International Journal of Bifurcation and Chaos

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 the present survey, the contemporary state of the art of the problem of the justification of nonstationary linearizations is presented.

Many useful and well-known image processing templates for cellular neural networks (CNN's) can be derived from neural field models, thereby providing a neural basis for the CNN paradigm. The potential ability of multitasking image processing is investigated by using these templates. Many visual illusions are simulated via CNN image processing. The ability of the CNN to mimic such high-level brain functions suggests possible applications of the CNN in cognitive engineering. Furthermore, two kinds of painting-like image processings, namely, texture generation and illustration style transformation are investigated.

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L₁ between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L₁ and the other collinear libration points L₂, L₃ is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 < δ < 1) and non-Shil'nikov assumptions (δ ≥ 1). As it is well known for the case of differentiable systems, under Shil'nikov assumptions there exist infinitely many periodic orbits accumulating to the homoclinic loop. We also prove that this behavior persists at δ = 1. Moreover, for δ > 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit.

This paper is principally concerned with the effect of splitting and permuting, or shifting, the branches of an onto piecewise linear map, with particular regard to the effect on the autocorrelation of the associated chaotic sequence. This is shown to be a chaotic function of the shifting parameter of the map, and its sensitivity with respect to minute changes in this parameter is termed autocorrelation chaos. This paper presents both analytical and computational studies of the phenomenon.

In a broad sense, model reduction means producing a low-dimensional dynamical system that replicates either approximately, or more strictly, exactly and topologically, the output of a dynamical system. Model reduction has an important role in the study of dynamical systems and also with engineering problems. In many cases, there exists a good low-dimensional model for even very high-dimensional systems, even infinite dimensional systems in the case of a PDE with a low-dimensional attractor. The theory of global attractors approaches these issues analytically, and focuses on finding (depending on the question at hand), a slow-manifold, inertial manifold, or center manifold, on which a restricted dynamical system represents the interesting behavior of the dynamical system; the main issue depends on defining a stable invariant manifold in which the dynamical system is invariant. These approaches are analytical in nature, however, and are therefore not always appropriate for dynamical systems known only empirically through a dataset. Empirically, the collection of tools available are much more restricted, and are essentially linear in nature. Usually variants of Galerkin's method, project the dynamical system onto a function linear subspace spanned by modes of some chosen spanning set. Even the popular Karhunen–Loeve decomposition, or POD, method is exactly such a method. As such, it is forced to either make severe errors in the case that the invariant space is intrinsically a highly nonlinear manifold, or bypass low-dimensionality by retaining many modes in order to capture the manifold. In this work, we present a method of modeling a low-dimensional nonlinear manifold known only through the dataset. The manifold is modeled as a discrete graph structure. Intrinsic manifold coordinates will be found specifically through the ISOMAP algorithm recently developed in the Machine Learning community originally for purposes of image recognition.

We describe the fine structure of the global attractor of the Cahn–Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equilibrium solutions due to [Maier-Paape & Miller, 2002] with algebraic Conley index techniques. Using the information on the set of equilibria as assumption, we build Morse decompositions and connection matrices. The latter imply existence of heteroclinic connections between the equilibria inside the attractor. While path-following of the parameter range of Cahn–Hilliard, we find more and more complicated dynamical behavior. One of our main results describes the fine structure of the attractor for mean mass zero with four stable cosine structured equilibria and eight other stable equilibria that have a quarter circle nodal line. Besides that, we also study the attractor in symmetry fixed point spaces where we, for example, find nonunique connection matrices and saddle–saddle connections of Morse sets.

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue–Morse sequence on ℕ. This is done by taking the same automaton-structure as in the one-dimensional case, but using binary number systems in ℤ^m instead of in ℕ. It is shown that the corresponding ±1-valued Thue–Morse sequences are either periodic or have a singular continuous spectrum, dependent on the binary number system. Specific results are given for dimensions up to six, with extensive illustrations for the one-, two- and three-dimensional case.

