International Journal of Bifurcation and Chaos

2D k-totalistic CA are life-like Artificial Micro-Worlds (AMWs), which contain a large variety of self-reproducing species. In this paper we present a gallery of these systems, with simple and complex behavior. Firstly, we present two simple self-reproducer species, with a linear behavior and then two self-reproducer species with a rhombic development. A curious double-strand self-reproducer completes this first part of the self-reproducing species taxonomy. We present then other interesting new species: their reproductive phenomena are more complex, resulting in space-time diagrams of great variety.

It is well-known that binary-valued cellular automata, which are defined by simple local rules, have the amazing feature of generating very complex patterns and having complicated dynamical behaviors. In this paper, we present a new type of cellular automaton based on real-valued states which produce an even greater amount of interesting structures such as fractal, chaotic and hypercyclic. We also give proofs to real-valued cellular systems which have fixed points and periodic solutions.

In this paper, the generalized synchronization of spatial Julia sets is investigated by applying linear coupling. We achieve the generalized synchronization of two different spatial Julia sets and also obtain the stable domain of the coupling strength based on the stable domains given of the spatial Julia sets. The relationship between the complex fixed plane and the synchronization of the spatial Julia sets is also analyzed and shows that the anticipative objective is accomplished. Finally, simulations are presented to demonstrate the effectiveness and feasibility of linear coupling.

For third-order nilpotent critical points of a planar dynamical system, the problem of characterizing its center and focus is completely solved in this article by using the integral factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic systems under small perturbations, it is proved that there exist eight small-amplitude limit cycles created from a nilpotent critical point.

We consider the dynamic behaviors of a mathematical chemostat model with state dependent impulsive perturbations. By using the Poincaré map and analogue of Poincaré's criterion, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three. Numerical simulation are carried out to illustrate the feasibility of our main results, thus implying that the presence of pulses makes the dynamic behavior more complex.

Fault diagnosis has played an important role in the identification of fault mechanisms and the subsequent successful isolation of faults in electronic circuits. In this paper, we propose a novel procedure for fault diagnosis in analog circuits. We first generate a set of fault patterns from fault simulation, and our main task is to develop a practical description of the way in which these fault patterns interact. Our approach is based on the construction of a complex network that describes the inter-dependence of the various fault patterns. Analysis of this complex network shows that the degree distribution is scalefree-like and the connectivity is small-world. We henceforth identify a small number of fault patterns that are most highly connected (of highest degrees) with other fault patterns. Furthermore, we study the connection between this network of fault patterns and the original circuit, the purpose being to relate the information of the high-degree fault patterns with the physical circuit topology, thus allowing the physical fault locations and circuit elements to be identified. Our proposed approach will find applications in automatic fault diagnosis of large-scale electronic circuits.

In this paper, we analyze a rather simple system in which some substance is being stored, released, and replenished simultaneously in some interdependent way. We investigate the dynamic behavior of such a system, using a two-dimensional map-based discrete-time model, and derive an integrated dynamical scene for this model. More specifically, we show the existence of an invariant curve induced by the well-known Neimark–Sacker bifurcation corresponding to the presence of a periodically oscillating behavior in this model.

The control of spiral wave breakup due to Doppler instability has been investigated. It is found that by applying the spatiotemporal modulation method with suitable control parameters, spiral wave breakup can be prevented. Further numerical simulations show that preventing spiral wave breakup is a result of the decrease of meandering motion of the spiral tip.

A recent proposal of q-deformation scheme for the nonlinear maps has stimulated new directions in the studies of various nonlinear dynamical systems. Such studies are advantageous in the analytical modeling of several nonlinear physical phenomena which cannot be perfectly represented by the standard or canonical models. This paper attempts to numerically analyze the behavior of q-deformed version of Henon map, which is one of the prototypical models exhibiting strange chaotic attractor. We particularly investigate the effect of q-deformation on the various kinds of long-term asymptotic dynamics of the canonical Henon map and also characterize the complete deformation parameter space into different regions corresponding to the periodic and strange chaotic motions. We also identify the route to chaos and new structures in the chaotic strange attractor of q-deformed Henon map.

For differential equation in the Chazy class IX, their corresponding three-dimensional system is studied in this paper by using dynamical system methods and Cosgrove's results. In a level set, the exact explicit parametric representations of a heteroclinic cycle family and uncountably infinitely many periodic solutions as well as quasi-periodic solutions, are all obtained.

This paper investigates global robust synchronization of a class of nonautonomous chaotic systems with parameter mismatch under the master–slave variable substitution control. A criterion of linear matrix inequality (LMI) for the global robust synchronization is rigorously proven and the corresponding synchronization error bound is analytically estimated. The LMI criterion is then applied to the gyrostat systems, obtaining some simple and optimized algebraic criteria for global robust synchronization of the master–slave gyrostat systems with time-varying phase mismatch under single-variable substitution configurations, further verified in a numerical example.

