Chaos Prediction Research Articles

Chaos prediction in an MMIC frequency divider in millimetric band

A technique is proposed for the analysis of possible quasi-periodic routes to chaos in microwave circuits. This is based on the systematic application of a Nyquist stability analysis along the solution paths in self-oscillating mixer operation, as a function of a relevant parameter. The steady quasi-periodic solutions, with two fundamental frequencies, are determined from the harmonic balance method. The stability analysis allows the detection of possible asynchronous instabilities, leading to a three-fundamental quasi-periodic solution, from which the transition to chaotic behavior will occur. This technique has allowed the theoretical prediction of the onset of chaos that was experimentally observed in a monolithic microwave integrated circuit (MMIC) frequency divider by two in the millimetric band.

Chaos in the Portevin–Le Châtelier Effect

We report the verification of the prediction of chaos in the Portevin–Le Châtelier effect or the jerky flow by analyzing the stress signals obtained from samples of polycrystalline Al–Mg alloys subjected to a constant strain rate test. Particular care is taken to obtain reasonably long and accurate stress signals. The analysis of these signals is carried out by using several complementary methods such as calculation of correlation dimension, singular value decomposition and the spectrum of Lyapunov exponents. The analysis shows the existence of a finite correlation dimension and a positive Lyapunov exponent. Using the existence of a positive Lyapunov exponent and finite correlation dimension as a discriminator, we also carry out a surrogate analysis of the time series to ascertain that the signals are not from a power law stochastic process. The analysis provides an unambiguous support for the existence of chaos in Portevin–Le Châtelier effect thus verifying the prediction of the model. Further, from the analysis we find that the minimum number of variables required for a dynamical description of the jerky flow appears to be four or five consistent with the model.

Recognizing randomness in a time series

We study the effects of intrinsic noise on low-dimensional chaotic systems and nonlinear oscillating systems. For the former, we show that adding increasing amounts of intrinsic noise increasingly destroys the characteristic short-term predictability of chaos, while for the latter, we show that phase points near the deterministic limit cycle execute Brownian-like motions. Quantitative methods are also proposed to estimate the amount of noise contained in the signal.

Wave chaos and travel time effects of ocean mesoscale structure at global ranges

A range-dependent ray-trace model and the broadband UMPE model are used to model sound propagation at fixed bearing through a field of mesoscale baroclinic modes in order to study the effects of the mesoscale structure on travel time at long ranges. The ray model exhibits chaos at ranges beyond a few Mm as manifested by an exponentially increasing number of triplications of the wavefront. In addition, the ray model predicts a mesoscale travel time bias in the direction of later time (‘‘cold bias’’) of the last axial arrival that is about 50–150 ms/Mm. At center frequency 75 Hz and bandwidth 50 Hz, the full-wave UMPE model qualitatively confirms the ray-trace predictions of chaos, that is now called ‘‘wave chaos’’ because it represents apparently random and unstable behavior of finite frequency wavefields. Also, the UMPE model quantitatively confirms the above mesoscale bias of the last axial arrival. Theories based on the adiabatic approximation are used to explain the modeled mesoscale bias in magnitude and sign. The steeper, early raylike arrivals are relatively stable in the presence of mesoscale structure and should be useful for long-range tomography. [Work supported by ONR.]

Travel time effects of mesoscale structure on rays and waves at global ranges

A range‐dependent ray trace model and a broadband PE model are used to model sound propagation at fixed bearing through a field of mesoscale baroclinic modes in order to study the effects of mesoscale structure on travel time at long ranges. The ray model exhibits chaos at ranges beyond a few Mm as manifested by an exponentially increasing number of eigenrays and triplications, especially in the late‐time near‐axial arrivals. In addition, the ray model predicts a mesoscale travel time bias, in the direction of later time, of the last axial arrival amounting to about 100–200 ms/Mm. At center frequency 75 Hz and bandwidth 50 Hz, the full‐wave PE model qualitatively confirms the ray trace predictions of chaos, and shows that the later near‐axial arrivals are smeared out into a continuum of unresolvable multipaths characterized by saturated (Gaussian) statistics at ranges of a few Mm, and also quantitatively confirms the mesoscale bias of 100–200 ms/Mm of the last axial arrival. The steeper, early raylike arrivals are relatively stable in the presence of mesoscale structure and may be useful for long‐range tomography. [Work supported by ONR.]

Chaos in Jerky Flow: Theory and Experiment

Recently, the prediction of chaos in jerky flow based on a nonlinear dynamical model has been verified by demonstrating the existence of correlation dimension for the stress signals obtained on samples of AlCu alloys. However, an unambiguous support of chaos requires the existence of a positive Lyapunov exponent. Here we consider a recent method which uses state space reconstruction by embedding the signals in higher dimension and obtaining the signature of chaos using the property of divergence of near by orbits. The method provides a reasonable estimate of the delay time and the embedding dimension. We first illustrate the method by using the time series obtained from a dynamical model and then apply it to the experimental signals. The analysis shows that the experimental signals are truly chaotic. The minimum number of variables required for the dynamical description of jerky flow appears to be five, consistent with the estimate of the correlation dimension

Identification of Nonlinear Vibration System by a Neural Network. The Prediction of Chaos by Learning of Periodic Response.

