Number Of Positive Lyapunov Exponents Research Articles

Structural stability and hyperbolicity violation in high-dimensional dynamical systems

This report investigates the dynamical stability conjectures of Palis and Smale and Pugh and Shub from the standpoint of numerical observation and lays the foundation for a stability conjecture. As the dimension of a dissipative dynamical system is increased, it is observed that the number of positive Lyapunov exponents increases monotonically, the Lyapunov exponents tend towards continuous change with respect to parameter variation, the number of observable periodic windows decreases (at least below numerical precision) and a subset of parameter space exists such that topological change is very common with small parameter perturbation. However, this seemingly inevitable topological variation is never catastrophic (the dynamic type is preserved) if the dimension of the system is high enough.

Lyapunov Spectrum in a Four-Coupled Rössler Oscillator

A four-coupled Rossler oscillator system is investigated. Various phases shown such as asynchronized, partially synchronized, intermittency and completely synchronized states appear as the coupling strength varies. Lyapunov spectrum is calculated for this system. The number of positive Lyapunov exponents decreases as synchronization becomes to be strong. Only one positive exponent exists in the completely synchronized state. The phase transitions occur when the value of second or third exponent changes from zero to negative. Time variation of Lyapunov exponents that take the values near zero plays an important role for intermittency states.

Enhancing chaoticity of spatiotemporal chaos

In some practical situations strong chaos is needed. This introduces the task of chaos control with enhancing chaoticity rather than suppressing chaoticity. In this paper a simple method of linear amplifications incorporating modulo operations is suggested to make spatiotemporal systems, which may be originally chaotic or nonchaotic, strongly chaotic. Specifically, this control can eliminate periodic windows, increase the values and the number of positive Lyapunov exponents, make the probability distributions of the output chaotic sequences more homogeneous, and reduce the correlations of chaotic outputs for different times and different space units. The applicability of the method to practical tasks, in particular to random number generators and secure communications, is briefly discussed.

TURBULENCE CONTROL AND SYNCHRONIZATION AND CONTROLLABLE PATTERN FORMATION

Pattern and turbulence control and synchronization are investigated by considering the model of the complex Ginzburg–Landau equation. It is shown that in fully-developed turbulence region, when a large number of positive Lyapunov exponents exists, in the presence of a large gradient force, we can drive the system to various desired spatiotemporal patterns, and control and synchronize violent turbulence by applying convenient boundary control technique.

Estimation of Lyapunov spectra from space–time data

A method to estimate Lyapunov spectra from spatio-temporal data is presented, which is well-suited to be applied to experimental situations. It allows to characterize the high-dimensional chaotic states, with possibly a large number of positive Lyapunov exponents, observed in spatio-temporal chaos. The method is applied to data from a coupled map lattice.

The control of high-dimensional chaos in time-delay systems to an arbitrary goal dynamics.

We present the control of high-dimensional chaos, with possibly a large number of positive Lyapunov exponents, of unknown time-delay systems to an arbitrary goal dynamics. We give an existence-and-uniqueness theorem for the control force. In the case of an unknown system, a formula to compute a model-based control force is derived. We give an example by demonstrating the control of the Mackey-Glass system toward a fixed point and a Rossler dynamics. (c) 1999 American Institute of Physics.

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.

Characterization of the spatial complex behavior and transition to chaos in flow systems

We introduce a “spatial” Lyapunov exponent to characterize the complex behavior of non-chaotic but convectively unstable flow sytems. This complexity is of spatial type and is due to sensitivity to the boundary conditions. We show that there exists a relation between the spatial-complexity index we define and the comoving Lyapunov exponents. In such systems the transition to chaos, i.e., the occurrence of a positive Lyapunov exponent, can manifest itself in two different ways. In the first case (from neither chaotic nor spatially complex behavior to chaos) one observes the typical scenario; i.e., as the system size grows up the spectrum of the Lyapunov exponents gives rise to a density. In the second case (when the chaos develops from a convectively unstable situation) one observes only a finite number of positive Lyapunov exponents.

Controlling Turbulence in the Complex Ginzburg-Landau Equation

Controlling turbulence in the complex Ginzburg-Landau equation (CGLE) is investigated. The CGLE is generalized to include gradient force. Local injections (pinnings) are applied for turbulence control. It is found that local injections are effective in eliminating turbulence. In particular, for large gradient force, it is possible to suppress fully developed turbulence by adding a few injections much less in number than the number of positive Lyapunov exponents of the system. The high efficiency of controlling is heuristically explained, based on the spatial correlation length and space-time-variable transformation.

Synchronization of chaotic nonlinear oscillators: study of two coupled CSTRs

The synchronization of two coupled chaotic CSTRs, in which isothermal cubic autocatalytic reactions taking place, is investigated. It has been shown that the synchronization is possible through a hyperchaos–chaos transition, when the coupled system evolves on a 3D sub-manifold embedded in 6D phase space. The synchronization of chaotic nonlinear oscillators can be achieved when the number of positive Lyapunov exponents of the coupled system equals to the number of positive Lyapunov exponents of a single system. The unsynchronized periodic oscillations show an interesting symmetric phase plane (mirror image) with respect to the diagonal.

