Double Scroll Research Articles

Discussion

The dynamics of wave front in two dimensional excitable media is described by the derivative Burgers’ equation. In this paper, we will carry out Lie symmetry analysis to the equation for constructing particular solutions associated with chemical patterns. The form of the variable coefficient in the reduced equation by using symmetries classifies the invariant solutions into three cases and the solutions include arc, circle, knee, spiral and double scroll patterns.

Performance analysis of locally recurrent neural networks

Many practical applications of neural networks require the identification of non‐linear deterministic systems or chaotic systems. In these cases the use of a network architecture known as locally recurrent neural network (LRNN) is often preferable in place of standard feedforward multi‐layer perceptron (MLP) networks, or of globally recurrent neural network. In this paper locally recurrent networks are used to simulate the behaviour of the Chua’s circuit that can be considered a paradigm for studying chaos. It is shown that such networks are able to identify the underlying link among the state variables of the Chua’s circuit. Moreover, they are able to behave like an autonomous Chua’s double scroll, showing a chaotic behaviour of the state variables obtainable through a suitable circuit elements choice.

Dynamics of the two-frequency torus breakdown in the driven double scroll circuit

In this work we numerically identify three scenarios for the two-frequency torus breakdown to chaos, using the driven double scroll circuit, with varying driving parameters. Two of these scenarios follow the Curry-Yorke route to chaos. For one scenario, we identify the transition to chaos through the onset of a heteroclinic tangle and its heteroclinic points. In the other scenario, chaos appears via period-doubling bifurcations. The third scenario is through the type-II intermittency for which a quasiperiodic torus grows in size, breaks by touching external saddle points, and forms a heteroclinic saddle connection. These dynamic scenarios have distinct structure evolutions: for the Curry-Yorke route, chaos appears softly and alternates with phase locking, while, for the type-II intermittency, chaos appears abruptly and is preserved for a large range of the varying driving parameter.

Easy-to-implement method to target nonlinear systems.

In this work we present a method to rapidly direct a chaotic system, to an aimed state or target, through a sequence of control perturbations, with few different amplitudes chosen according to the allowed control-parameter changes. We applied this procedure to the one-dimensional Logistic map, to the two-dimensional Henon map, and to the Double Scroll circuit described by a three-dimensional system of differential equations. Furthermore, for the Logistic map, we show numerically that the resulting trajectory (from the starting point to the target) goes along a stable manifold of the target. Moreover, using the Henon map, we create and stabilize unstable periodic orbits, and also verify the procedure robustness in the presence of noise. We apply our method to the Double Scroll circuit, without using any low-dimensional mapping to represent its dynamics, an improvement with respect to previous targeting methods only applied for experimental systems that are mapping-modeled. (c) 1998 American Institute of Physics.

Phase-Locking and Bifurcations of the Sinusoidally-Driven Double Scroll Circuit

The dynamic alterations of an electronic circuit in a chaotic regime, described by the Double Scroll attractor, subjected to sinusoidal perturbation are numerically investigated. Parameter diagrams of the circuit phase-locking oscillations in terms of the driving amplitude and frequency are computed. These diagrams have highly interleaved and complex structures, part of them Cantor-like fractals. However, a Cantor-like fractal structure is also observed. In addition, the power spectrum analysis is used to find and characterize three ways of phase-locking the Double Scroll circuit, and to determine how this process depends on the driving parameters. Furthermore, the dynamics of bifurcation phenomena, as chaotic attractor entrainment, Arnold's tongues, coexistence of attractors, and hysteresis are identified in the parameter space.

DYNAMICAL BEHAVIOR OF ACOUSTICALLY COUPLED CHAOS OSCILLATORS

We have found, for the first time, that Acoustically Coupling Chaos Oscillator (ACCO) also exhibits a chaotic attractor called the “double scroll Chua’s attractor” by using the modified Chua’s circuit. When each ACCO is in an oscillating periodic state, the interaction of sound waves with acoustic coupling of two ACCOs can also result in the appearance of chaotic sound waves.

