High-dimensional Chaotic Attractors Research Articles

Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns

We propose a general method for constructing a minimal cover of high-dimensional chaotic attractors by embedded unstable recurrent patterns. By minimal cover we mean a subset of available patterns such that the approximation of chaotic dynamics by a minimal cover with a predefined proximity threshold is as good as the approximation by the full available set. The proximity measure, based on the concept of a directed Hausdorff distance, can be chosen with considerable freedom and adapted to the properties of a given chaotic system. In the context of a spatiotemporally chaotic attractor of the Kuramoto–Sivashinsky system on a periodic domain, we demonstrate that the minimal cover can be faithfully constructed even when the proximity measure is defined within a subspace of dimension much smaller than the dimension of space containing the attractor. We discuss how the minimal cover can be used to provide a reduced description of the attractor structure and the dynamics on it.

Inferring symbolic dynamics of chaotic flows from persistence.

We introduce "state space persistence analysis" for deducing the symbolic dynamics of time series data obtained from high-dimensional chaotic attractors. To this end, we adapt a topological data analysis technique known as persistent homology for the characterization of state space projections of chaotic trajectories and periodic orbits. By comparing the shapes along a chaotic trajectory to those of the periodic orbits, state space persistence analysis quantifies the shape similarity of chaotic trajectory segments and periodic orbits. We demonstrate the method by applying it to the three-dimensional Rössler system and a 30-dimensional discretization of the Kuramoto-Sivashinsky partial differential equation in (1+1) dimensions.

Is a hyperchaotic attractor superposition of two multifractals?

In the context of chaotic dynamical systems with exponential divergence of nearby trajectories in phase space, hyperchaos is defined as a state where there is divergence or stretching in at least two directions during the evolution of the system. Hence the detection and characterization of a hyperchaotic attractor is usually done using the spectrum of Lyapunov Exponents (LEs) that measure this rate of divergence along each direction. Though hyperchaos arise in different dynamical situations and find several practical applications, a proper understanding of the geometric structure of a hyperchaotic attractor still remains an unsolved problem. In this paper, we present strong numerical evidence to suggest that the geometric structure of a hyperchaotic attractor can be characterized using a multifractal spectrum with two superimposed components. In other words, apart from developing an extra positive LE, there is also a structural change as a chaotic attractor makes a transition to the hyperchaotic phase and the attractor changes from a simple multifractal to a dual multifractal, equivalent to two inter-mingled multifractals. We argue that a cross-over behavior in the scaling region for computing the correlation dimension is a manifestation of such a structure. In order to support this claim, we present an illustrative example of a synthetically generated set of points in the unit interval (a Cantor set with a variable iteration scheme) displaying dual multifractal spectrum. Our results are also used to develop a general scheme to generate both hyperchaotic as well as high dimensional chaotic attractors by coupling two low dimensional chaotic attractors and tuning a time scale parameter.

Controlling Unstable Chaos: Stabilizing Chimera States by Feedback

We present a control scheme that is able to find and stabilize an unstable chaotic regime in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to classical delayed feedback control, the scheme is noninvasive, however only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effect. We demonstrate the control scheme for so-called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.

A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems

We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation.

Synchronization of spatiotemporal semiconductor lasers and its application in color image encryption

Optical chaos is a topic of current research characterized by high-dimensional nonlinearity which is attributed to the delay-induced dynamics, high bandwidth and easy modular implementation of optical feedback. In light of these facts, which add enough confusion and diffusion properties for secure communications, we explore the synchronization phenomena in spatiotemporal semiconductor laser systems. The novel system is used in a two-phase colored image encryption process. The high-dimensional chaotic attractor generated by the system produces a completely randomized chaotic time series, which is ideal in the secure encoding of messages. The scheme thus illustrated is a two-phase encryption method, which provides sufficiently high confusion and diffusion properties of chaotic cryptosystem employed with unique data sets of processed chaotic sequences. In this novel method of cryptography, the chaotic phase masks are represented as images using the chaotic sequences as the elements of the image. The scheme drastically permutes the positions of the picture elements. The next additional layer of security further alters the statistical information of the original image to a great extent along the three-color planes. The intermediate results during encryption demonstrate the infeasibility for an unauthorized user to decipher the cipher image. Exhaustive statistical tests conducted validate that the scheme is robust against noise and resistant to common attacks due to the double shield of encryption and the infinite dimensionality of the relevant system of partial differential equations.

