International Journal of Bifurcation and Chaos

The paper stresses the universal role that Cellular Nonlinear Networks (CNNs) are assuming today. It is shown that the dynamical behavior of 3D CNN-based models allows us to approach new emerging problems, to open new research frontiers as the generation of new geometrical forms and to establish some links between art, neuroscience and dynamical systems.

We discuss efficient continuation algorithms for solving nonlinear eigenvalue problems. First, we exploit the idea of symmetry reductions and discretize the problem on a symmetry cell by the finite element method. Then we incorporate the multigrid V-cycle scheme in the context of continuation method to trace solution branches of the discrete problems, where the preconditioned Lanczos method is used as the relaxation scheme. Next, we apply the symmetry reduction technique to the two-grid finite element discretization scheme [Chien & Jeng, 2005] to solve some nonlinear eigenvalue problems in physical science. The two-grid centered difference discretization scheme described therein was also implemented for comparison. Sample numerical results are reported.

In this work, we study the realization and bifurcation of Boolean functions of four variables via a Cellular Neural Network (CNN). We characterize the basic relations between the genes and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set. Based on the analysis, we have rigorously proved that there are exactly 1882 linearly separable Boolean functions of four variables, and found an effective method for realizing all linearly separable Boolean functions via an uncoupled CNN. Consequently, any kind of linearly separable Boolean function can be implemented by an uncoupled CNN, and all CNN genes that are associated with these Boolean functions, called the CNN gene bank of four variables, can be easily determined. Through this work, we will show that the standard CNN invented by Chua and Yang in 1988 indeed is very essential not only in terms of engineering applications but also in the sense of fundamental mathematics.

Stochastic effects on convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with constant transmission delays are studied. Without assuming the boundedness, monotonicity and differentiability of the activation functions, nor symmetry of synaptic interconnection weights, by skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, M-matrix properties, inequality technique, stochastic analysis and non-negative semimartingale convergence theorem, delay independent and easily verifiable sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, respectively. The results are compared with the previous results derived in the literature for discrete delayed RNNs without diffusion or stochastic perturbation. Two examples are also given to demonstrate our results.

Some delay independent and delay dependent conditions are derived for the global stability of the bidirectional associative memory neural networks with delayed self-feedback. Regarding the self-connection delay as the parameter to be varied, the linear stability and Hopf bifurcation analysis are carried out. An algorithm to determine the direction and stability of the Hopf bifurcations is also worked out. Some examples and numerical simulations are presented.

This paper investigates different types of chaotic synchronization in a system of two coupled sine maps. Due to the bimodal nature of the individual map, there is a range of parameters in which two synchronized chaotic states coexist along the main diagonal. In certain parameter regions, various (regular or chaotic) asynchronous states coexist with the synchronized chaotic states, and the basins of attraction become quite complicated. We determine the regions of stability for the so-called principal cycles that arise through transverse period-doubling bifurcations of synchronized saddle cycles. Particular emphasis is paid to the occurrence of chaotic antisynchronization, the coexistence of antisynchronous chaotic states, and the presence of narrow regions of parameter space in which states of chaotic synchronization and antisynchronization exist simultaneously. For each of these cases we provide detailed pictures of the associated basin structures.

The paper describes the slow evolution of two adaptive traits that regulate the interactions between two mutualistic populations (e.g. a flowering plant and its insect pollinator). For frozen values of the traits, the two populations can either coexist or go extinct. The values of the traits for which populations extinction is guaranteed are therefore of no interest from an evolutionary point of view. In other words, the evolutionary dynamics must be studied only in a viable subset of trait space, which is bounded due to the physiological cost of extreme trait values. Thus, evolutionary dynamics experience so-called border collision bifurcations, when a system invariant in trait space hits the border of the viable subset. The unfolding of standard and border collision bifurcations with respect to two parameters of biological interest is presented. The algebraic and boundary-value problems characterizing the border collision bifurcations are described together with some details concerning their computation.

In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and shown to exhibit ten small limit cycles: five around each of the focus points. It will be proved in this paper that the system can have twelve small limit cycles. The major tasks involved in the proof are to compute the focus values and solve coupled enormous large polynomial equations. A computationally efficient perturbation technique based on multiple scales is employed to calculate the focus values. Moreover, the focus values are perturbed to show that the system can exactly have twelve small limit cycles.

The triple physical pendulum with barriers is used to model a piston — connecting rod — crankshaft system of a mono-cylinder combustion engine. Among other results, a peculiar motion of the piston — connecting rod — crankshaft model is reported, i.e. six sliding stages of the piston along the cylinder per one engine cycle are illustrated. In spite of its simplicity, the introduced model can serve as the first step in a more advanced modeling taking into account various technological details. In particular, the proposed model can be considerably useful for the investigations of impacts between the piston and the cylinder aiming at the reduction or total elimination of this source of harmful noise.

