International Journal of Bifurcation and Chaos

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.

We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form , and thus check certain sufficient conditions for chaotic behavior of dynamical systems in a rigorous way.

Our proof of the underlying abstract theorem avoids referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem.

The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach.

Abstract results and computational methods are demonstrated by finding embedded copies of the standard horseshoe dynamics in iterates of the classical Hénon mapping.

In this tutorial, we present optimal Cellular Nonlinear Network (CNN) templates for implementing linearly-separable one-dimensional (1-D) Cellular Automata (CA). From the gallery of CNN templates presented in this paper, one can calculate any of the 256 1-D CA Rules studied by Wolfram using a CNN Universal Machine chip that is several orders of magnitude faster than conventional programming on a digital computer.

In this paper, we consider a delayed discrete neural network of two identical neurons with excitory interactions. After investigating the stability of the given system, we establish a new scheme, and use the scheme to analyze the possible bifurcations occurring in the model. The process from its stable equilibrium to its multiple periodic patterns is explored clearly. A clarification for the asymptotically synchronous/asynchronous regions of such a system with ℤ₂ symmetry is included.

We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell.

Bifurcations of homoclinic orbit connecting the strong stable and strong unstable directions are investigated for four-dimensional system. The existence, numbers, co-existence and incoexistence of 1-homoclinic orbit, 2ⁿ-homoclinic orbit, 1-periodic orbit and 2ⁿ-periodic orbit are obtained, and the bifurcation surfaces (including codimension-1 homoclinic bifurcation surfaces, double periodic orbit bifurcation surfaces, homoclinic-doubling bifurcation surfaces, period-doubling bifurcation surfaces and codimension-2 triple periodic orbit bifurcation surface, and homoclinic and double periodic orbit bifurcation surface) and the existence regions are also located.

The Bonhöffer–van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

The Shilnikov-type multipulse orbits and chaotic dynamics for a simply supported rectangular thin plate under combined parametric and external excitations are studied in this paper for the first time. The rectangular thin plate is subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously. The formulas of the rectangular thin plate are derived from the von Kármán equation and Galerkin's method. The method of multiple scales is used to find the averaged equation in the case of 1:2 internal resonance. Based on the averaged equation, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of purely imaginary eigenvalues using the Maple program. The dissipative version of the energy-phase method is utilized to analyze the multipulse global bifurcations and chaotic dynamics of a parametrically and externally excited rectangular thin plate. The global dynamical analysis indicates that there exist the multipulse jumping orbits in the perturbed phase space of the averaged equation for a parametrically and externally excited rectangular thin plate. These results show that the chaotic motions of the multipulse Shilnikov-type can occur for a parametrically and externally excited rectangular thin plate. Numerical simulation results are presented to verify the analytical predictions. It is also found from the results of numerical simulation that the Shilnikov-type multipulse orbits exist for a parametrically and externally excited thin plate.

This paper is devoted to the study of the properties of basins of attraction and the domains of feasible trajectories (discrete trajectories having an ecological sense) generated by a family of two-dimensional map T related to a discrete model of populations generation. The inverse of T has vanishing denominator giving rise to nonclassical singularities: a nondefinition line, a focal point and a prefocal line. Furthermore, the differences and relations between the feasible set, the feasible domains and the basins of attraction are presented. A phenomena of coexistence of attractors is shown. The structure of a chaotic repellor is interpreted by use of the singularities.

The discretization of ordinary differential equations is investigated for the case of the conservative Hénon–Heiles system. Starting from a discrete Hamiltonian function, which is invariant under time reversal, discrete equations of motion are analytically obtained using three different discretization schemes recently proposed and investigated in the literature. In the case where the discretization scheme successfully provide discrete systems in which the trace of the Jacobian matrix corresponding to the property required by a conservative system is preserved, it is shown that they are not necessarily invariant to time reversal. Such models are however quite robust when the time step is increased. For the schemes where the trace of Jacobian matrix does not match the condition required by conservative systems, it is shown that energy conservation is not achieved and the original dynamics is lost. Steps toward the solution to this problem are given.

