Efficient algorithm for detecting unstable periodic orbits in chaotic systems

Route to hyperbolic hyperchaos in a nonautonomous time-delay system

We consider a self-oscillator whose excitation parameter is varied. The frequency of the variation is much smaller than the natural frequency of the oscillator so that oscillations in the system are periodically excited and decayed. Also, a time delay is added such that when the oscillations start to grow at a new excitation stage, they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to nonlinearity, the seeding from the past arrives with a doubled phase so that the oscillation phase changes from stage to stage according to the chaotic Bernoulli-type map. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between the delay time and the excitation period, we found a coupling strength between these subsystems as well as intensity of the phase doubling mechanism responsible for the hyperbolicity. Due to this, a transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following steps of the transition scenario are revealed and analyzed: (a) an intermittency as an alternation of long staying near a fixed point at the origin and short chaotic bursts; (b) chaotic oscillations with frequent visits to the fixed point; (c) plain hyperchaos without hyperbolicity after termination visiting the fixed point; and (d) transformation of hyperchaos to the hyperbolic form.

Symmetry restoring bifurcations and quasiperiodic chaos induced by a new intermittency in a vibro-impact system

Quasiperiodic chaos (QC), which is a combination of quasiperiodic sets and a chaotic set, is uncovered in the six dimensional Poincaré map of a symmetric three-degree of freedom vibro-impact system. Accompanied by symmetry restoring bifurcation, this QC is the consequence of a novel intermittency that occurs between two conjugate quasiperiodic sets and a chaotic set. The six dimensional Poincaré map P is the 2-fold composition of another virtual implicit map Q, yielding the symmetry of the system. Map Q can capture two conjugate attractors, which is at the core of the dynamics of the vibro-impact system. Three types of symmetry restoring bifurcations are analyzed in detail. First, if two conjugate chaotic attractors join together, the chaos-chaos intermittency induced by attractor-merging crisis takes place. Second, if two conjugate quasiperiodic sets are suddenly embedded in a chaotic one, QC is induced by a new intermittency between the three attractors. Third, if two conjugate quasiperiodic attractors connect with each other directly, they merge to form a single symmetric quasiperiodic one. For the second case, the new intermittency is caused by the collision of two conjugate quasiperiodic attractors with an unstable symmetric limit set. As the iteration number is increased, the largest finite-time Lyapunov exponent of the QC does not converge to a constant, but fluctuates in the positive region.

Detection meeting control: Unstable steady states in high-dimensional nonlinear dynamical systems

We articulate an adaptive and reference-free framework based on the principle of random switching to detect and control unstable steady states in high-dimensional nonlinear dynamical systems, without requiring any a priori information about the system or about the target steady state. Starting from an arbitrary initial condition, a proper control signal finds the nearest unstable steady state adaptively and drives the system to it in finite time, regardless of the type of the steady state. We develop a mathematical analysis based on fast-slow manifold separation and Markov chain theory to validate the framework. Numerical demonstration of the control and detection principle using both classic chaotic systems and models of biological and physical significance is provided.

Finite-time braiding exponents

Topological entropy of a dynamical system is an upper bound for the sum of positive Lyapunov exponents; in practice, it is strongly indicative of the presence of mixing in a subset of the domain. Topological entropy can be computed by partition methods, by estimating the maximal growth rate of material lines or other material elements, or by counting the unstable periodic orbits of the flow. All these methods require detailed knowledge of the velocity field that is not always available, for example, when ocean flows are measured using a small number of floating sensors. We propose an alternative calculation, applicable to two-dimensional flows, that uses only a sparse set of flow trajectories as its input. To represent the sparse set of trajectories, we use braids, algebraic objects that record how trajectories exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid stretches loops over a finite time interval is the Finite-Time Braiding Exponent (FTBE). We study FTBEs through numerical simulations of the Aref Blinking Vortex flow, as a representative of a general class of flows having a single invariant component with positive topological entropy. The FTBEs approach the value of the topological entropy from below as the length and number of trajectories is increased; we conjecture that this result holds for a general class of ergodic, mixing systems. Furthermore, FTBEs are computed robustly with respect to the numerical time step, details of braid representation, and choice of initial conditions. We find that, in the class of systems we describe, trajectories can be re-used to form different braids, which greatly reduces the amount of data needed to assess the complexity of the flow.

