International Journal of Bifurcation and Chaos

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

It is shown that 8 of all 256 elementary cellular automata have the property of chaos synchronization using a binary driving sequence. The binary synchronization signal ensures good noise robustness. A simple measure for characterizing long periodic cycles as being "chaotic" was introduced. It is also shown that certain relations between the profile of attractors shaped by the local rule in the finite state space and the property of chaos synchronization exists. A simple modulation–demodulation scheme is proposed allowing to build spread spectrum communication links at low implementation costs while the cellular automata state space (number of cells) can be taken as large as required by cryptographic reasons.

A mutualism is an interaction where the involved species benefit from each other. We study a two-dimensional hexagonal three-state cellular automaton model of a two-species mutualistic system. The simple model is characterized by four parameters of propagation and survival dependencies between the species. We map the parametric set onto the basic types of space-time structures that emerge in the mutualistic population dynamic. The structures discovered include propagating quasi-one-dimensional patterns, very slowly growing clusters, still and oscillatory stationary localizations. Although we hardly find such idealized patterns in nature, due to inreased complexity of interaction phenomena, we recognize our findings as basic spatial patterns of mutualistic systems, which can be used as baseline to build up more complex models.

In the conventional spatial formulation of the iterated prisoner's dilemma, only the results generated in the last round are taken into account in deciding the next choice. Historic memory of past iterations can be taken into account by featuring players by a summary of their previous winnings and choices. The effect of such a memory implementation when the players are allowed to follow a simple full-deterministic structurally coevolving dynamic is assessed in this work.

This paper is concerned with the asymptotic behavior of the solutions u(x,t) of the Swift–Hohenberg equation with quintic nonlinearity on a one-dimensional domain (0, L). With α and the length L of the domain regarded as bifurcation parameters, branches of nontrivial solutions bifurcating from the trivial solution at certain points are shown. Local behavior of these branches are also studied. Global bounds for the solutions u(x,t) are established and then the global attractor is investigated. Finally, with the help of a center manifold analysis, two types of structures in the bifurcation diagrams are presented when the bifurcation points are closer, and their stabilities are analyzed.

In this paper, the axisymmetric flow of a viscous fluid through a porous annular tube with walls kept at different temperatures is studied theoretically. The physical properties of the fluid remain constant, notably its specific mass, its dynamic viscosity and its thermal diffusivity. The nondimensional parameters which the solutions of the problem depend on are defined. A numerical integration using the shooting method is applied for solving the Navier–Stokes equations and the energy equation. Bifurcation diagrams are presented and enable to highlight significant properties of the flow. Some thermal behaviors corresponding to specific values of the parameters are performed. Asymmetric solutions of the steady flow are described and some results about velocity components are also analyzed.

In this paper deep ocean sound propagation through random media is considered. The study is conducted within a ray theory framework, which facilitates the assessment of ray stability. Model ocean environments where there is a gradual transition between two ambient sound speed profiles, a single duct Munk profile and a double duct profile taken in the Eastern North Atlantic are examined. We build on the finding that the ambient sound speed structure controls ray stability [Beron-Vera & Brown, 2003], and extend this statement for sound speed profiles with transition. It is shown that launching basins, plots constructed by the Maximal Lyapunov Exponent and indicating desirable ray launching parameters, can be predicted by the unperturbed ray system using the nonlinearity parameter.

Parametrized contact-equivalence is a successful theory for the understanding and classification of the qualitative local behavior of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view making explicit the singular behavior due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter. We show how to use path formulation to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the nondegenerate umbilics singularities are the generic cores in four situations: the general or gradient problems, with or without ℤ₂ symmetry where ℤ₂ acts on the second component of ℝ² via κ(x,y) = (x,-y). The universal unfolding of the umbilic singularities have an interesting "Russian doll" type of structure of miniversal unfoldings in all those categories.

With the path formulation approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance, some internal hierarchy of parameters). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new. We discuss some applications to the bifurcation of a loaded cylindrical panel. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.

In the paper, the mixing property of ℤ² subshift of finite type (SFT) is studied, some checkable conditions are provided to examine the property of mixing. Once the mixing property is checked, a class of examples can be found to show it possesses an entropy conjugacy property proposed in [Quas & Trow, 2000]. Finally, we construct an example of two shift spaces X and Y that are entropy conjugate but not conjugate.

