International Journal of Bifurcation and Chaos

Active Walk (AW) is a paradigm for self-organization and pattern formation in simple and complex systems, originated by Lam in 1992. In an AW, the walker changes the deformable landscape as it walks, and is influenced by the changed landscape in choosing its next step. Active walk models have been applied successfully to various biological, physical, geological and economic systems from both the natural and social sciences. More recently, it has been used to model human history. In Part I of this review, the birth of the AW paradigm, its basic concepts and formulations, a solvable two-site model, and the experiments and AW modeling of surface-reaction filamentary patterns are presented. Part II here continues with properties of AW, and applications of AW in nonliving and living systems — including those from the social sciences and human history. (In particular, unsuspected quantitative laws and a prediction about the Chinese history are given.) A comment on the relationship between physics, social science and complex systems is provided. The review concludes with open problems in the form of workable research projects and general discussions.

In this paper, we study synchronization and asynchronization in a Coupled Lorenz-type Map Lattice (CLML). Lorenz-type map forms a chaotic system with an appropriate discontinuous function. We prove that in a CLML with suitable coupling strength, there is a subset of full measure in the phase space such that chaotic synchronization occurs for any orbit starting from this subset and there is a dense subset of measure zero in the phase space such that synchronization will never occur. We also provide numerical observations to explain our results.

This paper deals with the problem of master-slave synchronization for uncertain Lur'e systems via time-varying delay feedback control. The parametric uncertainty is assumed to be norm bounded. Several new and sufficient conditions are presented such that the uncertain Lur'e master and slave systems are synchronous for all admissible uncertainties. These synchronization criteria are dependent on the size of time delay, which can be expressed by means of matrix inequalities. The adopted method is based on defining a new Lyapunov–Krasovskii function and using some inequalities techniques. Our results obtained here extend and improve some previously related results. Finally, two numerical examples are provided to demonstrate the applications of our proposed results.

In this article, we use a variant of a recently introduced algebraic state estimation method obtained from a fast output signal time derivatives computation process. The fast time derivatives calculations are entirely based on the consequences of using the "algebraic approach" in linear systems description (basically, module theory and non-commutative algebra). Here, we demonstrate, through computer simulations, the effectiveness of the proposed algebraic approach in the accurate and fast (i.e. nonasymptotic) estimation of the chaotic states in some of the most popular chaotic systems. The proposed state estimation method can then be used in a coding–decoding process of a secret message transmission using the message-modulated chaotic system states and the reliable transmission of the chaotic system observable output. Simulation examples, using Chen's chaotic system and the Rossler system, demonstrate the important features of the proposed fast state estimation method in the accurate extraction of a chaotically encrypted messages. In our simulation results, the proposed approach is shown to be quite robust with respect to (computer generated) transmission noise perturbations. We also propose a way to evade computational singularities associated with the local lack of observability of certain chaotic system outputs and still carry out the encrypting and decoding of secret messages in a reliable manner.

We propose a periodic feedback scheme for the stabilization of periodic orbits for discrete time chaotic systems. We first consider one-dimensional discrete time systems and obtain some stability results. Then we extend these results to higher dimensional discrete time systems. The proposed scheme is quite simple and we show that any hyperbolic periodic orbit can be stabilized with this scheme. We also present some simulation results.

In this paper, the nonlinear oscillations and chaotic dynamics of a parametrically excited simply supported symmetric cross-ply laminated composite rectangular thin plate are further investigated. Considering geometric nonlinearity and nonlinear damping, a two-degree-of-freedom nonlinear system under parametric excitation is obtained to give the nonlinear governing equations of motion for laminated composite plate subjected to in-plane load. The method of multiple scales is utilized to obtain the averaged equations that are numerically solved to obtain the steady bifurcation responses and analysis of stability for laminated composite thin plate. It is illustrated that under certain conditions laminated composite thin plate may have the multiple steady bifurcation solutions and jumping may occur. The chaotic motion of rectangular symmetric cross-ply laminated composite thin plate is also found by using numerical simulation. It is found that the occurrence of the periodic, quasi-periodic and chaotic motions for a parametrically excited four-edges simply supported rectangular symmetric cross-ply laminated composite thin plate depends on the parametric excitation.

A two variable model describing the circadian fluctuation of two proteins (PER and TIM) in cells is considered. The original model was set up by Leloup and Goldbeter [1998], the present form was developed by Tyson et al. [1999]. Periodic solutions with 24-h period were investigated in those papers. Here the possible phase portraits and bifurcations are studied in detail. The saddle-node and Hopf bifurcation curves are determined in the plane of two parameters by using the parametric representation method [Simon et al., 1999]. It is shown that there are four cases according to their mutual position (as the remaining parameters of the system are varied). Using these curves the number and type of the stationary points are determined in all four cases. The global bifurcation diagram, which is a system of bifurcation curves that divide the given parameter plane into regions according to topological equivalence of global phase portraits, is also determined. Finally, the so-called constant period curves are computed numerically. These curves consist of those parameter pairs on the parameter plane for which the system has a limit cycle with a given period. It turns out that in a wide range of parameters the system has limit cycles with period close to 24-h.

