International Journal of Bifurcation and Chaos

This paper presents the basin tree diagrams of all hyper Bernoulli σ_τ-shift rules for string lengths L = 3, 4, …, 8. These diagrams have revealed many global and time-asymptotic properties that we have subsequently proved to be true for all L < ∞. In particular, we have proved that local rule has no Isles of Eden for all L, and that local rules and are inhabited by a dense set (continuum) of Isles of Eden if, and only if, L is an odd integer. A novel and powerful graph-theoretic tool, called Isles-of-Eden digraph, has been developed and can be used to test the existence of dense Isles of Eden of any local rule which satisfies certain constraints, such as rules , , , , as well as all invariant local rules, such as rules , , and , subject to no constraints.

Classical and new results concerning the topological structure of skew-products semiflows, coming from nonautonomous maps and differential equations, are combined in order to establish rigorous conditions giving rise to the occurrence of strange nonchaotic attractors on 𝕋^d × ℝ. A special attention is paid to the relation of these sets with the almost automorphic extensions of the base flow. The scope of the results is clarified by applying them to the Harper map, although they are valid in a much wider context.

A new conjugate Lorenz-type system is introduced in this paper. The system contains as special cases the conjugate Lorenz system, conjugate Chen system and conjugate Lü system. Chaotic dynamics of the system in the parametric space is numerically and thoroughly investigated. Meanwhile, a set of conditions for possible existence of chaos are derived, which provide some useful guidelines for searching chaos in numerical simulations. Furthermore, some basic dynamical properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions and the compound structure of the system are demonstrated with various numerical examples.

In this paper, the generation of n × m-scroll attractors under a Chua-circuit framework is presented. By using a sawtooth function, f₁(x), and a staircase function, f₂(y), n × m-scroll attractors can be generated and observed from a third-order circuit. Its dynamical behaviors are investigated by means of theoretical analysis as well as numerical simulation. Moreover, two electronic circuits are designed for its realization, and experimental observations of n × m-scroll attractors based on Chua's circuit are reported, for the first time in the literature.

This paper discusses the stair function approach for the generation of scroll grid attractors of fractional differential systems. The one-directional (1-D) n-grid scroll, two-directional (2-D) (n × m)-grid scroll and three-directional (3-D) (n × m × l)-grid scroll attractors are created from a fractional linear autonomous system with a simple stair function controller. Being similar to the scroll grid attractors of classical differential systems, the scrolls of 1-D n-grid scroll, 2-D (n × m)-grid scroll and 3-D (n × m × l)-grid scroll attractors are located around the equilibria of fractional differential system on a line, on a plane or in 3D, respectively and the number of scrolls is equal to the corresponding number of equilibria.

In a two-dimensional cellular automaton model of retained excitation every excited cell stays excited if the number of excited neighbors belong to some interval, the cell takes refractory state otherwise. Every resting cell is excited if the number of excited cells in its neighborhood belong to some other interval; cell-state transition from refractory to resting state is unconditional. We classify 1296 rules of retained excitation based on how dynamics of excitable lattices develop after initial stimulation. Several modes of space-time activity dynamics are discovered: not growing but persistent domains of activity, domains with rectangular, octagonal and almost circular growth, amoeba-like growing patterns, mobile and still localizations.

We consider the system consisting of two identical n-neural loops which are coupled via a single neuron of each loop with discrete time delay. A framework describing the linearized stability region in the parameter space for the trivial steady state solution is constructed. In the boundary of the stability region, a periodic oscillation may arise from the Hopf bifurcation while these periodic oscillations could be dead owing to the change of coupling strength, delay or meeting a pitchfork bifurcation. The examples of n = 4 and n = 5 show the phenomena of oscillation arising or death, and the effects of the coupling strengths and delay on the stability and oscillation. The criticalities of pitchfork and Hopf bifurcations are investigated by using the normal form method for retarded functional differential equations.