This paper describes a convenient method of identification of a nonlinear vibration system by a neural network using time series data. A neural network using a suitable system of inputs and outputs is applicable for system identification of nonlinear vibration by learning periodic response. By learning only time serie data of periodic response in coexisting regions of chaos and periodic response, it is possible to predict both forced vibration chaos response and periodic response on predicting conditions near learning data by the neural network. The effectiveness of the method presented is demonstrated by numerical simulations for the Duffing system.

Observation of dynamical localization in atomic momentum transfer: A new testing ground for quantum chaos.

We report a direct observation of dynamical localization. This effect is a quantum suppression of diffusion in a system that is classically chaotic. Our experiment measures the momentum transferred from a modulated standing wave of a near-resonant laser to a sample of ultracold atoms and is a realization of a periodically driven rotor. The conceptual simplicity and experimental control over the time dependent Hamiltonian make this system an ideal testing ground for the predictions of quantum chaos.

Manifestation of quantum chaos in electronic band structures.

We use semiconductors as an example to show that quantum chaos manifests itself in the energy spectrum of crystals. We analyze the {\it ab initio} band structure of silicon and the tight-binding spectrum of the alloy $Al_xGa_{1-x}As$, and show that some of their statistical properties obey the universal predictions of quantum chaos derived from the theory of random matrices. Also, the Bloch momenta are interpreted as external, tunable, parameters, acting on the reduced (unit cell) Hamiltonian, in close analogy to Aharonov-Bohm fluxes threading a torus. They are used in the investigation of the parametric autocorrelator of crystal velocities. We find that our results are in good agreement with the universal curves recently proposed by Simons and coworkers.

Properties of cross-well chaos in an impacting system

In this paper we present results on chaotic motions in a periodically forced impacting system which is analogous to the version of Duffing’s equation with negative linear stiffness. Our focus is on the prediction and manipulation of the cross-well chaos in this system. First, we develop a general method for determining parameter conditions under which homoclinic tangles exist, which is a necessary condition for cross-well chaos to occur. We then show how one may manipulate higher harmonics of the excitation in order to affect the range of excitation amplitudes over which fractal basin boundaries between the two potential wells exist. We also experimentally investigate the threshold for cross-well chaos and compare the results with the theoretical results. Second, we consider the rate at which the system crosses from one potential well to the other during a chaotic motion and relate this to the rate of phase space flux in a Poincare map defined in terms of impact parameters. Results from simulations and experiments are compared with a simple theory based on phase space transport ideas, and a predictive scheme for estimating the rate of crossings under different parameter conditions is presented. The main conclusions of the paper are the following: (1) higher harmonics can be used with some effectiveness to extend the region of deterministic basin boundaries (in terms of the amplitude of excitation) but their effect on steady-state chaos is unreliable; (2) the rate at which the system executes cross-well excursions is related in a direct manner to the rate of phase space flux of the system as measured by the area of a turnstile lobe in the Poincare map. These results indicate some of the ways in which the chaotic properties of this system, and possibly others such as Duffing’s equation, are influenced by various system and input parameters. The main tools of analysis are a special version of Melnikov’s method, adapted for this piecewise-linear system, and ideas of phase space transport.

Analysis and prediction of chaos in rainfall and stream flow time series

The time delay embedding method of reconstructing the phase-space diagram of a time series is examined for uncovering possible chaotic behaviour and making non-linear predictions. The reliability of the various parameters that characterise chaotic time series are verified using artificially generated data from random, Autoregressive Moving Average (ARMA) and chaotic series with additive noise. Non-linear predictions are made using a local approximation method and applied to some daily rainfall and stream flow data in Hong Kong. There is convincing statistical evidence to believe that the stream flow and rainfall data series are better modelled by the time delay embedding approach than by the traditional linear ARMA approach.

Prediction of bifurcations and chaos for an asymmetric elastic oscillator

The nonlinear chaotic response of a harmonically forced elastic oscillator with quadratic and cubic nonlinearities is studied. The possibility of obtaining reliable prediction of bifurcations and chaos through combined use of stability analysis of low-order approximate solutions and accurate localized point-by-point and cell mapping computer simulations is examined. Some satisfactory results are obtained for the bifurcation predictive capability and the location of regions where chaos actually occurs.

