Narrow Results

Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

Chua Oscillator exhibits a wide variety of nonlinear behavior and has become a paradigm for theoretical and experimental investigations of chaotic systems. An initial exploration of the parameter space for the circuit shows that the system and its generalizations generates a broad range of very different strange attractors. In the work described in this paper, we constructed "a gallery" of these attractors, including patterns that have never previously been observed. We identified the regions of parameter space occupied by each attractor and the initial conditions leading to production of the attractor. System behavior was characterized using time series, FFT graphs and in some cases Lyapunov exponents. In this way we created a complex picture of chaos, which we divided into six parts. The first, we publish here. The rest of our work will be published in subsequent issues of this journal. In this first paper, we describe how to build Chua Oscillator and some of its generalizations, as proposed in the recent literature. We introduce the main features that characterize the chaotic behavior of each of these systems. Finally, we offer hints on the mechanisms underlying synchronization between pairs of coupled Chua Oscillators. The investigation of chaos allows for the emergence of a complex picture, which could improve our knowledge of these challenging phenomena in contemporary science.

Chua's circuit is a physical system which can be used to investigate chaotic processes. One of its identifying features is the ability to produce a huge variety of strange attractors, each with its own characteristic form, size and model. These characteristics extend to a range of different systems derived from the original circuit.

In the first paper A Gallery of Chua's Attractors. Part I, we presented physical circuits and some generalizations based on Chua's oscillator, together with techniques for building the circuit and a summary description of its chaotic behavior.

In this second part of our work, we present an overview of forms which can only be produced by the physical circuit, using novel techniques of scientific visualization to explore, discover, analyze and validate our large collection of data. Starting with cases already known in the literature, we show that the circuit can produce an infinite set of three-dimensional patterns. A small sample is included in our paper. More specifically, we present 195 strange attractors generated by the circuit. For each attractor we provide three-dimensional images, time series and FFTs. Finally, we provide Lyapunov exponents for a subset of "base attractors".

The visualization of patterns related to chaos is a challenge for those who are part of today's dynamical systems community, especially when we consider the aim of providing users with the ability to visually analyze and explore large, complex datasets related to chaos. Thus visualization could be considered a useful element in the discovery of unexpected relationships and dependencies that may exist inside the domain of chaos, both in the phase and the parameter spaces. In the second part of "A Gallery of Chua attractors", we presented an overview of forms which can only be produced by the physical circuit. In Part III, we illustrate the variety and beauty of the strange attractors produced by the dimensionless version of the system. As in our earlier work, we have used ad hoc methods, such as bifurcation maps and software tools, allowing rapid exploration of parameter space. Applying these techniques, we show how it is possible, starting from attractors described in the literature, to find new families of patterns, with a special focus on the cognitive side of information seeking and on qualitative processes of change in chaos, thus demonstrating that traditional categories of chaos exploration need to be renewed.

After a brief introduction to dimensionless equations for Chua's oscillator, we show 150 attractors, which we represent using three-dimensional images, time series and FFT diagrams. For the most important patterns, we also report Lyapunov exponents. To show the position of dimensionless attractors in parameter space, we use parallel coordinate techniques that facilitate the visualization of high dimensional spaces. We use Principal Components Analysis (PCA) and Mahalanobis Distance to provide additional tools for the exploration and visualization of the structure of the parameter space.

The literature on Chua Oscillator includes more than a thousand papers, describing an astonishing variety of chaotic behavior. Gallery of Chua attractors, Part IV presents a collection of 101, previously unknown attractors, generated by a generalization of Chua circuit with a smooth nonlinear function. The gallery is the result of an extensive exploration of the parameter space for Chua cubic system, in which we used PCA and Hausdorf Distances to guide us. During this exploration we sensed the beauty of the chaotic patterns and recorded this beauty for the nonlinear community.

The attractors we describe here represent only a small proportion of those we discovered during our exploration of phase space: much intensive research remains to be done. However, the very number of attractors we found suggests it might be possible not only to detect the morphogenetic processes which determine the points in phase space ("catastrophe points") where a family of attractors disappears and another one comes to life, but to identify more general "laws of morphogenesis" governing the behavior of these systems. In this paper, we outline five such rules.

In this article, we conclude our series of papers on the analysis and visualization of Chua attractors and their generalizations. We present a gallery of 144 n-scroll, 15 hyperchaotic and 37 synchronized systems. Along with time series and FFT we provide 3D visualizations; for some attractors we also supply Lyapunov coefficients and fractal dimensions.