In this paper, a well-known multistability phenomenon is discovered and observed in two-parameter families of nonlinear discrete-time neuron networks. A specific strategy of controlling multistability (multiple oscillation patterns) to unimode oscillation pattern (synchronization) by linear asymmetric diffusion coupling is proposed and the numerical simulation provides solid evidence for its effectiveness.

In this paper certain identities are generated for minimal polynomials Θ_n(z) of numbers 2cos(2π/n) and for Chebyshev polynomials of the first kind. Against the background of these identities, sets of cyclic points of some modified versions of these polynomials are derived. The discussion leads to the analysis of a certain new counting sequence of numbers connected with Carmichael function.

The "Q-curves have long been observed and studied as the shadowy curves which appear illusively — not explicitly drawn — in the familiar orbit diagram of Myrberg's map f_c(x) = x² + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x² + c as well — by "bifurcation diagram" we mean the collection of all periodic points of f_c (attracting, indifferent and repelling) — these collections form what we call "P-curves". We show Q-curves and P-curves intersect in one of two ways: At a superattracting periodic point on a P-curve, the infinite family of Q-curve s which intersect there are all tangent to the P-curve. At a Misiurewicz point, no tangencies occur at these intersections; the slope of the P-curve is the fixed point of a linear system whose iterates give the slopes of the Q-curves.

We also introduce some new phenomena associated with c sin x illustrating briefly how its two different families of Q-curves interact with P-curves.

Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix.

In this paper we demonstrate the power of the computer algebra package conley, which enables one to compute connection and transition matrices, two of the main algebraic tools of the CONLEY index theory. In particular, we study the CAHN–HILLIARD equation on the unit square and extend the results obtained in [Maier-Paape et al., 2007] to a bigger range of the bifurcation parameter. Besides providing several explicit computations using conley, the definition of connection matrices is reconsidered, simplified, and presented in a self-contained manner in the language of CONLEY index theory. Furthermore, we introduce so-called energy induced bifurcation intervals, which can be utilized by conley to differential equations with a parameter. These bifurcation intervals are used to automatically path-follow the set of connection matrices at bifurcation points of the underlying set of equilibria.

A method of greatly decreasing the activation time of a control method based on stable manifold information is proposed.

In an already classical paper, May [1976] pointed out that simple nonlinear models in ecology can possess complicated dynamics. A long debate concerning the possible existence of chaotic ecologies has followed the appearance of this paper. Morris [1990] stressed that several problems exist when making conclusions regarding chaotic or nonchaotic dynamics as models are fitted to ecological data. Takens' theorem [1981] should, in principle, provide tools for solving some of these problems, but in reality essential assertions are most often violated. First, limits regarding the length of the used time-series must be assumed. Second, the theorem can be applied for evaluating positive Lyapunov exponents only, and is hence not applicable for distinguishing nonchaotic dynamics from chaotic.

We use a deterministic food-chain model that possesses a wide range of different dynamical patterns to demonstrate the existence of cases that are misclassified with respect to chaotic or nonchaotic motion as models are fitted to data in the various dynamical regimes. Our results are valid regardless of what finite data size is assumed. The results are best understood for high-periodic cases in the noise-free limit. If the relevant phase-space is not sufficiently well-populated with data in the vicinity of the periodic orbit, then sufficient complexity of the models are not supported by standard model selection criteria like AIC or GCV. Thus, the fitted models only in the best cases contain information concerning both the location of the periodic orbit and the eigenvalues of the Jacobians evaluated along the periodic orbit. If the space is moderately well populated with data, then the data is often described by an unstable periodic orbit of the fitted model. The attractor that was to be described by the fitted model was destabilized and is now a repeller. We call this phenomenon stability switches. It reminds about the noise-induced phenomenon reported by Rand and Wilson [1992], but we point out that the problem reported here is caused by the fitting procedure itself, not by the added noise.

The repeller that has been created by the model-fitting procedure can be located in the basins of attraction of some fixed point, the infinity, or some periodic or chaotic attractors of the fitted models. The situation seems similar when periodic attractors are not well enough populated with data for describing their location and when chaotic attractors are to be described by data. Similar stability switches occur and the dynamics of models fitted to data may differ or coincide with the dynamics of the attractor that generated the data in an unpredictable manner.

The center problem and bifurcations of limit cycles for the Lyapunov system are continuously studied. We shall prove that we can construct successively a formal series such that the Lyapunov system is reduced a half-normal form. From the coefficients of the half-normal form, we obtain directly the Lyapunov constants of the origin. As examples, for two classes of cubic systems, the center and focus problem, and multiple bifurcations of limit cycles are studied.