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.

Macro- and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons

We study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, and evaluation of Lyapunov exponents in bi-parameter diagrams. Such an aggregated approach allows for detecting regions of simple and chaotic dynamics, and demarcating borderlines-exact bifurcation curves. We demonstrate how the organizing centers-points corresponding to codimension-two homoclinic bifurcations-along with fold and period-doubling bifurcation curves structure the biparametric plane, thus forming macro-chaotic regions of onion bulb shapes and revealing spike-adding cascades that generate micro-chaotic structures due to the hysteresis.

Computing periodic orbits with arbitrary precision

This paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision. These very high requirements are useful, for example, in the studies of complex pole location in many physical systems. The algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation (ODE) solver, tides, that uses a Taylor-series method. Nowadays, this methodology is the only one capable of reaching precision up to thousands of digits for ODEs. The method is shown to be quadratically convergent. Some numerical tests for the paradigmatic Lorenz model and the Hénon-Heiles Hamiltonian are presented, giving periodic orbits up to 1000 digits.

Parametric evolution of unstable dimension variability in coupled piecewise-linear chaotic maps

In the presence of unstable dimension variability numerical solutions of chaotic systems are valid only for short periods of observation. For this reason, analytical results for systems that exhibit this phenomenon are needed. Aiming to go one step further in obtaining such results, we study the parametric evolution of unstable dimension variability in two coupled bungalow maps. Each of these maps presents intervals of linearity that define Markov partitions, which are recovered for the coupled system in the case of synchronization. Using such partitions we find exact results for the onset of unstable dimension variability and for contrast measure, which quantifies the intensity of the phenomenon in terms of the stability of the periodic orbits embedded in the synchronization subspace.

On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows

We explore the possibility of extending the stabilizing transformations approach [J. J. Crofts and R. L. Davidchack, SIAM J. Sci. Comput. (USA) 28, 1275 (2006)]. to the problem of locating large numbers of unstable periodic orbits in high-dimensional flows, in particular those that result from spatial discretization of partial differential equations. The approach has been shown to be highly efficient when detecting large sets of periodic orbits in low-dimensional maps. Extension to low-dimensional flows has been achieved by the use of an appropriate Poincare surface of section [D. Pingel, P. Schmelcher, and F. K. Diakonos, Phys. Rep. 400, 67 (2004)]. For the case of high-dimensional flows, we show that it is more efficient to apply stabilizing transformations directly to the flows without the use of the Poincare surface of section. We use the proposed approach to find many unstable periodic orbits in the model example of a chaotic spatially extended system-the Kuramoto-Sivashinsky equation. The performance of the proposed method is compared against other methods such as Newton-Armijo and Levenberg-Marquardt algorithms. In the latter case, we also argue that the Levenberg-Marquardt algorithm, or any other optimization-based approach, is more efficient and simpler in implementation when applied directly to the detection of periodic orbits in high-dimensional flows without the use of the Poincare surface of section or other additional constraints.

Finding periodic orbits of higher-dimensional flows by including tangential components of trajectory motion

Methods to search for periodic orbits are usually implemented with the Newton-Raphson type algorithms that extract the orbits as fixed points. When used to find periodic orbits in flows, however, many such approaches have focused on using mappings defined on the Poincaré surfaces of section, neglecting components perpendicular to the surface of section. We propose a Newton-Raphson based method for Hamiltonian flows that incorporates these perpendicular components by using the full monodromy matrix. We investigated and found that inclusion of these components is crucial to yield an efficient process for converging upon periodic orbits in high dimensional flows. Numerical examples with as many as nine degrees of freedom are provided to demonstrate the effectiveness of our method.

Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation

Many chaotic dynamical systems of physical interest present a strong form of nonhyperbolicity called unstable dimension variability (UDV), for which the chaotic invariant set contains periodic orbits possessing different numbers of unstable eigendirections. The onset of UDV is usually related to the loss of transversal stability of an unstable fixed point embedded in the chaotic set. In this paper, we present a new mechanism for the onset of UDV, whereby the period of the unstable orbits losing transversal stability tends to infinity as we approach the onset of UDV. This mechanism is unveiled by means of a periodic orbit analysis of the invariant chaotic attractor for two model dynamical systems with phase spaces of low dimensionality, and seems to depend heavily on the chaotic dynamics in the invariant set. We also described, for these systems, the blowout bifurcation (for which the chaotic set as a whole loses transversal stability) and its relation with the situation where the effects of UDV are the most intense. For the latter point, we found that chaotic trajectories off, but very close to, the invariant set exhibit the same scaling characteristic of the so-called on-off intermittency.

A note on chaotic unimodal maps and applications

Based on the word-lift technique of symbolic dynamics of one-dimensional unimodal maps, we investigate the relation between chaotic kneading sequences and linear maximum-length shift-register sequences. Theoretical and numerical evidence that the set of the maximum-length shift-register sequences is a subset of the set of the universal sequence of one-dimensional chaotic unimodal maps is given. By stabilizing unstable periodic orbits on superstable periodic orbits, we also develop techniques to control the generation of long binary sequences.

Stabilizing long-period orbits via symbolic dynamics in simple limiter controllers

We present an efficient approach to determine the control parameter of simple limiter controllers by using symbolic dynamics of one-dimensional unimodal maps. By applying addition- and subtraction-symbol rules for generating an admissible periodic sequence, we deal with the smallest base problem of the digital tent map. The proposed solution is useful for minimizing the configuration of digital circuit designs for a given target sequence. With the use of the limiter controller, we show that one-dimensional unimodal maps may be robustly employed to generate the maximum-length shift-register sequences. For an arbitrary long Sarkovskii sequence, the control parameters are analytically given.

Transition to intermittent chaotic synchronization

Coupled chaotic oscillators can exhibit intermittent synchronization in the weakly coupling regime, as characterized by the entrainment of their dynamical variables in random time intervals of finite duration. We find that the transition to intermittent synchronization can be characteristically distinct for geometrically different chaotic attractors. In particular, for coupled phase-coherent chaotic attractors such as those from the Rössler system, the transition occurs immediately as the coupling is increased from zero. For phase-incoherent chaotic attractors such as those in the Lorenz system, the transition occurs only when the coupling is sufficiently strong. A theory based on the behavior of the Lyapunov exponents and unstable periodic orbits is developed to understand these distinct transitions.

On Jacobian matrices for flows

We present a general method for constructing numerical Jacobian matrices for flows discretized on a Poincaré surface of section. Special attention is given to Hamiltonian flows where the additional constraint of energy conservation is explicitly taken into account. We demonstrate the approach for a conservative dynamical flow and apply the technique for the general detection of periodic orbits.

Estimating a generating partition from observed time series: symbolic shadowing

We propose a deterministic algorithm for approximating a generating partition from a time series using tessellations. Using data generated by Hénon and Ikeda maps, we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order, and provide good estimates of the metric and topological entropies. The algorithm gives useful results even with a short noisy time series.

Variational method for finding periodic orbits in a general flow

A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems. An initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop. The "Newton descent" partial differential equation that governs this evolution is an infinitesimal step version of the damped Newton-Raphson iteration. The feasibility of the method is demonstrated by its application to the Hénon-Heiles system, the circular restricted three-body problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.