Synchronization of time-delay systems

We present the linear-stability analysis of synchronised states in coupled time-delay systems. There exists a synchronisation threshold, for which we derive upper bounds, which does not depend on the delay time. We prove that at least for scalar time-delay systems synchronisation is achieved by transmitting a single scalar signal, even if the synchronised solution is given by a high-dimensional chaotic state with a large number of positive Lyapunov-exponents. The analytical results are compared with numerical simulations of two coupled Mackey-Glass equations.

Transmission of Signals via Synchronization of Chaotic Time-Delay Systems

Secure communication via chaotic synchronization is demonstrated using dynamical systems governed by delay deferential equations. Strange attractors of such systems can have an arbitrarily large number of positive Lyapunov exponents giving rise to very complex time signals. This features can provide high security of masked messages.

Modelling chaos and hyperchaos with 3-D maps

Two generalizations on R 3 of the Hénon nonconservative map are defined. Although, in many respects, they are closely related to each other, one of them can produce strange attractors with only one positive Lyapunov exponent, while the other has strange attractors with two positive Lyapunov exponents, thus showing a “hyperchaotic” behaviour. The results of numerical tests to be discussed, show, that apart from obvious differences in the number of positive Lyapunov exponents and the shape of the chaotic atractor, the hyperchaos may not be much different from regular chaos.

Synchronization of spatiotemporal chaos and its applications

Synchronization of spatiotemporal chaos is investigated, based on the models of one-way coupled map lattice systems. It is found that under certain conditions spatiotemporal hyperchaos with a huge number of positive Lyapunov exponents can be synchronized by simply driving a single site, and the mechanism for this efficiency is analyzed. This approach of spatiotemporal chaos synchronization is applied for multichannel secure communication and secure information storage.

Quasiattractors and homoclinic tangencies

Recent results describing nontrivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction, with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors.

Synchronization of chaos and hyperchaos using linear and nonlinear feedback functions

Using the method of variable feedback, synchronization of chaotic and hyperchaotic systems is presented. The robustness of the method based on the flexibility of choices of feedback functions is exemplified. Linear and nonlinear feedback functions and their linear superpositions are used for synchronization. Calculations with model systems indicate that functions constructed by linear superposition of feedback functions that independently synchronize a system are more efficient in achieving synchronization than the functions from which they are made. We have not noticed any difference in the technique of synchronization based on the number of positive Lyapunov exponents.

Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits.

Recent results describing non-trivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather non-trivial features which are in a sharp distinction with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors. (c) 1996 American Institute of Physics.

Design of hyperchaotic flows.

We present a design strategy in terms of ordinary differential equations which creates chaotic attractors with an increasing number of positive Lyapunov exponents as the (finite) dimension of the system is increased. First, we introduce the most simple abstract equation containing only one nonlinearity. Second, we suggest a piecewise linear version of the abstract equation. Third, we propose a set of chemical reactions and demonstrate that the corresponding rate equations produce hyperchaotic behavior equivalent to the abstract system.

Towards a statistical mechanics of spatiotemporal chaos.

Coupled H\'enon maps are introduced to model in a more appropriate way chaos in extended systems. An effective technique allows the extraction of spatiotemporal periodic orbits, which are then used to approximate the invariant measure. A further implementation of the \ensuremath{\zeta}-function formalism reveals the extensive character of entropies and dimensions, and allows the computation of the associated multifractal spectra. Finally, the analysis of short chains indicates the existence of distinct phases in the invariant measure, characterized by a different number of positive Lyapunov exponents.

Calculated Attractor Dimensions for Low-Order Spectral Models

Advancing knowledge about the phase space topologies of nonlinear hydrodynamic or dynamical systems has raised the question of whether the structure of the attractors in which the solutions are eventually confined can be characterized rigorously and economically. It is shown by applying the Lyapunov exponents, Lyapunov dimension, and correlation dimension to several low-order truncated spectral models that these quantities give useful information about the phase space structure and predictability characteristics of such attractors. The Lyapunov exponents measure the average exponential rate of convergence or divergence of nearby solution trajectories in an appropriate phase space. The Lyapunov dimension dL incorporates the dynamical information of the Lyapunov exponents to give an estimate of the dimension of the system attractor, while the correlation dimension v is a more geometrically motivated measure that is simple to compute and related to more classical dimensions. The Lyapunov exponents detect bifurcations between solution regimes and also subtle predictability differences between attractors. As measures of chaotic attractor dimension, v>dL in all cases, and the ratio v/dL is smallest at values of the forcing just above the transition to chaos. Changes in the Lyapunov dimension are concentrated in a small range of forcing values, while the correlation dimension varies more uniformly. The value of dL is tied closely to the number of positive Lyapunov exponents, while v is more sensitive to the magnitude of the chaotic component of the system. Variations in these measures for a hierarchy of convection models support the idea that the appearance of strong chaos in two-dimensional models is truncation-related, and can be delayed to arbitrarily large forcing if enough modes are included.