Periodic Orbits In Chaotic Systems Research Articles

Detecting unstable periodic points of chaotic maps by stability transformation of reservoir

Detecting unstable periodic orbits in chaotic systems based on the time series is a fundamental problem in nonlinear dynamics, but it often becomes extremely challenging one. In this study, we propose a new approach for detecting unstable periodic points using reservoir computing and stability transformation method. We connects reservoir computing, which is a well-known machine learning technique, and stability transformation method, which can detect unstable periodic points in chaotic dynamical systems, to perform unstable periodic points detection in a data-driven and model-free process. In this paper, we use an example of the Hénon map to demonstrate detecting unstable fixed point and unstable 2-periodic points.

Topological classification of periodic orbits in the generalized Lorenz-type system with diverse symbolic dynamics

Periodic orbits play important roles in the analysis of dynamic behaviors in chaotic systems, and they are fundamental keys to understanding the properties of the strange attractor. In this paper, the unstable periodic orbits of a three-dimensional (3D) autonomous Lorenz-type system, the so-called generalized Lorenz-type system (GLTS), are investigated using the variational method. Taking the parameters of the GLTS as (a,b,c)=(10,100,10.4), we use two cycles as basic building blocks to establish 1D symbolic dynamics, and all short unstable cycles up to topological length 5 are found. With typical parameters (a,b,c)=(10,42.72,1), the correlation between the Burke-Shaw system (BSS) and the GLTS is discussed, symbolic dynamics for four letters are identified, and the periodic orbits are labeled inside or outside of the attractor based on whether the symbol sequence of the cycle contains the building blocks with the self-linking number 1. The variational method verifies the effectiveness of locating the periodic orbits, and topological classification provides a novel way to build appropriate symbolic dynamics. We also utilize the homotopy evolution approach to continuously deform the found periodic orbits while varying different parameters, and analyze the various bifurcations in the GLTS. The current work further expands the approach of establishing symbolic dynamics while analyzing the periodic orbits in chaotic systems, and can be applied to the study of turbulence and other complex problems.

Sensitivity of long periodic orbits of chaotic systems

The properties of long, numerically determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents, and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

Fundamental cupolets of chaotic systems

Cupolets are a relatively new class of waveforms that represent highly accurate approximations to the unstable periodic orbits of chaotic systems, and large numbers can be efficiently generated via a control method where small kicks are applied along intersections with a control plane. Cupolets exhibit the interesting property that a given set of controls, periodically repeated, will drive the associated chaotic system onto a uniquely defined cupolet regardless of the system's initial state. We have previously demonstrated a method for efficiently steering from one cupolet to another using a graph-theoretic analysis of the connections between these orbits. In this paper, we discuss how connections between cupolets can be analyzed to show that complicated cupolets are often composed of combinations of simpler cupolets. Hence, it is possible to distinguish cupolets according to their reducibility: a cupolet is classified either as composite, if its orbit can be decomposed into the orbits of other cupolets or as fundamental, if no such decomposition is possible. In doing so, we demonstrate an algorithm that not only classifies each member of a large collection of cupolets as fundamental or composite, but that also determines a minimal set of fundamental cupolets that can exactly reconstruct the orbit of a given composite cupolet. Furthermore, this work introduces a new way to generate higher-order cupolets simply by adjoining fundamental cupolets via sequences of controlled transitions. This allows for large collections of cupolets to be collapsed onto subsets of fundamental cupolets without losing any dynamical information. We conclude by discussing potential future applications.

Harmonic balance-based approach for optimal time delay to control unstable periodic orbits of chaotic systems

As a classical technique for chaos suppression, the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits (UPOs) embedded in chaotic systems. A critical issue for achieving high controlling precision is to search for an appropriate time delay. This paper proposes a simple yet effective approach, based on incremental harmonic balance method, to determine the optimal time delay in the delayed feedback controller. The time delay is adjusted within the iterative scheme provided by the proposed method, and finally converges to the period of the target UPO. As long as the optimal time delay is fixed, moreover, the attained solution makes it quite convenient to analyze its stability according to the Floquet theory, which further provides the effective interval of the feedback gain.

Stabilization of chaotic systems via act-and-wait delayed feedback control using a high-precision direct integration method

A technique for the numerical analysis of the stability problem in chaotic systems via act-and-wait delayed feedback control is developed and justified. Recently, act-and-wait modification of a delayed feedback control method is proposed to stabilize unstable periodic orbits in dynamical systems. To overcome the difficulties in obtaining conditions under which the state of closed-loop system converges toward a periodic solution via the act-and-wait scheme, this paper presents a high-precision direct integration method for calculating the monodromy matrix corresponding to the closed-loop system based on the theory of Peano–Baker series. The capabilities of the proposed schemes in stabilization of unstable periodic orbits of chaotic systems are illustrated by numerical examples.