Conditions for efficient chaos-based communication

We find the conditions for a chaotic system to transmit a general source of information efficiently. Transmission of information with very low probability of error is possible if the topological entropy of the transmitted wave signal is greater than or equal to the Shannon entropy of the source message minus the conditional entropy coming from the limitations of the channel (such as equivocation by the noise). This condition may not be always satisfied both due to dynamical constraints and due to the nonoptimal use of the dynamical partition. In both cases, we describe strategies to overcome these limitations.

Entropy and bifurcations in a chaotic laser

We compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection. We consider the Poincaré return map to a fixed plane, and are able to compute the stable and unstable manifolds of periodic points globally, even though it is impossible to find a plane on which the Poincaré map is globally smoothly defined. In this way, we obtain the information that forms the input of the entropy calculations, and characterize the boundary crisis in which the chaotic attractor is destroyed. This boundary crisis involves a periodic point with negative eigenvalues, and the entropy associated with the chaotic attractor persists in a chaotic saddle after the bifurcation.

Control of hyperchaos

A general method for controlling chaotic systems with one or more positive Lyapunov exponents is investigated analytically and numerically. The method retains the formal features of the adaptive adjustment mechanism and can be equally applied to various types of the unstable fixed points. It is shown that the method proposed here neither asks for any prior analytical knowledge of the system, nor any internal or external controlling parameters in advance.

Analyses of transient chaotic time series

We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits, from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.

Catastrophic bifurcation from riddled to fractal basins

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

Stabilizing unstable discrete systems

A general method for stabilizing unstable discrete systems to a fixed point or high-period orbit is developed analytically and numerically in this paper. It is shown that the method can be equally applied to the systems with one or more positive Lyapunov exponents. Moreover, the method does not require a prior analytical knowledge of the system under investigation, nor any additional control parameters.

Detecting unstable periodic orbits in chaotic continuous-time dynamical systems

We extend the recently developed method for detecting unstable periodic points of chaotic time-discrete dynamical systems to find unstable periodic orbits in time-continuous systems, given by a set of ordinary differential equations. This is achieved by the reduction of the continuous flow to a Poincaré map which is then searched for periodic points. The algorithm has global convergence properties and needs no a priori knowledge of the system. It works well for both dissipative and Hamiltonian dynamical systems which is demonstrated by exploring the Lorenz system and the hydrogen atom in a strong magnetic field. The advantages and general features of the approach are discussed in detail.

Local and global control of high-period unstable orbits in reversible maps

We study the nonlinear dynamics of a complex system, described by a two-dimensional reversible map. The phase space of this map exhibits elements typical of Hamiltonian systems (stability islands) as well as of dissipative systems (attractor). Due to the interaction between the stability islands and the attractor, the transition to chaos in this system occurs through the collapse of the stability island and stochastization of the limiting-cycles orbits. We show how to apply the method of discrete parametric control to stabilize unstable high-period orbits. To achieve highly efficient control we introduce the concepts of local and global control. These concepts are useful in situations where there are "dangerous" points on the target orbit, i.e., the points where the probability of breakdown of control is high. As a result, the dangerous points turn out to be much more sensitive to external noise than other points on the orbit, and only the dangerous points determine how effective the control is.

Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems

A topological approach and understanding to the detection of unstable periodic orbits based on a recently proposed method [Phys. Rev. Lett. 78, 4733 (1997)] is developed. This approach provides a classification of the set of transformations necessary for finding the orbits. Applications to the Ikeda and Henon map are performed, allowing a study of the distributions of Lyapunov exponents for high periods. In particular, the properties of the least unstable orbits up to period 36 are investigated and discussed.

Estimating generating partitions of chaotic systems by unstable periodic orbits

An outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1. It has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition. Here we present a general, dimension-independent, and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits. Our algorithm allows us to obtain the approximate location of the generating partition for the Ikeda-Hammel-Jones-Moloney map.