Prediction of horseshoe chaos in BVP and DVP oscillators

An analytical threshold condition for the prediction of horseshoe chaos is obtained for the Bonhoeffer-van der Pol (BVP) oscillator, through the equivalent Duffing-van der Pol (DVP) oscillator, using the Melnikov method. Numerical and analytical predictions of the tangential intersection of stable and unstable orbits of saddle have been compared for both the DVP and BVP oscillators. The analytical threshold of the BVP oscillator, derived by extrapolating that of the DVP oscillator, is found to be in good agreement with the actual numerical analysis of the system.

A HARMONIC BALANCE APPROACH FOR CHAOS PREDICTION: CHUA’S CIRCUIT

The paper proposes a practical engineering approach for predicting chaotic dynamics in an important class of nonlinear systems. The aim of this approach is to provide a heuristic method of analysis which can give reasonably accurate answers but is far simpler to apply than other more rigorous methods based on nonlinear dynamics. Our approach is founded on the harmonic balance principle and uses standard describing function techniques well known to design engineers. Our method consists of a synergism of two independent techniques, each one constituting a possible mechanism for chaos. These two techniques are combined into a single algorithm which is highly efficient computationally, taking only a fraction of the time normally required by other more exact procedures. In order to present and illustrate our algorithm clearly, and in order to compare its predictions with readily available results obtained by rigorous methods, we have chosen Chua’s circuit as a vehicle to demonstrate the effectiveness of this approach. Chua’s circuit was chosen not only for the huge amount of results already published concerning the dynamics of this system, but also because it represents a real physical system easily built in the laboratory, and whose simple mathematical model has proven to be realistic and mathematically tractable. Although our algorithm does not guarantee its prediction is fully reliable, any more than the widely used describing function method does, its significance is based entirely on the empirical evidence that it yields qualitatively correct, though not exact, results for all of the chaotic phenomena that we have investigated in Chua’s circuit, as well as in many other chaotic systems. We hope therefore that our chaos prediction algorithm will find practical uses among engineers and scientists not familiar with more specialized mathematical approaches, in search of a simple and practical, although not rigorous, tool for analyzing systems with complex dynamics.

Chaos prediction in nonlinear feedback systems

The paper investigates the chaotic behaviour of nonlinear feedback systems. A heuristic model of this phenomenon is proposed and applied. Conditions for the existence and the location of chaotic motions are derived in terms of simple relations among the parameters of the system. Two examples show the application of the method and its approximation is discussed.

Dynamics of computational ecosystems.

Recently, Huberman and Hogg [in The Ecology of Computation, edited by B. A. Huberman (North-Holland, 1988), pp. 77--115] analyzed the dynamics of resource allocation in a model of computational ecosystems which incorporated many of the features endemic to large distributed processing systems, including distributed control, asynchrony, resource contention, and cooperation among agents and the concomitant problems of incomplete knowledge and delayed information. In this paper we supplement an analysis of several simple examples of computational ecosystems with computer simulations to gain insight into the effects of time delays, cooperation, multiple resources, inhomogeneity, etc. The simulations verify Huberman and Hogg's prediction of persistent oscillations and chaos, and confirm the Ceccatto-Huberman [Proc. Natl. Acad. Sci. U.S.A. 86, 3443 (1989)] prediction of extremely long-lived metastable states in computational ecosystems. Extending the analysis to inhomogeneous systems, we show that they can be more stable than homogeneous systems because agents with different computational needs settle into different strategic niches, and that overly clever local decision-making algorithms can induce chaotic behavior.

Prediction of chaos in Josephson junction by the Melnikov function technique

Prediction of chaos in Josephson junction by the Melnikov function technique

Chaotic behaviour of a pendulum with variable length

The Melnikov function for the prediction of Smale horseshoe chaos is applied to a driven damped pendulum with variable length. Depending on the parameters, it is shown that this dynamical system undertakes heteroclinic bifurcations which are the source of the unstable chaotic motion. The analytical results are illustrated by new numerical simulations. Furthermore, using the averaging theorem, the stability of the subharmonics is studied.

Prediction of chaos in a Josephson junction by the Melnikov-function technique.

The Melnikov function for prediction of Smale horseshoe chaos is applied to the rf-driven Josephson junction. Linear and quadratic damping resistors are considered. In the latter case the analytic solution including damping and dc bias is used to obtain an improved threshold curve for the onset of chaos. The prediction is compared to new computational solutions. The Melnikov technique provides a good, but slightly low, estimate of the chaos threshold.

Pattern Formation and Chaos in Synergetic Systems

We first briefly sketch a general approach how to reduce the equations of systems composed of many subsystems to equations for, in general, few order parameters at instability points. As special case generalized Ginzburg-Landau equations are obtained. Recent results based on these equations showing the pattern formation in the convection instability and flames are presented. Bifurcations from tori to other tori are treated and some general conclusions are drawn. Analogies between fluid dynamics and lasers which for the first time led to the prediction of laser light chaos are pointed out. Finally the suspension of a class of discrete one-dimensional maps is discussed and explicitly presented for a typical case.