We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R⁴ and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features.

A four-node network consisting of a negative circuit controlling a positive one is studied. It models some of the features of the p53 gene network. Using piecewise linear dynamics with thresholds, the allowed dynamical classes are fully characterized and coded. The biologically relevant situations are identified and conclusions drawn concerning the effectiveness of the p53 network as a tumor inhibitor mechanism.

Neuronal firing is one of the main information carriers in nervous systems. It has been reported that highly fluctuating, aperiodic "noise-like" inputs, similar to those occurring in areas of nervous systems, evoke reliable firing. To explain this surprising result it has been suggested that (i) the high frequency components of such inputs or a resonance phenomenon are responsible for reliable firing, with (ii) internal biophysical noise being the potential major inherent source of unreliability. Conversely, the present study shows that such fluctuating inputs do not always lead to reliability, even in the absence of noise. Using the noise-free FitzHugh–Nagumo model, we demonstrate a regime in which reliable firing never occurs. To analyze the changes responsible for unreliable firing we use methods from the Random Dynamical Systems theory. This is done through careful numerical investigations since no analytical solutions are known to our system. More precisely, from the observation that the stationary distributions remained qualitatively unchanged throughout the area of interest we conclude that the switch from reliable to unreliable firing is not associated to a phenomenological stochastic bifurcation. However, this change of behavior corresponded to a dynamical stochastic bifurcation, with the leading Lyapunov exponent of the system becoming positive. The bifurcation resulted also in a change of the stochastic attractor from a single stochastic equilibrium to one reminiscent of a strange attractor. The combination of these results lead us to argue that the unreliable firing results from stochastic chaos, and that such a phenomenon may limit the reliability of neurons in nervous systems.

The Local Activity Lemma establishes that "A homogeneous nonconservative medium cannot exhibit complexity unless it is locally active". In this communication, starting from the previous remark we report examples that prove systems with complex local active behavior like Reaction–Diffusion Cellular Nonlinear Networks (RD-CNNs) can work as image processors also in the 3D domain.

This Letter considers the problem of chaotic secure communication in the drive-response framework. The drive system can be augmented into a higher order system, and then a sliding mode observer based response system can be constructed to synchronize this augmented system. If they satisfy certain conditions, the hidden message can be recovered directly by the concept of equivalent control. Theoretical analysis and numerical simulation verify the effectiveness of the proposed method.

In the presence of multiplicative and additive colored noise with colored cross-correlation, a method of parameter-induced aperiodic stochastic resonance (SR) for baseband binary pulse amplitude modulated (PAM) signal processing is investigated in detail. Firstly, the dynamic bit error rate (DBER) is defined by the nonstationary solution of the Fokker–Planck equation (FPE). Because the output probability densities of the nonlinear system are unsymmetrical, the decision level of the binary signal is decided by minimizing DBER. Then we take this minimal DBER (MDBER) as a quantity to optimize in the aperiodic SR problem by tuning the system parameters. With theoretical and numerical simulation methods, the effect of noise correlation times on MDBER is discussed. Finally, by comparison between theoretical and numerical simulation results, we find that the agreement is good for not so large noise correlation times.

We study position dependent random maps on the unit interval with holes where the possible laws of motion are piecewise monotonic transformations. The main result of this note is proving the existence of absolutely continuous conditionally invariant measures.

We improve the least squares (LS) method for building models of a nonlinear dynamical system given finite time series which are contaminated by observational noise. When the noise level is low, the LS method gives good estimates for the parameters, however, the models selected as the best by information criteria often tend to be over-parameterized or even degenerate. We observe that the correct model is not selected as the best model despite belonging to the chosen model class. To overcome this, we propose a simple but very effective idea to use the LS method more appropriately. We apply the method for model selection. Numerical studies indicate that the method can be used to apply information criteria more effectively, and generally avoid over-fitting and model degeneracy.

In this Letter, we study the stability of differential equations with time-dependent delay. Several theorems are established for stability on a finite time interval, called "interval stability" for simplicity, and Liapunov stability. These theorems are applied to the generalized Gauss-type predator–prey models, and satisfactory results are obtained.

In this paper, the problem of limit cycles bifurcated from the equator for a cubic polynomial system is investigated. The best result so far in the literature for this problem is six limit cycles. By using the method of singular point value, we prove that a cubic polynomial system can bifurcate seven limit cycles from the equator. We also find that a rational system has an isochronous center at the equator.

The feedback controllers are designed to modify the amplitude of limit cycles in van der Pol oscillator and generalized van der Pol oscillator. Bifurcation control equations of weakly nonlinear systems are obtained by using the method of multiple scales. Gain-amplitude curves of controlled systems are drawn. Based on numerical study, the brief results of controlling amplitude of limit cycle are given for strongly nonlinear system.