We present a computer-assisted analysis of the phase space features and bifurcations of a noninvertible, discrete-time system. Our focus is on the role played by noninvertibility in generating disconnected basins of attraction and the breakup of invariant circle solutions. Transitions between basin of attraction structures are identified and organized according to "levels of complexity," a term we define in this paper. In particular, we present an algorithm that provides a computational approximation to the boundary (in phase space) separating points with different preimage behavior. The interplay between this boundary and other phase space features is shown to be crucial in understanding global bifurcations and transitions in the structure of the basin of attraction.

This Letter presents an effective edge detection technique based on the cellular neural network paradigm. The approach exploits a rigorous model of the image contours and takes into account some electrical restrictions of existing hardware implementations. The method yields accurate results, better than the ones achievable by other cellular neural network-based techniques.

Oscillating chemical bulk reactions of the Belousov–Zhabotinsky (BZ) type usually show a well-defined periodical color change representing the underlying chemical reactions. Models including up to more than 50 coupled reactions have been developed to describe the BZ-reaction in detail, however, typically a much simpler set of equations is used (i.e. the Brüsselator, the FKN, or the FitzHugh–Nagumo model). There are various types of oscillating reactions with color changes, for example, from transparent to yellow (Ce catalyzed), or from blue to red (Ferroin catalyzed). In equilibrium systems and in models each color-change is usually a simple curve with one maximum, and the oscillation is stable in time.

We have investigated the detailed structure of the intrinsic optical signal (IOS) of a BZ-reaction in-depth andfound, in a nonstationary ferroin catalyzed BZ-reaction, a much more complex optical behavior.

Amazingly, this IOS with its detailed time dependence very much resembles electrical and optical signals from important neuronal processes. These are especially the sodium current during action potentials (AP) and the retinal spreading depression (SD). The time-scale of all these processes can be different by decades. This principal similarity, however, points to a deeper common theoretical basis of all these processes as assumed up to now, and makes the presented type of BZ-reaction an ideal model for a variety of neurophysiological signals, although the mechanisms are strictly different.

This Letter presents a general approach to projective synchronization that features a linear observer with an ability to arbitrarily scale a drive system attractor. The technique can be applied to wide classes of chaotic and hyperchaotic systems, namely time-delay systems described by functional differential equations (FDEs), continuous-time systems described by ordinary differential equations (ODEs) and discrete-time systems described by difference equations (DEs). Theoretical and simulation results demonstrate that a linear observer can duplicate chaotic system states in any desired scale using only a scalar synchronizing signal. The proposed approach is readily implemented in hardware.

This paper deals with the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator. The amplitude of the oscillatory states both in the autonomous and nonautonomous case are derived. The interaction between the amplitudes of the external excitation and the limit-cycles are also analyzed. The domain of the admissible values on the amplitude for the external excitation is found. The effects of the control parameter on the behavior of a driven multi-limit-cycle Van der Pol model are analyzed and it appears that with the appropriate selection of the coupling parameter, the quenching of chaotic vibrations takes place.

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

In this paper, we study Hopf bifurcation of a second-order nonlinear differential equation with time delay by using the Lyapunov–Schmidt reduction. The approximate analytical expressions of the periodic solutions bifurcated from the trivial solution are given. We also discuss the stability of these periodic solutions. The numerical simulations line up with the theoretical results.

Stability analysis of spatially discretized approximations to invariant circle solutions to a noninvertible map is shown to produce seemingly spurious eigenvalues, found to exist for all truncation numbers. Numerical analysis of the spectral Galerkin projection procedure used to compute the solutions and the eigenvalues of the linearization shows that the set of discrete eigenvalues converges in terms of its mean value. Further analysis shows that the spurious eigenvalues — those that are farthest from the mean value — correspond to eigenfunctions which produce the largest residual in the eigenvalue problem used to compute the eigenvalues and eigenfunctions.