The goal in constructing our Gallery has been to make the general public aware of the enormous variety of chaotic phenomena and to change the widespread impression that they are isolated rarities. The Gallery provides a valuable collection of images and technical data which can be used to analyze these phenomena and to reproduce them in future studies.

From a scientific point of view, we have tried to identify new methodological approaches to the study of chaos, opening nontraditional perspectives on the complexity of this domain. In our papers, we have discussed a broad range of topics, ranging from techniques for visualizing Chua attractors to computational methods allowing us to make a statistical classification of attractors' positions in phase space and to describe the evolutionary processes through which their shapes change over time. We see these processes as analogous to population dynamics in artificial environments. Within these environments, we use experimental methods to identify the models which guide morphogenetic change and which organize genetic landscapes in parameter space.

This paper is organized as follows. First, we provide formal descriptions of the attractors generated by n-scroll, hyperchaotic and synchronized systems. The next section describes a Gallery of Chua attractors, generated by gradually varying the parameters and analyzing the resulting bifurcation maps. We then describe software tools allowing us to perform statistical analyses on selected sets of attractors, to visualize them, to explore their organization in phase space, and to conduct experimental investigations of the morphogenetic processes through which a small set of base attractors can generate a broad range of different forms. In the last section, we describe the creation of a Virtual 3D Gallery displaying some of the attractors we have presented in our six papers. The attractors are organized by theme, as they might be in a museum. The environment allows users to explore the attractors, interact with shapes, listen to music and sounds generated by the attractors, change their spatial organization, and create new shapes. To complete the paper — and the series — we propose a number of general conclusions.

We report experimental observation of cycle-cycle bursting near homoclinic bifurcation in a self-oscillating Chua oscillator when it is diffusively coupled to another excitable Chua oscillator. The excitable oscillator induces an asymmetry in the self-oscillating Chua oscillator via coupling. The coupling strength controls the strength of the asymmetry that plays a key role in transition to bursting near homoclinic bifurcation.

A unidirectional coupling scheme is investigated in double scroll type chaotic oscillators that reveal interesting multiscroll dynamics. Instead of using self-oscillatory systems, in this scheme, double scroll chaos from one oscillator is forced into another similar oscillator in a resting state. This coupling scheme is explored in the Chua oscillator, a modified Chua oscillator and the Lorenz oscillator. We have modified the Chua oscillator by simply changing its piecewise linear function slightly, thereby deriving a new 3-scroll attractor. We have observed 4-scroll, 6-scroll attractors in the driven Chua oscillator and the modified Chua oscillator respectively in an intermittency regime of weaker coupling. We have extended the coupling scheme to the Lorenz system when even more interesting multiscroll dynamics (3-, 4-, 5-, 6-scroll) is observed with decreasing coupling strength. It appears as if a hidden multiscroll structure unfolds with weakening coupling interactions. One after another, additional scrolls appear in the driven Lorenz system when the coupling strength is gradually decreased in the weaker coupling regime. The origin of such multiscroll dynamics is explained using eigenvalue analysis and a bifurcation diagram. A schematic diagram of the multiscroll trajectories is presented to further elucidate the evolution of the scrolls. Experimental evidence is also presented using the Chua circuit and an electronic analog of the Lorenz system.

Starting from previous analytical results assuring the existence of a saddle-node bifurcation curve of periodic orbits for continuous piecewise linear systems, numerical continuation is done to get some primary bifurcation curves for the piecewise linear Chua's oscillator in certain dimensionless parameter plane. The primary period doubling, homoclinic and saddle-node of periodic orbits' bifurcation curves are computed. A Belyakov point is detected in organizing the connection of these curves. In the parametric region between period-doubling, focus-center-limit cycle and homoclinic bifurcation curves, chaotic attractors coexist with stable nontrivial equilibria. The primary saddle-node bifurcation curve plays a leading role in this coexistence phenomenon.

The study lies in the field of physical reservoir computing and aims to develop a classifier using Fisher Iris dataset for benchmark tasks. Single Chua chaotic oscillator acts as a physical reservoir. The study was performed using computer simulation. The features of Iris flowers are represented as the consequence of short pulses at a constant level of a control parameter, which is fed to the oscillator, changing its dynamics. During the classification of flowers, the oscillator works without being reset, so each pulse on the input changes the phase trajectory and makes it unique for each Iris flower. Finally, the estimation of the symmetry of an attractor makes it possible to connect each species of Iris with the properties of the attractor.

The resulting architecture of the classifier includes a single-node externally-driven Chua oscillator with time-delayed input. The classifier shows two mistakes in classifying the dataset with 75 samples working in chaotic mode.