In this communication, synchronization of two diffusively coupled Chua's circuit is studied from the analytical and experimental points of view. The conditions under which complete synchronization is ensured are derived by applying a strategy based on the Master Stability Function. The experimental realization, that makes use of the State-Controlled Cellular Nonlinear Network based implementation of Chua's circuit, shows the synchronized behavior of two circuits coupled with a passive resistor as diffusion coefficient. The results obtained indicates diffusive coupling as a mutually reduced order state observer, in the sense that one circuit observes the other and vice versa. Moreover, the concept of synchronization by using passive elements can be extended to spatially extended reaction–diffusion systems.

We experimentally demonstrate that both Voronoi diagram and its dual graph Delaunay triangulation are simultaneously constructed — for specific conditions — in cultures of plasmodium, a vegetative state of Physarum polycephalum. Every point of a given planar data set is represented by a tiny mass of plasmodium. The plasmodia spread from their initial locations but, in certain conditions, stop spreading when they encounter plasmodia originated from different locations. Thus space loci not occupied by the plasmodia represent edges of Voronoi diagram of the given planar set. At the same time, the plasmodia originating at neighboring locations form merging protoplasmic tubes, where the strongest tubes approximate Delaunay triangulation of the given planar set. The problems are solved by plasmodium only for limited data sets, however the results presented lay a sound ground for further investigations.

Coupled threshold elements with self-inhibition display a phase transition to an oscillating state where the elements fire in synchrony with a period T that is of the order of the dead-time caused by self-inhibition. This transition is noise-activated and therefore displays strong collectively enhanced stochastic resonance. For an exponentially decaying distribution of dead-times the transition to the oscillating state occurs, coming from high noise temperatures, via a Hopf bifurcation and coming from low temperatures, via a saddle node bifurcation. The transitions can be triggered externally by noise and oscillating signals. This opens up new possibilities for controlling slow wave sleep.

In this paper we model a computer network consisting of one standard Internet connection and one nonstandard connection that transmits uniformly sized bursts of packets at regular intervals. The nonstandard connection can represent probing activity of either a diagnostic measurement or attack. Using bifurcation diagrams, we study how the network's behavior changes as a function of the probing frequency. These diagrams reveal interesting, nonintuitive behavior. We present a series of models of increasing simplicity that capture the significant features of the network's behavior. Our simplest model is a piecewise linear, discontinuous one-dimensional map. This map helps explain the structure of the bifurcation diagram, and allows us to directly determine Lyapunov exponents, which give a measure of the system's predictability. As a result, we are able to more precisely describe and categorize the dynamics, including chaos, exhibited by this network.

The global asymptotic stability (GAS) for a class of nonlinear impulsive neural networks with multiple delays are discussed. By skillfully constructing suitable Lyapunov functionals and the linear matrix inequality (LMI) technique, a new sufficient delay-dependent stability criterion is derived. The proposed stability results are less conservative than some recently known ones in the literature, which is demonstrated via two examples with simulation.

For Cohen–Grossberg neural networks with time-varying delays, by fixed point and contract mapping theorems, a sufficient condition ensuring the existence and uniqueness of an equilibrium is proposed. To guarantee the delay independent global stability of the equilibrium, two sufficient conditions are proposed by means of a time delay differential inequality and contradiction tricks, respectively. By virtue of a special Lyapunov functional as well as properties of M-matrices, a sufficient condition undertaking the delay dependent global stability of the equilibrium is introduced. Compared with known literatures, the presented results place slack restrictions on the activation functions, and are suitable for the networks with time-varying delays. Furthermore, most of the obtained results are independent of the amplification functions, making their applicability more far-reaching. Finally, two examples are numerically simulated to illustrate the validity as well as novelty of the criteria.

Multiscale entropy (MSE) discloses the intrinsic multiple scales in the complexity of physical and physiological signals, which are usually featured by heavy-tailed distributions. Most of these research results are pure experimental search, till Costa et al. made the first attempt to the theoretical basis of MSE. However, the analysis only supports the Gaussian distribution [Phys. Rev. E71, 021906 (2005)]. In this paper, we present the theoretical basis of MSE under the inverse Gaussian distribution, which is a typical model for physiological, physical and financial data sets. The analysis is applicable to uncorrelated inverse Gaussian process and 1/f noise with the multivariate inverse Gaussian distribution, providing a reliable foundation for potential applications of MSE to explore complex physical and physiological time series.