Double scroll in a simple ‘2D’ chaoticoscillator

A very simple RC oscillator exhibiting the double-scroll chaotic attractor is described. The circuit contains two opamps, a Wien bridge, and two diodes used as nonlinear devices. The typical waveforms and the phase portraits of the attractor are presented to illustrate the double scroll in a quasi-two-dimensional oscillator.

Control of a recurrent neural network emulator for the double scroll

A recurrent neural network, that behaves as the double scroll, is stabilized and controlled by means of linear dynamic feedback. The controller design is based upon stability criteria for neural control systems and involves the solution of a nonlinear optimization problem.

Chaotic systems with a null conditional Lyapunov exponent under nonlinear driving.

Control of a chaotic system by homogeneous nonlinear driving, when a conditional Lyapunov exponent is zero, may give rise to special and interesting synchronizationlike behaviors in which the response evolves in perfect correlation with the drive. Among them, there are the amplification of the drive attractor and the shift of it to a different region of phase space. In this paper, these synchronizationlike behaviors are discussed, and demonstrated by computer simulation of the Lorentz model [E. N. Lorenz, J. Atmos. Sci. 20 130 (1963)] and the double scroll [T. Matsumoto, L. O. Chua, and M. Komuro, IEEE Trans. CAS CAS-32, 798 (1985)]. \textcopyright{} 1996 The American Physical Society.

RETRIEVING DYNAMICAL INVARIANTS FROM CHAOTIC DATA USING NARMAX MODELS

This paper is concerned with the estimation of dynamical invariants from relatively short and possibly noisy sets of chaotic data. In order to overcome the difficulties associated with the size and quality of the data records, a two-step procedure is investigated. Firstly NARMAX models are fitted to the data. Secondly, such models are used to generate longer and cleaner time sequences from which dynamical invariants such as Lyapunov exponents, correlation dimension, the geometry of the attractors, Poincaré maps and bifurcation diagrams can be estimated with relative ease. An additional advantage of this procedure is that because the models are global and have a simple structure, such models are amenable for analysis. It is shown that the location and stability of the fixed points of the original systems can be analytically recovered from the identified models. A number of examples are included which use the logistic and Hénon maps, Duffing and modified van der Pol oscillators, the Mackey-Glass delay system, Chua’s circuit, the Lorenz and Rössler attractors. The identified models of these systems are provided including discrete multivariable models for Chua’s double scroll, Lorenz and Rössler attractors which are used to reconstruct the trajectories in a three-dimensional state space.

Learning a simple recurrent neural state space model to behave like Chua's double scroll

The authors present a simple discrete time autonomous neural state space model (recurrent network) that behaves like Chua's double scroll. The model is identified using Narendra's dynamic back propagation procedure. Learning is done in packets of increasing time horizon. >

Chua's circuit can be generated by CNN cells

A new realization for the unfolded Chua's circuit is introduced. It has been derived from the connection of three simple generalized cellular neural network (CNN) cells. The main theoretical implication of this result is that the CNN cell represents the primitive for realizing high complex dynamics. The circuit implementation of the introduced system and experimental results referring to the double scroll attractor are reported. >

EXPERIMENTAL VERIFICATION OF PYRAGAS’S CHAOS CONTROL METHOD APPLIED TO CHUA’S CIRCUIT

We have built an experimental setup to apply Pyragas’s [1992, 1993] control method in order to stabilize unstable periodic orbits (UPO) in Chua’s circuit. We have been able to control low period UPO embedded in the double scroll attractor. However, experimental results show that the control method is useful under some restrictions we will discuss.

Chaos-Hyperchaos Intermittency

In this paper we describe chaos-hyperchaos intermittency in two unidirectionally coupled Chua's (double scroll) circuits. This intermittent behaviour can be considered as a generalization of classical intermittency to higher-dimensional systems.

