International Journal of Bifurcation and Chaos

Since their inception at Macy conferences in later 1940s, complex systems have remained the most controversial topic of interdisciplinary sciences. The term "complex system" is the most vague and liberally used scientific term. Using elementary cellular automata (ECA), and exploiting the CA classification, we demonstrate elusiveness of "complexity" by shifting space-time dynamics of the automata from simple to complex by enriching cells with memory. This way, we can transform any ECA class to another ECA class — without changing skeleton of cell-state transition function — and vice versa by just selecting a right kind of memory. A systematic analysis displays that memory helps "discover" hidden information and behavior on trivial — uniform, periodic, and nontrivial — chaotic, complex — dynamical systems.

The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper. It is shown that the model has a Bogdanov–Takens singularity (cusp case) of codimension at least 4 for some parameter values. Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary. Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop. These results present far richer dynamics compared to the model with no harvesting. Numerical simulations, including the repelling and attracting Bogdanov–Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis.

We perform a systematic comparison between different algorithms for solving the Blind Signal Separation problem. In particular, we compare five well-known algorithms for Independent Component Analysis (ICA) with a recently proposed algorithm based on linear state space modeling (IC–LSS). The comparison is based on simulated mixtures of six source signals, five of which are generated by nonlinear deterministic processes evolving on chaotic attractors. The quality of the reconstructed sources is quantified by two measures, one based on a distance measure implemented by a Frobenius norm, and one based on residual mutual information. We find that the IC–LSS modeling algorithm offers several advantages over the ICA algorithms: it succeeds in unmixing Gaussian sources, on short time series it performs, on average, better than static ICA algorithms, it does not try to remove coincidental dependencies resulting from finite data set size, and it shows the potential to reconstruct the sources even in the case of noninvertible mixing. As expected, for the case of non-Gaussian sources, invertible mixing and sufficient time series length, the ICA algorithms typically outperform IC–LSS modeling.

A model is considered for turbulent diffusion which consists of a Riesz space fractional derivative to describe the turbulent phenomenon and also includes advection and classical diffusion. We present a first order explicit numerical method and a second order implicit numerical method to solve our problem and prove convergence results for both methods, including the derivation of stability constraints needed for the explicit numerical method to converge. In the end, to give some insights into the phenomenon of turbulent diffusion described by the Riesz fractional derivative, we show the behavior of the solution when we consider a Gaussian initial condition.

The prime merit of the paper is to visualize hyperfractals via support functions. More precisely, address sets of hyperfractals, i.e. fractals living in hyperspaces, will be visualized in this way.

For a class of nonlinear diffusion–convection–reaction equations, corresponding to two families of heteroclinic orbits connecting two nodes of the traveling wave system, the existence of uncountably infinite many global monotonic and nonmonotonic wavefront solutions is discussed. By using the method of planar dynamical systems, the dynamical behavior of the corresponding traveling wave system is studied. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink wave solutions are given.

Impulsively synchronized chaos with criterion from conditional Lyapunov exponent is often interrupted by desynchronized bursts. This is because the Lyapunov exponent cannot characterize local instability of synchronized attractor. To predict the possibility of the local instability, we introduce a concept of supreme local Lyapunov exponent (SLLE), which is defined as supremum of local Lyapunov exponents over the attractor. The SLLE is independent of the system trajectories and therefore, can characterize the extreme expansion behavior in all local regions with prescribed finite-time interval. It is shown that the impulsively synchronized chaos can be kept forever if the largest SLLE of error dynamical systems is negative and then the burst behavior will not appear. In addition, the impulsive synchronization with negative SLLE allows large synchronizable impulsive interval, which is significant for applications.

In this paper, we propose a ring-scroll Chua system by introducing a generalized ring transformation. Some basic dynamical properties of this generalized ring transformation are discussed. The parameter regions and the periodic orbits, which are embedded in Chua chaotic attractor mapping to those in ring-scroll Chua chaotic attractor, are investigated, too. Moreover, the topological horseshoe of this system is investigated. Finally, the ring-scroll Chua chaotic attractor is physically implemented by using digital signal processors.

Although variability is a ubiquitous characteristic of the nervous system, under appropriate conditions neurons can generate precisely timed action potentials. Thus considerable attention has been given to the study of a neuron's output in relation to its stimulus. In this study, we consider an increasingly popular spiking neuron model, the adaptive exponential integrate-and-fire neuron. For analytical tractability, we consider its piecewise-linear variant in order to understand the responses of such neurons to periodic stimuli. There exist regions in parameter space in which the neuron is mode locked to the periodic stimulus, and instabilities of the mode locked states lead to an Arnol'd tongue structure in parameter space. We analyze mode locked solutions and examine the bifurcations that define the boundaries of the tongue structures. The theoretical analysis is in excellent agreement with numerical simulations, and this study can be used to further understand the functional features related to responses of such a model neuron to biologically realistic inputs.

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous center.

An essential issue for the validation of ciphers is cryptanalysis, that is, the study of attacks in order to reveal possible weaknesses. In this paper, we propose a systematic and unified methodology for cryptanalyzing known chaos-based cryptosystems borrowed from the open literature, namely shift keying, discrete parameter modulation, switched message-embedding, time-varying delay-based schemes. It is shown that all of them have a hybrid feature. Based on this consideration, the cryptanalysis leads to a specific parameter identification. The method is extended in straightforward way for hybrid cryptosystems defined over finite fields, involving in particular Boolean functions. The complexity of the cryptanalysis is derived regardless of the exhibited dynamics and provides a new quantitative measure for security assessment, thus allowing to show, by the way, that most of the chaos-based cryptosystems proposed so far in the literature are weak.