In this work, it is proved that any degenerate center is limit of a linear type center and when the degenerate center has an analytic first integral then it is limit of an analytic linear type center. A new method to detect integrability developed in [Giné & Santallusia, 2011] is applied to the degenerate center problem. Moreover, a review of the most important recent contributions to the degenerate center problem is given.

An effective and simple method, called perturbation scheme (PS), is proposed to study fold-Hopf bifurcation for delayed neural systems qualitatively and quantitatively when time delay and connection weight are considered as two bifurcation parameters. As an illustration, the proposed method is employed to investigate a delayed bidirectional associative memory (BAM) neural network. Dynamics arising from fold-Hopf bifurcation are classified qualitatively and expressed approximately in a closed form for periodic solution. We also give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the delayed system. Our investigations reveal that secondary Hopf bifurcation and pitchfork bifurcation of limit cycle may emanate from the pitchfork-Hopf point. In addition, the secondary Hopf bifurcation can lead to multistability between equilibrium points and periodic solution in some region of parameter space. The validity of analytical predictions is shown by their consistency with the results from the center manifold reduction (CMR) with normal form and numerical simulation. As an analytical tool, the advantage of the PS also lies in its simplicity and ease of implementation.

A compositive pancreatic β-cells model is developed by focusing on two important slow negative feedback mechanisms, where three slow processes C_ss, C_er and a with different time scales can interact to drive fast, medium and slow electrical bursting typically observed in pancreatic β-cells. We found that diverse patterns of electrical bursting with shapes varying from only waves to the coexistence of waves and spikes to only spikes, which are robust to fast, medium and slow oscillation periods. Therefore, dynamic bifurcation mechanisms of these bursting patterns could be surveyed by fast/slow analysis of only fast bursting. Here, the faster slow variable C_ss is regarded as a bifurcation parameter and C_er has no effect on the fast subsystem, while the slower variable a is nearly constant. In the light of all bifurcations associated with bursting, three topological types of bursting are attained, that is, "fold/fold" point–point hysteresis loop bursting, "fold/Hopf" bursting via the "fold/fold" hysteresis loop and "fold/homoclinic" bursting via the "fold/homoclinic" hysteresis loop.

We provide a local saddle-node bifurcation result for quasiperiodically forced interval maps. As an application, we give a rigorous description of saddle-node bifurcations of 3-periodic graphs in the quasiperiodically forced logistic map with small forcing amplitude.

The question of spectral analysis for deterministic chaos is not well understood in the literature. In this paper, using iterates of chaotic interval maps as time series, we analyze the mathematical properties of the Fourier series of these iterates. The key idea is the connection between the total variation and the topological entropy of the iterates of the interval map, from where special properties of the Fourier coefficients are obtained. Various examples are given to illustrate the applications of the main theorems.

This paper is a continuation of Part I where the authors treated the Fourier analysis of chaotic time series generated by a chaotic interval map. Here, we perform multiresolution analysis by using wavelet coefficients and characterize some necessary and sufficient conditions for the occurrence of chaos by the exponential growth with respect to the number of iterations n of certain sums of the wavelet coefficients.

Chaotic circuits have been widely used in the teaching of nonlinear dynamics disciplines, where a common practice is pedagogically based on the circuit point of view. Chua's circuit is the most popular platform for the demonstration of its components with rich dynamical patterns. However, engineering students majoring in control systems are more familiar with feedback systems rather than physical electronics with nonlinear components. This may lead to some difficulty in understanding the nonlinear properties of Chua's circuit, at least on first sight. This paper provides an alternative approach to teaching and learning chaotic oscillators by using the inherent understanding of feedback systems with simple modules at the undergraduate level. Utilizing the idea of chaotification, which means to make a nonchaotic module chaotic, the modules consist of only four blocks yet can generate more than thirty types of chaotic patterns via their various combinations. Conceptually, the blocks can be assembled as various mixed-mode chaotic circuits. Functionally, the blocks are easy-to-use in a Simulink-like fashion. Structurally, they are a LEGO-like educational kit. With low-cost implementation employing a few op-amps for each block, the newly proposed modules are simple, self-instructional and suitable for teaching and training students in school laboratories and in experimental environments.

In this paper, we consider chaotic synchronization in coupled map lattices (CMLs) with periodic boundary conditions. We give a rigorous proof of the occurrence of synchronization for 1D such CMLs with lattice size n = 5 for suitable parameters in the chaotic regime by Lyapunov method.