FROM THE CHUA CIRCUIT TO THE GENERALIZED CHUA MAP

We analytically derive a one-dimensional map from an ODE which produces a double scroll very similar to the Chua double scroll. Our analysis leads us to suggest a generalization of the Chua circuit to an n-dimensional system of ODEs that we will call the generalized Chua equations. The third order system of ODEs in this class contains the Chua equations as a special case. Parallel to the generalized Chua equations we define the generalized Chua maps. An important feature of these equations and maps is that the source of their nonlinearity is a sigmoid function, and functions very similar in their properties to the sigmoid function. We show that this class of equations contains examples of maps that reproduce the Lorenz and Rössler dynamics. We suggest that a general theory of these equations and maps, and their relationship to one-dimensional maps, is possible. A benefit of our analysis shows that the dynamics of the maps of Rössler, Chua, and Lorenz maps can be traced to a common set of building blocks, and we conclude that the Chua map is the simplest of the three maps and therefore understanding the complexity in the Chua map provides a foundation for understanding chaos in a large class of n-dimensional equations that includes the maps of Rössler and Lorenz.

BISPECTRAL ANALYSIS OF CHUA'S CIRCUIT

Bispectral analysis (which isolates quadratic nonlinear interactions among triads of Fourier components) is used to investigate bifurcations in Chua's circuit. For period-doubled limit cycles, the dominant interactions of the circuit are quadratically nonlinear, and bicoherence spectra isolate the phase coupling between increasing numbers of triads of Fourier components as the nonlinearity of the system is increased. For circuit parameters that result in a chaotic, Rössler-type attractor, bicoherence spectra indicate that quadratic nonlinear interactions are important to the dynamics. For parameters that lead to the double scroll chaotic attractor the bispectrum is zero, suggesting that nonlinear interactions of order higher than quadratic dominate the dynamics. Higher-than-second order spectra (e.g. trispectra) are required to isolate the individual nonlinear interactions for the double scroll.

Bifurcation and chaos in cellular neural networks

Bifurcation phenomena and chaotic behavior in cellular neural networks are investigated. In a two-cell autonomous system, Hopf-like bifurcation has been found, at which the flow around the origin, an equilibrium point of the system, changes from asymptotically stable to periodic. As the parameter grows further, by reaching another bifurcation value, the generated limit cycle disappears and the network becomes convergent again. Chaos is also presented in a three-cell autonomous system. It is shown that the chaotic attractor found here has properties similar to the famous double scroll attractor. >

CONTROL OF CHUA'S CIRCUIT

This paper considers the control of a polynomial variant of the original Chua's circuit. Both state space techniques and input-output techniques are presented. It is shown that standard control theory approaches can easily accommodate a chaotic system. Furthermore, it is shown that a harmonic balance approach could predict the period doubling phenomenon and onset of the double scroll chaos, as well as providing a control approach.

HIGHER-ORDER SPECTRAL ANALYSIS TO DETECT NONLINEAR INTERACTIONS IN MEASURED TIME SERIES AND AN APPLICATION TO CHUA’S CIRCUIT

Higher-order spectra have been used to investigate nonlinear interactions between the Fourier components of measured time series in a remarkably wide range of random processes. The basic techniques of detecting and isolating nonlinear phase coupling in observed data using higher-order spectral analysis are reviewed here. These techniques are then used to investigate nonlinear interactions in time series of voltages measured from a realization of Chua’s circuit. For period-doubled limit cycles, quadratic and cubic nonlinear interactions result in phase coupling and energy exchange between increasing numbers of triads and quartets of Fourier components as the nonlinearity of the system is increased. For circuit parameters that result in a chaotic, Rössler-type attractor, bicoherence and tricoherence spectra indicate that both quadratic and cubic nonlinear interactions are important to the dynamics. For parameters that lead to the double-scroll chaotic attractor the bispectrum is zero, but the tricoherences are high, consistent with the importance of higher-than-second order nonlinear interactions during chaos associated with the double scroll.

Higher order spectral analysis of Chua's circuit

Higher order spectral analysis is used to investigate nonlinearities in time series of voltages measured from a realization of Chua's circuit. For period-doubled limit cycles, quadratic and cubic nonlinear interactions result in phase coupling and energy exchange between increasing numbers of triads and quartets of Fourier components as the nonlinearity of the system is increased, For circuit parameters that result in a chaotic, Rossler-type attractor, bicoherence and tricoherence spectra indicate that both quadratic and cubic nonlinear interactions are important to the dynamics. When the circuit exhibits a double-scroll chaotic attractor the bispectrum is zero, but the tricoherences are high, consistent with the importance of higher-than-second order nonlinear interactions during chaos associated with the double scroll.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>