Extensive chaotic dynamics in neural networks in the balanced state

The irregular activity of neurons in the cortex [1] is thought to arise from strong input fluctuations [2], which result from a balance between excitatory and inhibitory synaptic inputs [3]. This is called the balanced state of cortical networks. However, recent studies of the underlying dynamics of the balanced state have led to contrary results, strongly depending on the used single neuron models [3, 4]. Here, we study the network dynamics of sparsely coupled theta neurons in the balanced state. The theta neuron model has an active spike generating mechanism and is the standard form of type I neurons [5]. In a random matrix approximation of the Jacobian, we derive an expression for the mean Lyapunov exponent. By analyzing the full set of Lyapunov exponents and the maximal Lyapunov vector in a numerically exact way, we reveal extensive spatio-temporal deterministic chaos. The studied networks exhibit high-dimensional chaotic attractors, giving rise to many dynamical degrees of freedom to encode information. At the same time, the intrinsic entropy production is surprisingly high, limiting information processing to the immediate stimulus response. Purely inhibitorily coupled networks

High-dimensional chaotic attractors in a gyrotron with nonfixed field structure

The chaotic dynamics of a gyrotron with nonfixed field structure is numerically simulated and the Lyapunov exponents of chaotic attractors are calculated. The dimensions of chaotic attractors estimated using the Kaplan-Yorke formula prove to be anomalously high. This fact is related to the presence of a large number of high-Q eigenmodes in the gyrotron resonator operating in the vicinity of a critical frequency of the electrodynamic system.

Study on the communication method for chaotic encryption in remote monitoring systems

In chaotic cryptosystems, it is recognized that using (very) high dimensional chaotic attractors for encrypting a given message may improve the privacy of chaotic encoding. In this paper, we study a kind of hyperchaotic systems by using some classical methods. The results show that besides the high dimension, the sub-Nyquist sampling interval (SI) is also an important factor that can improve the security of the chaotic cryptosystems. We use the method of time series analysis to verify the results.

TRANSITION TO HIGH-DIMENSIONAL CHAOS THROUGH QUASIPERIODIC MOTION

In this contribution we report on a transition to high-dimensional chaos through three-frequency quasiperiodic behavior. The resulting chaotic attractor has a one positive and two null Lyapunov exponents. The transition occurs at the point at which two symmetry related three-dimensional tori merge in a crisis-like bifurcation. The route can be summarized as: 2D torus → 3D torus → high-dimensional chaotic attractor.

CHAOTIC RESONANCE — METHODS AND APPLICATIONS FOR ROBUST CLASSIFICATION OF NOISY AND VARIABLE PATTERNS

A fundamental tenet of the theory of deterministic chaos holds that infinitesimal variation in the initial conditions of a network that is operating in the basin of a low-dimensional chaotic attractor causes the various trajectories to diverge from each other quickly. This "sensitivity to initial conditions" might seem to hold promise for signal detection, owing to an implied capacity for distinguishing small differences in patterns. However, this sensitivity is incompatible with pattern classification, because it amplifies irrelevant differences in incomplete patterns belonging to the same class, and it renders the network easily corrupted by noise. Here a theory of stochastic chaos is developed, in which aperiodic outputs with 1/f2 spectra are formed by the interaction of globally connected nodes that are individually governed by point attractors under perturbation by continuous white noise. The interaction leads to a high-dimensional global chaotic attractor that governs the entire array of nodes. An example is our spatially distributed KIII network that is derived from studies of the olfactory system, and that is stabilized by additive noise modeled on biological noise sources. Systematic parameterization of the interaction strengths corresponding to synaptic gains among nodes representing excitatory and inhibitory neuron populations enables the formation of a robust high-dimensional global chaotic attractor. Reinforcement learning from examples of patterns to be classified using habituation and association creates lower dimensional local basins, which form a global attractor landscape with one basin for each class. Thereafter, presentation of incomplete examples of a test pattern leads to confinement of the KIII network in the basin corresponding to that pattern, which constitutes many-to-one generalization. The capture after learning is expressed by a stereotypical spatial pattern of amplitude modulation of a chaotic carrier wave. Sensitivity to initial conditions is no longer an issue. Scaling of the additive noise as a parameter optimizes the classification of data sets in a manner that is comparable to stochastic resonance. The local basins constitute dynamical memories that solve difficult problems in classifying data sets that are not linearly separable. New local basins can be added quickly from very few examples without loss of existing basins. The attractor landscape enables the KIII set to provide an interface between noisy, unconstrained environments and conventional pattern classifiers. Examples given here of its robust performance include fault detection in small machine parts and the classification of spatiotemporal EEG patterns from rabbits trained to discriminate visual stimuli.

