolds. Under an area-preserving map, this property assures that any initial deviation from the stable (unstable) manifold collapses onto them under inverse (forward) iterations of the map. Using a set of judiciously chosen auxiliary points on the manifolds, long orbit segments can be calculated using the stable and unstable manifold intersections of the heteroclinic (homoclinic) tangle. Detailed calculations using the example of the kicked rotor are provided along with verification of the relation between action differences and certain areas bounded by the manifolds.

Controlling chaos faster

Predictive feedback control is an easy-to-implement method to stabilize unknown unstable periodic orbits in chaotic dynamical systems. Predictive feedback control is severely limited because asymptotic convergence speed decreases with stronger instabilities which in turn are typical for larger target periods, rendering it harder to effectively stabilize periodic orbits of large period. Here, we study stalled chaos control, where the application of control is stalled to make use of the chaotic, uncontrolled dynamics, and introduce an adaptation paradigm to overcome this limitation and speed up convergence. This modified control scheme is not only capable of stabilizing more periodic orbits than the original predictive feedback control but also speeds up convergence for typical chaotic maps, as illustrated in both theory and application. The proposed adaptation scheme provides a way to tune parameters online, yielding a broadly applicable, fast chaos control that converges reliably, even for periodic orbits of large period.

Computer-Assisted Proofs and Symbolic Computations

We discuss some main points of computer-assisted proofs based on reliable numerical computations. Such so-called self-validating numerical methods in combination with exact symbolic manipulations result in very powerful mathematical software tools. These tools allow proving mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying theorems fast finite precision arithmetic is used. The results are absolutely rigorous. To demonstrate the power of reliable symbolic-numeric computations we investigate in some details the verification of very long periodic orbits of chaotic dynamical systems. The verification is done directly in Maple, e.g. using the Maple Power Tool intpakX or, more efficiently, using the C++ class library C-XSC.

Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation

In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

General approach to the localization of unstable periodic orbits in chaotic dynamical systems

We present a method to detect the unstable periodic orbits of a multidimensional chaotic dynamical system. Our approach allows us to locate in an efficient way unstable cycles of, in principle, arbitrary length with a high accuracy. Based on a universal set of linear transformations the originally unstable periodic orbits are transformed into stable ones, and can consequently be detected and analyzed easily. This method is applicable to dynamical systems of any dimension, and requires no preknowledge with respect to the solutions of the original chaotic system. As an example of application of our method, we investigate the Ikeda attractor in some detail.

Complexity, randomness, discretization: Some remarks on a program of J. Ford

We discuss certain aspects of the complexity of periodic orbits of chaotic dynamical systems in a symbolic setting. Our theory is based on a generalization of the notion of random sequences in the sense of Kolmogorov and Martin-Löf. Applying our results to the systems studied by Ford et al. leads to completely different conclusions.

Directing orbits of chaotic systems in the presence of noise: feedback correction

In thispaper, we consider the directing of orbits of chaotic dynamicalsystems towards desired targets in the presence of random dynamicalnoise. Our aim is to significantly reduce the time needed toreach a target region by applying only small, bounded perturbations.We outline a technique for deriving open-loop controllers basedon optimal control theory, and then propose methods for improvingthe performance of these controllers to counter the effects ofnoise. The improvements are illustrated by examples.

GEOMETRY OF TARGETING OF CHAOTIC SYSTEMS

In this paper, we analyze the geometry of directing of orbits of chaotic dynamical systems. The geometric approach enables us to interpret the obtained results so as to complement some of the existing ideas about minimum-time targeting. The analysis is illustrated by an example.

Stabilizing chaotic-scattering trajectories using control

The method of stabilizing unstable periodic orbits in chaotic dynamical systems by Ott, Grebogi, and Yorke (OGY) is applied to control chaotic scattering in Hamiltonian systems. In particular, we consider the case of [ital nonhyperbolic] chaotic scattering, where there exist Kolmogorov-Arnold-Moser (KAM) surfaces in the scattering region. It is found that for short unstable periodic orbits not close to the KAM surfaces, both the probability that a particle can be controlled and the average time to achieve control are determined by the initial exponential decay rate of particles in the hyperbolic component. For periodic orbits near the KAM surfaces, due to the stickiness effect of the KAM surfaces on particle trajectories, the average time to achieve control can greatly exceed that determined by the hyperbolic component. The applicability of the OGY method to stabilize intermediate complexes of classical scattering systems is suggested.