Exact relations between homoclinic and periodic orbit actions in chaotic systems.

Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulas expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well-estimated errors. This enables an explicit study of relations between periodic orbits, which results in an analytic expression for the action differences between long periodic orbits and their shadowing decomposed orbits in the cycle expansion.

Clustering of Periodic Orbits and Ensembles of Truncated Unitary Matrices

Periodic orbits in chaotic systems form clusters, whose elements traverse approximately the same points of the phase space. The distribution of cluster sizes depends on the length n of orbits and the parameter p which controls closeness of orbits actions. We show that counting of cluster sizes in the baker's map can be turned into a spectral problem for an ensemble of truncated unitary matrices. Based on the conjecture of the universality for the eigenvalues distribution at the spectral edge of these ensembles, we obtain asymptotics of the second moment of cluster distribution in a regime where both n and p tend to infinity. This result allows us to estimate the average cluster size as a function of the number of encounters in periodic orbits.

Stabilization of discrete-time chaotic systems via improved periodic delayed feedback control based on polynomial matrix right coprime factorization

Delayed feedback control, proposed by Pyragas,is one of useful control methods of stabilizing unstable periodic orbits of chaotic systems without their exact information. This paper discusses an improved periodic delayed feedback control for the stabilization of the discrete-time chaotic systems. A technique to solve the generalized Sylvester matrix equation, based on polynomial matrix right coprime factorization, is introduced in order to give an explicit solution to the periodic gains of the closed-loop system. Moreover, some examples are demonstrated to show the effectiveness of the proposed method.

Survey on finding the periodic orbits in chaotic systems

The periodic orbits provide a skeleton for the organization of complex chaotic systems,for many important characteristics and dynamic properties of these systems can be determined by solving the periodic orbits,such as the accurate calculation of Lyapunov exponents,estimation of topological entropy and description of a chaotic invariant set. First,the paper reviewed the current commonly used four methods to find periodic orbits,which are NR algorithm,Broyden algorithm,SD algorithm and DL algorithm,and analyzed their characteristics and mutual relations.Second,the paper discussed the advantages,disadvantages and scope of each method with specific examples in detail.Finally,the paper pointed out that DL algorithm is more ideal among the four algorithms, and suggested the future research direction.

Clustering of periodic orbits in chaotic systems

In the framework of the semiclassical approach, the universal spectral correlations in Hamiltonian systems with classical chaotic dynamics can be attributed to the systematic correlations between the actions of periodic orbits which (up to the switch in the momentum direction) pass through approximately the same points of the phase space. By considering symbolic dynamics of the system one can introduce a natural ultrametric distance between periodic orbits and organize them into clusters of orbits approaching each other in the phase space. We study the distribution of cluster sizes for the baker's map in the asymptotic limit of long trajectories. This problem is equivalent to the one of counting degeneracies in the length spectrum of the de Bruijn graphs. Based on this fact, we derive the probability that k randomly chosen periodic orbits belong to the same cluster. Furthermore, we find asymptotic behaviour of the largest cluster size and derive the probability P(t) that a random periodic orbit belongs to a cluster smaller than , t ∈ [0, 1].

Extended resonant feedback technique for controlling unstable periodic orbits of chaotic system

An improvement of the recently described resonant chaos control method is suggested. Negative feedback loop containing a notch-rejection filter, tuned to the main harmonic of the unstable periodic orbit, is supplemented with a set of notch filters tuned to the higher harmonics. The extended method is applied to an electrical circuit representing the Duffing–Holmes type non-autonomous two-well chaotic oscillator. Stabilization of the period-1 orbit is achieved with very small control force. The residual control signal is about 1% compared to the main variable. Mathematical model based on a two-well piecewise parabolic potential is presented and numerical simulation is performed. Numerical results are confirmed by hardware experiments.

Global stabilization of periodic orbits in chaotic systems by using symbolic dynamics

In this report, a control method for the stabilization of periodic orbits for a class of discrete-time systems that are topologically conjugate to symbolic dynamics is proposed and applied to a population model in an ecosystem and the Smale horseshoe map. A periodic orbit is assigned as a target by giving a sequence in which symbols have periodicity. As a consequence, it is shown that any periodic orbits can be globally stabilized by using arbitrarily small control inputs. This work is the first attempt to systematically design a control system based on symbolic dynamics in the sense that one estimates the magnitude of control inputs and analyzes the Lyapunov stability.