Current studies in neurophysiology award a key role to collective behaviors in both neural information and image processing. This fact suggests to exploit phase locking and frequency entrainment in oscillatory neural networks for computational purposes. In the practical implementation of artificial neural networks delays are always present due to the non-null processing time and the finite signal propagation speed. This manuscript deals with networks composed by delayed oscillators, we show that either long delays or constant external inputs can elicit oscillatory behavior in the single neural oscillator. Using center manifold reduction and normal form theory, the equations governing the whole network dynamics are reduced to an amplitude-phase model (i.e. equations describing the evolution of both the amplitudes and the phases of the oscillators). The analysis of a network with a simple architecture reveals that different kind of phase locked oscillations are admissible, and the possible coexistence of in-phase and anti-phase locked solutions.

The existence of solitary wave, kink wave and periodic wave solutions of a class of singular reaction–diffusion equations is obtained using some effective methods from the dynamical systems theory. Specially, for a class of nonlinear wave equations, fundamental properties of profiles of traveling wave solutions determined by some bounded orbits of the traveling wave systems are rigorously proved. Parametric conditions that guarantee the existence of the aforementioned solutions are derived and given explicitly.

The creation of an outer layer of chaotic magnetic field lines in a tokamak is useful to control plasma-wall interactions. Chaotic field lines (in the Lagrangian sense) in this region eventually hit the tokamak wall and are considered lost. Due to the underlying dynamical structure of this chaotic region, namely a chaotic saddle formed by intersections of invariant stable and unstable manifolds, the exit patterns are far from being uniform, rather presenting an involved fractal structure. If three or more exit basins are considered, the respective basins exhibit an even stronger Wada property, for which a boundary point is arbitrarily close to points belonging to all exit basins. We describe such a structure for a tokamak with an ergodic limiter by means of an analytical Poincaré field line mapping.

There is a kind of correspondence between stochastic resonance and bifurcation of the moment equations of a noisy nonlinear system with the same noise intensity as the resonance independent variable and the bifurcation parameter, respectively. In this paper, this correspondence is examined and revealed in the noisy one-dimensional bistable system and the noisy two-dimensional Duffing oscillator. The bifurcation of the moment equations of each noisy system is the bifurcation with double-branch of fixed-point shift. Besides classical stochastic resonance, a kind of complex stochastic resonance corresponds to the bifurcation of moment equations. This complex stochastic resonance is induced by the stochastic transitions of system motion among the three fixed point attractors on both sides of the bifurcation point of the original system, which is predicted semi-analytically. Finally, due to this correspondence being examined, the mechanism of stochastic resonance can be provided through analyzing the change of the energy transfer induced by the bifurcation of the moment equations.

We consider the existence and stability of multibump solutions of a class of integral-differential equations modeling a single layer of homogeneous neural network with both excitatory and inhibitory neurons. The existence results are obtained by combining several shifts of a one-bump solution. Dynamical properties are obtained by considering the equation as an infinite-dimensional dynamical system and the spectrum of multibump solutions in terms of the weight functions. The center manifold theory and its foliation are used to show exponential stability with asymptotic phase for multibump solutions. Numerical results for some possible bifurcation phenomena are also presented.

This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows that H(4) ≥ 20, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem. It is well known that H(2) ≥ 4, and it has been recently proved that H(3) ≥ 12. The number of limit cycles obtained in this paper greatly improves the best existing result, H(4) ≥ 15, for fourth-degree polynomial planar systems.

In this paper, we study fingerprint feature extraction via CNN with Von Neumann neighborhood. The extraction was implemented by using CNN with nine input variables, and we find that the process could also be implemented with only five variables, and an easier algorithm without compromising the effectiveness. According to the CNN model with five input variables and the corresponding CNN gene bank done by Chen et al. [2006, Http1], we can determine the CNN gene easily. Simultaneously, we also find some results in one of the references are incorrect.

In this letter, a three-dimensional continuous-time smooth autonomous system with quadratic nonlinear terms is proposed for generating multiscroll chaotic attractors. Observation of 2 × 2-scroll attractors generated from this kind of system is reported for the first time. The result is confirmed by both numerical simulations and electronic circuit experiments.

This paper deals with the master-slave synchronization problem of Lur'e systems based on time-varying delay feedback control. The time-varying delay is only assumed to be bounded. Delay-dependent conditions are derived such that the controlled slave system can track the master system. The synchronization criteria are expressed in terms of linear matrix inequality, which can be checked readily by using some standard numerical packages. A simulation example is provided to demonstrate the effectiveness of the proposed approach.

In this paper we investigate the drive-response type synchronization of Hopfield-type neural networks with connection time delays for both discrete and distributed cases. By employing the Lyapunov functional method, we propose a sufficient condition to ensure the occurrence of synchronization with exponential rates. This criterion is independent of time delays as well as the types of delay. We prove that the exponential synchronization occurs provided a certain weighted sum of the connection and coupling strengths is negative enough, no matter that the connection time delay is of discrete or distributed case. Numerical examples are provided to illustrate the results.

A new form of cluster synchronization is explored in cellular arrays of chaotic oscillators. Previously, it was shown in the literature that symmetries of the coupling topology with uniform interaction weights lead to several coexisting clusters of synchronized cells. In this study a new phenomenon is presented where highly asymmetric interaction weights can give rise to cluster synchronization regimes with partial synchronization. In addition, cluster or partial synchronization regimes corresponding to asymmetric interaction patterns can break the underlying symmetries of the network topology and boundary conditions at the expense of some residual synchronization error.

We propose a simple method for secure digital signal transmission by making some modifications in the single-step parameter modulation technique propo sed earlier to overcome certain inherent deficiencies. In the modified method, the parameter modulation is effectively regulated or controlled by the chaotic signal obtained from the transmitting chaotic system so that it has the maximum security. Then, the same idea is also extended to the multistep parameter modulation technique. It is found that both the methods are secure against ciphertext (return map) and plaintex attacks. We have illustrated these methods by means of the Lorenz system.

A new uniform methodology to study the fast subsystem stability of general two-time scale nonlinear systems is developed. It consists of a direct method that provides estimates of the stability region of the fast subsystem that are uniform with respect to the slow variables, which are treated as uncertainties. The methodology is illustrated on small power systems leading to much improved results in estimating the stability region, critical clearing times as compared to traditional methods. As a by-product, it gives the required theoretical support to justify and to correct the heuristic approaches used in power system stability analysis literature.

The well-known Matsumoto–Chua–Kobayashi (MCK) circuit is of significance for studying hyperchaos, since it was the first experimental observation of hyperchaos from a real physical system. In this paper, we discuss the existence of hyperchaos in this circuit by virtue of topological horseshoe theory. The two disjoint compact subsets producing a horseshoe found in a specific 3D cross-section, both expand in two directions under the fourth Poincaré return map, this fact means that there exists hyperchaos in the circuit.

In this paper we study the localization problem of compact invariant sets of the optically injected laser system by applying the first order extremum conditions and cylindrical coordinates. Our main results consist in finding localization sets of simple forms which can be easily computed. Special attention is focussed on the case where we obtain a compact localization set.

In this paper we present a precise description of the connectedness locus of the family of polynomials (z² + x)² + y, where x, y are real numbers.

We study effects of additive spatiotemporal random variations, introduced to the payoffs of a spatial hawk–dove game, on the evolution of the reconciliatory and the aggressive strategy. We show that uncertainties facilitate aggressive behavior for a broad range of resource values. In particular, aggressors thrive best if stochastic influences are of the order of magnitude of deterministic payoff values. We argue that random payoff variations are potent and plausible promoters of aggressive behavior in human as well as animal societies if only the hawk–dove game payoff ranking applies.