This paper develops a theory of global exponential synchronization using two Chua circuits as the platform, for which five theorems are established by constructing new Lyapunov functions and employing the comparison principle in conjunction with the stability theory with respect to partial components.

This paper investigates the chaos synchronization problem for a class of uncertain master-slave chaotic systems. Based on the variable structure control theory, a strategy is proposed to guarantee the occurrence of a sliding mode motion of error states when the proposed control law is applied. As expected, the error state is able to drive to zero with match external uncertainties or into a predictable neighborhood of zero with mismatch external uncertainties. Furthermore, a modified continuous sliding mode controller is also proposed to avoid the chattering. Examples of Lorenz system and Chua's circuit are presented to demonstrate the obtained results.

In this Letter, attention is focused on the dynamics of a second-order nonlinear circuit driven by two sinusoidal signals. The early results reported herein show that the application of signals with slightly different frequencies enables the phenomenon of chaotic beats to be generated. In particular, the beats dynamics are analyzed both in time-domain and state-space, confirming the chaotic behavior of the proposed circuit.

Two novel circuit-independent RC nonautonomous chaotic oscillator structures are presented. Both structures rely on a periodic pulse-train as the driving force and on a comparator as the only source of nonlinearity.

In this letter, a symbolic method is proposed for analyzing the bifurcation in switching power converters. This method focuses on the cyclic operation of the system which characterizes its bifurcation behavior. The concept of block switching sequence is introduced, which, in conjunction with the waveform periodicity of the system, can be used to distinguish the various types of bifurcation behavior, e.g. "smooth" period doubling and "nonsmooth" border collision. This concept explicitly exposes the mechanism of border collision from a circuit operational viewpoint and leads to a direct detection criterion for this global bifurcation. The proposed method is applied, as an illustrative example, to develop two-dimensional bifurcation diagrams for a voltage-mode controlled buck converter.

In this paper we revisit a switching circuit designed by the authors and present a theoretical analysis on the existence of chaos in this circuit. For the ordinary differential equations describing this circuit, we give a computer-aided proof in terms of cross-section and Poincare map, by applying a modern theory of topological horseshoes theory to the obtained Poincare map, that this map is semiconjugate to the two-shift map. This implies that the corresponding differential equations exhibit chaos.

This note focuses on the stability analysis of some classes of nonlinear time-delay models, encountered as fluid models for TCP/AQM network. By combining analytical and numerical tools, the attractors of these models, as well as the local and global behaviors of the solutions are studied. Among others, the presence of a chaotic attractor is shown, which supports the proposition that TCP itself as a deterministic process can cause or contribute to chaotic behavior in a network.

The main goals of the paper are firstly to provide qualitative and quantitative information on the dynamics of the models under consideration, and secondly to illustrate the capabilities of computational tools for stability and bifurcation analysis of delay differential equations to analyze fluid flow models.

We investigate small two-dimensional arrays of locally coupled phase oscillators which are shown to exhibit a surprising variety of stable structures which include: single spiral waves, spiral pairs and spirals with secondary periodic core motion. This periodic core motion is not the core meander familiar to many models of active media, but is in fact induced by the boundary of the small domain. Such boundary motion was investigated by Sepulchre and Babloyantz [1993] for the complex Ginzburg–Landau equation and for the Brusselator model in a relaxation oscillation parameter regime. The current model confirms the findings in [Sepulchre & Babloyantz, 1993] and sheds new light on the origin of such motion. The model also exhibits other patterns, as well as a chaotic regime. We discuss the transition between patterns as the form of the coupling is changed as well as implications for pattern formation in general oscillatory media.

In this paper, by using the element decomposition and the covering property of a polyhedron by balls of radii ε in the finite dimensional space, we obtain an upper bound of the Kolmogorov's ε-entropy of the global attractors for the first- and second-order lattice systems.

In this paper, we are concerned with the complete and generalized synchronization problem for some kind of PDE chaotic systems. The criteria for the existence of synchronization manifold is given in a look-like general framework by invariant manifold method. The corresponding optimization calculation problems and procedures are proposed.

The discrete dynamics of a blood cell population model are investigated in terms of bistability, instability and chaos. For the first time, the nonlinear model of blood cell production and destruction is investigated with a feedback mechanism incorporated. Numerical simulations demonstrate that the population of blood cells is dependent upon the history of the system, and upon whether parameters (destruction and production rates) are increasing or decreasing. The system also exhibits complex dynamics, such as period-bubbling, and period-doubling and undoubling routes to and from chaos. Some clinical examples are cited in the paper.