Automatic control and tracking of periodic orbits in chaotic systems

Based on an automatic feedback adjustment of an additional parameter of a dynamical system, we propose a strategy for controlling periodic orbits of desired periods in chaotic dynamics and tracking them toward the set of unstable periodic orbits embedded within the original chaotic attractor. The method does not require information on the system to be controlled, nor on any reference states for the targets, and it overcomes some of the difficulties encountered by other techniques. Assessments of the method's effectiveness and robustness are given by means of the application of the technique to the stabilization of unstable periodic orbits in both discrete- and continuous-time systems.

Generating an adaptive multiresolution image analysis with compact cupolets

We present an efficient control scheme for stabilizing unstable periodic orbits of chaotic systems. The resulting orbits are called cupolets and have been proven to be useful in the representation of oscillatory or quasi periodic signals such as appear in music and image compression (Short et al., AES 118th Convention preprint 6446, May 2005; Short et al., AES 119th Convention preprint 6588, October 2005). In this paper we show that these cupolets can be used effectively to produce an adaptive basis for the space of real-valued functions of a discrete variable. From this basis, we construct a multiresolution analysis which allows for the approximation of signals at different resolution levels and apply it to image compression. This adaptive multiresolution analysis provides an interesting continuum between Fourier analysis and wavelet analysis.

Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control

In this paper by using a combination of fuzzy identification and the sliding mode control a fuzzy adaptive sliding mode scheme is designed to stabilize the unstable periodic orbits of chaotic systems. The chaotic system is assumed to have an affine form x ( n) = f( X) + g( X) u where f and g are unknown functions. Using only the input–output data obtained from the underlying dynamical system, two fuzzy systems are constructed for identification of f and g. Two distinct methods are utilized for fuzzy modeling, the least squares and the gradient descent techniques. Based on the estimated fuzzy models, an adaptive controller, which works through the sliding mode control, is designed to make the system track the desired unstable periodic orbits. The stability analysis of the overall closed loop system is presented in the paper and the effectiveness of the proposed adaptive scheme is numerically investigated. As a case of study, modified Duffing system is selected for applying the proposed method to stabilize its 2 π and 4 π periodic orbits. Simulation results show the high performance of the method for stabilizing the unstable periodic orbits of unknown chaotic systems.

Efficient Detection of Periodic Orbits in Chaotic Systems by Stabilizing Transformations

An algorithm for detecting periodic orbits in chaotic systems [R. L. Davidchack and Y.-C. Lai, Phys. Rev. E (3), 60 (1999), pp. 6172-6175], which combines the set of stabilizing transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighboring periods, has been shown to be highly efficient when applied to low-dimensional systems. The difficulty in applying the algorithm to higher-dimensional systems is mainly due to the fact that the number of the stabilizing transformations grows extremely fast with increasing system dimension. Here we analyze the properties of stabilizing transformations and propose an alternative approach for constructing a smaller set of transformations. The performance of the new approach is illustrated on the four-dimensional kicked double rotor map and the six-dimensional system of three coupled Henon maps.

A multi-dimensional scheme for controlling unstable periodic orbits in chaotic systems

The control of unstable periodic orbits (UPOs) in chaotic systems is herein investigated by applying a weighted sum of external periodic inputs. These inputs are drawn from a “generic” set of basis functions and used to stabilize different UPOs in a chaotic attractor. All parameters associated with this multi-dimensional control scheme are experimentally computed without a priori knowledge of the system, thus making it easily applicable to a wide variety of dynamical systems. The performance of this control scheme is illustrated with several well-established chaotic systems.

Detecting unstable periodic orbits in chaotic systems by using an efficient algorithm

Abstract How to detect unstable periodic orbits (UPOs) which reside on the chaotic attractors is an important problem in diverse scientific fields. In this paper, we present an efficient algorithm for computing UPOs. It can be applied to any finite dimensional chaotic systems including hyperchaotic systems. We also discuss numerically the converging regions of the algorithm here. The chaotic Henon map is taken as numerical example in the text.

Analytical properties and optimization of time-delayed feedback control.

Time-delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. If the equations governing the system dynamics are known, the success of the method can be predicted by a linear stability analysis of the desired orbit. Unfortunately, the usual procedures for evaluating the Floquet exponents of such systems are rather intricate. We show that the main stability properties of the system controlled by time-delayed feedback can be simply derived from a leading Floquet exponent defining the system behavior under proportional feedback control. Optimal parameters of the delayed feedback controller can be evaluated without an explicit integration of delay-differential equations. The method is valid for low-dimensional systems whose unstable periodic orbits are originated from a period doubling bifurcation and is demonstrated for the Rössler system and the Duffing oscillator.