Generating novel memories by integration of chaotic neural network modules

A principle of integrating neural network modules based on chaotic dynamics was studied on our two-moduled Nozawa model. Chaotic neural networks represent each embedded pattern as a low-dimensional periodic orbit, and the others are shown as high-dimensional chaotic attractors. This is equivalent to W. Freeman’s “I don’t know” and “I know” states. In particular, we noted that the combination of two-way inputs to each neural network module conflicted with embedded Hebbian correspondence. It was found that the interaction between the modules generated a novel “I know” state in addition to the embedded representation. Chaotic neural network modules can autonomously generate novel memories or functions by this interaction. The result suggests a functional integration in neural networks as it ought to be, e.g., feature binding and gestalt.

Extreme parametric uncertainty and instant chaos in coupled structural dynamics

Bifurcation to high-dimensional hyperchaos is observed in a driven coupled pendulum-flexible rod system. When the rod is in resonance with the pendulum, the system changes from a low dimensional periodic attractor to a high-dimensional chaotic attractor abruptly. It is shown that high-dimensional chaotic dynamics is hysteretic, and exhibits extreme sensitivity with respect to small parameter changes.

Long-term properties of time series generated by a perceptron with various transfer functions

We study the effect of various transfer functions on the properties of a time series generated by a continuous-valued feed-forward network in which the next input vector is determined from past output values. The parameter space for monotonic and non-monotonic transfer functions is analyzed in the unstable regions with the following main finding; non-monotonic functions can produce robust chaos whereas monotonic functions generate fragile chaos only. In the case of non-monotonic functions, the number of positive Lyapunov exponents increases as a function of one of the free parameters in the model, hence, high dimensional chaotic attractors can be generated. We extend the analysis to a combination of monotonic and non-monotonic functions.

Instant chaos and hysteresis in coupled linear-nonlinear oscillators

Abstract A novel bifurcation to high-dimensional hyperchaos is observed in a driven coupled pendulum-flexible rod system. When the rod is in resonance with the pendulum, the system changes from a low-dimensional periodic attractor to a high-dimensional chaotic attractor abruptly. The bifurcation, which is hysteretic, is conjectured to stem from an unstable global invariant manifold of slow frequency motions.

Transverse dynamics of a phase-conjugate resonator II: fast nonlinear medium

We investigate the transverse dynamics of a high-diffraction phase-conjugate resonator in which the phase-conjugate mirror is a diffusive Kerr medium with instantaneous response time and operates in the Raman–Nath scattering regime. A period-doubling scenario from regular behavior to low-dimensional chaos is found and the route to high-dimensional chaotic-attractors is analyzed by means of the spectrum of the Lyapunov exponents. We also compute both the Lyapunov dimension and the correlation dimension for a specific attractor and find different values.

Study of a high-dimensional chaotic attractor

The nature of a very high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high-dimensional attractor in the phase space. We introduce two complementary bases from which the attractor is observed, one the Lyapunov basis composed of the Lyapunov vectors and the another the Fourier basis composed of the Fourier modes. We introduce the “exterior” subspaces on the basis of the Lyapunov vectors and observe the chaotic motion projected onto these exteriors. It is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace “goes out” of the attractor. The origin of such a phenomenon is attributed to more fundamental features of our attractor, which become manifest when the attractor is observed from the Lyapunov basis. A counterpart of the phenomenon can be observed also on the Fourier basis because there is a statistical one-to-one correspondence between the Lyapunov vectors and the Fourier modes. In particular, a statistical property of the high-pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor.