International Journal of Bifurcation and Chaos

Cluster synchronization modes of continuous time oscillators that are diffusively coupled in a three-dimensional (3-D) lattice are studied in the paper via the corresponding linear invariant manifolds. Depending in an essential way on the number of oscillators composing the lattice in three volume directions, the set of possible regimes of spatiotemporal synchronization is examined. Sufficient conditions of the stability of cluster synchronization are obtained analytically for a wide class of coupled dynamical systems with complicated individual behavior. Dependence of the necessary coupling strengths for the onset of global synchronization on the number of oscillators in each lattice direction is discussed and an approximative formula is proposed. The appearance and order of stabilization of the cluster synchronization modes with increasing coupling between the oscillators are revealed for 2-D and 3-D lattices of coupled Lur'e systems and of coupled Rössler oscillators.

We briefly review the synchronization properties of cross-coupled spatially extended dynamical systems, with particular emphasis on elementary cellular automata and Kauffman networks subject to stochastic coupling. We also discuss the main results for the joint evolution of deterministically cross-coupled Ginzburg–Landau equations and neural networks. Both numerical and analytical approaches are addressed, and the main differences with the synchronization of zero-dimensional systems are highlighted. New results are presented characterizing the critical behavior at the synchronization transition of coupled Kauffman networks.

In this paper, we establish the existence and describe the global structure of traveling waves for a class of lattice delay differential equations describing cellular neural networks with distributed delayed signal transmission. We describe the transition of wave profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. We also describe an interval of the wave speed where the structure of the wave solution is unknown since the corresponding profile equation involves distributed argument of both advanced and retarded types, and we present some preliminary numerical simulation to illustrate the complexity.

Rescaled evolution sets of linear cellular automata on a lattice ℤ ⊕ G, where G is a finite Abelian group, with states in the finite field are defined and investigated.

Two-degree-of-freedom autonomous system with friction is analyzed numerically. The friction coefficient has been smoothened using arc tan function. The standard, but slightly modified chaos identification tools have been applied for the analyzed discontinuous system. Some interesting examples of stick-slip regular and chaotic dynamics have been illustrated and discussed.

This paper presents a new approach to the design of a composite sliding mode control for a class of chaotic systems with uncertainties. A significant feature of this control scheme is the incorporation of a new complementary sliding variable to the conventional sliding variable in order that a high-performance controller can be obtained. It has been shown that the guaranteed steady-state error bounds are reduced by half, as compared with the conventional sliding control. Moreover, the dynamic responses during the reaching phase are also significantly improved. We used a controlled uncertain Lorenz system and a controlled uncertain Chua's circuit as illustrative examples to demonstrate the effectiveness of the design.

Local bifurcation control designs have been addressed in the literature for stationary, Hopf, and period doubling bifurcations. This paper addresses the local feedback control of the Neimark–Sacker bifurcation, in which an invariant closed curve emerges from a nominal fixed point of a discrete-time system as a parameter is slowly varied. The analysis of this bifurcation is more involved than for previously considered bifurcations. The paper develops the stability and amplitude equations for the bifurcated invariant curves of the Neimark–Sacker bifurcation, and then proceeds to apply these relationships in the design of nonlinear feedbacks. The feedback controllers are applied to two examples: the delayed logistic map and a model reference adaptive control system model.

Given an arbitrary positive integer n, it is shown that there exist planar piecewise linear differential systems with at least n limit cycles.

Invariant areas generated by two-dimensional endomorphisms are studied using the method of critical curves. The invariant areas considered in this paper are obtained by iterating the noninvariant set constituted by the connected basin of an attracting set or the immediate basin of a nonconnected basin. This new kind of invariant area is of mixed type in the sense that its boundary is made up of critical curves arcs and arcs of saddle manifolds. The presentation is illustrated by three examples. A bifurcation changing the degree of connexity of an invariant area is described.

This paper introduces an analytical method for constructing spatial periodic orbits of specified periods. This result is then extended to generating spatial chaos in the sense of Li and Yorke.

Dissipative continuous dynamical systems with multiple external inputs are studied. When inputs are stochastically switched, the dynamics is characterized by trajectories with attractive fractal-like structures. We call this dynamics the fractal transition. This paper shows that a set of attractors obtained by periodic inputs can approximate trajectories of the fractal transition that have equal switching probabilities. The Hausdorff distance D is used to estimate the accuracy of approximation. We also present two numerical experiments, using harmonic and Duffing oscillators, corresponding to linear and nonlinear systems, respectively. We numerically show that D ≃ 0 given a large enough number of periodic inputs. For linear systems, under some conditions, we analytically derive the upper bound of D and show that D converges to 0.

Linear-plus-nonlinear feedback control is used to stabilize Hopf bifurcation in a rotor-magnetic bearing system, for which the linearized system possesses a double zero eigenvalues. The addition of nonlinear (quadratic) terms to the original linear feedback control formulation is used to modify the coefficients of the nonlinear terms in the reduced normal forms. It is found that feedback control incorporating certain quadratic terms renders the Hopf bifurcation supercritical. Finally, illustrative examples are given to verify the analytical results.

We report experimental control of chaos in an electronic circuit at 43.9 MHz, which is the fastest chaos control reported in the literature to date. Limiter control is used to stabilize a periodic orbit in a tuned collector transistor oscillator modified to exhibit simply folded band chaos. The limiter is implemented using a transistor to enable monitoring the relative magnitude of the control perturbation. A plot of the relative control magnitude versus limiter level shows a local minimum at period-1 control, thereby providing strong evidence that the controlled state is an unstable periodic orbit (UPO) of the uncontrolled system.

In this article the identification of AR system driven by chaotic sequences is addressed. This problem emerges in chaotic communication system, in which chaos-modulated signal passes through a channel described as an AR system. Two adaptive algorithms are presented to tackle this problem. Compared with the existing algorithms such as MPSV and MNPE, the proposed algorithms have very low computational complexities and can be used to track the system parameters in a slowly time-variant environment. The obtained simulation results illustrate that the proposed scheme can offer a better estimation accuracy than the conventional typical method in the high SNR case.

We subject the classical Volterra predator–prey ecosystem model with age structure for predator to periodic forcing, which in its unforced state has a globally stable focus as its equilibrium. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of prey. By using the Poincaré surface-of-section technique and numerical experiments, an abundance of periodic solutions, quasiperiodic solutions and chaotic solutions are revealed. The route of transitions to chaos were found to be very complex which include quasiperiodicity and frequency locking.

In this paper, we consider a switching system modeled by a discrete-time flow model, which has piecewise linear dynamics. In the system, it has been known that aperiodic oscillations occur. We qualitatively classify behaviors of the system into three modes, and define "blocks" consisting of the modes. We define a kind of Poincaré map by using the blocks, and show that the system has an invariant region in the Poincaré map. Moreover, we show that the Poincaré map can be reduced to a one-dimensional map. Finally, applying Lasota–Yorke's theorem to the one-dimensional map, we prove that the aperiodic oscillation is chaotic.

In this work we consider planar quasihomogeneous vector fields and we show, among other qualitative properties, how to calculate all the inverse integrating factors of such C¹ systems. Additionally, we obtain a necessary condition in order to have analytic inverse integrating factors and first integrals for planar positively semi-quasihomogeneous vector fields which are related to the existence of polynomial inverse integrating factors and first integrals for the quasihomogeneous cut. Examples are given and their relationship with Kovalevskaya exponents is shown.

We consider observer based synchronization of continuous-time chaotic systems. We present two message transmission schemes for such systems. The first one is based on chaotic masking and modulation, and the second one is based on only chaotic modulation. We show that in these schemes, the message may be recovered under certain conditions. We show that the proposed schemes are robust with respect to noise and parameter mismatch. We also present some simulation results.

The determination of bifurcation of equilibrium in micromachined elastic structures with liquid interface is posed as nonlinear boundary-value problems associated with spatially distributed models involving fluid forces. Explicit nonlinear algebraic equations for determining the bifurcation points associated with micromachined cantilever beam and membrane are derived. These equations permit bifurcation analysis involving any system parameter. It is shown that bifurcation of equilibrium in these structures is of the fold type. Numerical results are obtained for typical cantilever beam and membrane.