In the study of complex systems one of the major concerns is the detection and characterization of causal interdependencies and couplings between different subsystems. The nature of such dependencies is typically not only nonlinear but also asymmetric and thus makes the use of symmetric and linear methods ineffective. Moreover, signals sampled from real world systems are noisy and short, posing additional constraints on the estimation of the underlying couplings. In this article, we compare a set of six recently introduced methods for quantifying the causal structure of bivariate time series extracted from systems with complex dynamical behavior. We discuss the usefulness of the methods for detecting asymmetric couplings and directional flow of information in the context of uni- and bidirectionally coupled deterministic chaotic systems.

In this paper, a fast chaos-based pseudo-random number generator (PRNG) is proposed for secured communications. In order to achieve fast throughput and facilitate hardware realization, 32-bit fixed point representation and arithmetic are used. Even under such configuration with quantization errors which will make the normal chaos-based PRNG impractical, our scheme can pass all the statistical tests in the up-to-date National Institute of Standards and Technology (NIST) test suite with the output bit rate up to 134 Mbps in a 2.6 GHz Pentium-4 machine. With such a fast PRNG, a stream cipher is hence designed for the application of online secure voice communication system with User Datagram Protocol (UDP).

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flow-invariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Let Γ be a permutation group acting on the set of cells. We define the group network , whose architecture is entirely determined by the group orbits of Γ. We prove that if Γ has the "balanced extension property" then every balanced coloring of is a fixed-point coloring relative to the automorphism group of the group network. This theorem applies in particular when Γ is cyclic or dihedral, acting on cells as the symmetries of a regular polygon, and in these cases the automorphism group is Γ itself. In general, however, the automorphism group may be larger than Γ. Several examples of this phenomenon are discussed, including the finite simple group of order 168 in its permutation representation of degree 7. More dramatically, for some choices of Γ there exist balanced colorings of that are not fixed-point colorings. For example, there exists an exotic balanced 2-coloring when Γ is the symmetry group of the two-dimensional square lattice. This coloring is doubly periodic, and its reduction modulo 8 leads to a finite group with similar properties. Although these patterns do not arise from fixed-point spaces, we provide a group-theoretic explanation of their balance property in terms of a sublattice of index two.

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

We characterize the Rössler system by means of a map associating colors to various intervals of largest Lyapunov exponent values in parameter space. This color map allows quick access to quantitative information about the dynamics of the system. The map also permits parameter space navigation while intentionally maintaining the system in a desired state, and avoiding regions where the system's behavior would be undesirable. In addition, the map exhibits a rich structure of stability clusters composed of affine-similar repetitions of basic elementary cells called "swallows." These cells, until recently, have been known to be associated with discrete maps only.

In this paper n-dimensional discrete-time systems with static bifurcations are considered from the viewpoint of control theory. This paper presents an adaption of available formulas for bifurcation analysis in two-dimensional continuous-time systems to the case of smooth maps using a frequency domain approach. The analyzed bifurcations are the building blocks to understand other more complex singularities and to propose certain methods for controlling the bifurcation behavior in nonlinear maps in the future.

We analyze a two-dimensional Hindmarsh–Rose type model exhibiting properties of both Class 1 and Class 2 neurons. Although the system is two-dimensional and contains only four parameters, the obtained bifurcation diagrams show that the bifurcation structure satisfies conditions for emergence of both features with constant stimuli.

In this paper, we discuss exponential synchronization of nonlinear coupled dynamical networks. Sufficient conditions for both local and global exponential synchronization are given. These conditions indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold.

We investigated a choice of prototypical double-scroll attractors, namely trajectories resulting from Lorenz equations and from two sets of equations related to Chua's circuits. We found that the probability distribution for the residence times within a given scroll consists of prominent, exponentially decaying peaks. Similar peaks had been reported for stochastically resonant systems, which are driven periodically; however, in view of our results in autonomous systems, the appearance of such peaks has a higher degree of generality. In the systems we investigated, each peak corresponds to a subset of the attractor having a particular number of loops. Moreover, the sets of initial conditions leading to different peaks are interleaved in an analogous way as riddled or intermingled basins of attraction.