International Journal of Bifurcation and Chaos

We present a new continuation method for computing implicitly defined manifolds. The manifold is represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a boundary point. The boundary point is found using an expression for the boundary in terms of the vertices of a set of finite, convex polyhedra. The resulting algorithm is quite simple, allows adaptive spacing of the computed points, and deals with the problems of local and global overlap in a natural way. The algorithm is robust (the new points need only be near the boundary), and is well suited to problems with large embedding dimension, and small to moderate dimension.

In this paper we analyze the replication properties of additive or parity Cellular Automata (CA). The objective of the paper is twofold. Firstly, to review and extend the existing results in the cases of one- and two-dimensional CA. Secondly, to report a general result that states the replication properties of an n-dimensional CA, whose evolution law depends on an arbitrarym number of neighbor cells.

We introduce the standard Rössler oscillator and extend it from a system defined on ℝ³ to a system defined on . We extend the system in such a way that the simple Rössler type dynamics are preserved, whilst introducing an additional symmetry to the system such that when the oscillators are linearly coupled, there exists an embedding of symmetry forced invariant manifolds. We contrast our results with a previously considered model for a linearly coupled solid-state laser system which possessed a similar embedding of invariant manifolds. We discuss whether the mechanisms for loss of synchronization are system dependent or a more generic property of systems with embedded invariant manifolds.

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.

The imbalance of the boundary energy flow due to energy injection at one end and a nonlinear van der Pol boundary condition at the other end of the spatial one-dimensional interval can cause chaotic vibration of the linear wave equation [Chen et al., 1998b, 1998c]. However, such chaotic vibration is isotropic with respect to space and time because the two associated families of characteristics both propagate with the same speed and, thus, the "strength of chaos" is the same along both x and t directions. In this paper, we show that by including a mixed partial derivative linear energy transport term in the wave equation, nonlinearity in the van der Pol boundary condition can also cause chaotic vibration (without energy injection from the other end). Two new families of characteristics now travel with different speeds, leading to strong mixing of waves and nonisotropic spatiotemporal chaos. Parameter range for the route to chaos, including period-doubling, homoclinic orbits and Cantor-like invariant sets is classified. Numerical simulations of chaotic space-time profiles are also illustrated.

We consider the coupling of two nonidentical dynamical systems using an adaptive feedback linearization controller to achieve partial synchronization between the two systems. In addition we consider the case where an additional feedback signal exists between the two systems, which leads to bidirectional coupling. We demonstrate the stability of the adaptive controller, and use the example of coupling a Chua system with a Lorenz system, both exhibiting chaotic motion, as an example of the coupling technique. A feedback linearization controller is used to show the difference between unidirectional and bidirectional coupling. We observe that the adaptive controller converges to the feedback linearization controller in the steady state for the Chua–Lorenz example. Finally we comment on how this type of partial synchronization technique can be applied to modeling systems of coupled nonlinear subsystems. We show how such modeling can be achieved where the dynamics of one system is known only via experimental time series measurements.

Based on the variable structure model reference adaptive control (VS-MRAC) theory, a new control system for the control of chaos in Lorenz system, using only the measured output variable, is designed. For the derivation of the control law, it is assumed that the parameters of the model are unknown. Moreover, it is assumed that a disturbance input is present in the system. It is shown that in the closed-loop system, the output variable tracks a given reference trajectory, and the state vector converges to the equilibrium state. Digital simulation results show that the closed-loop system has good transient behavior and robustness to the uncertainties and disturbance input.

The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel–Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identified with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms.

We present a method for finding symbolic dynamics for a planar diffeomorphism with a homoclinic tangle. The method only requires a finite piece of tangle, which can be computed with available numerical techniques. The symbol space is naturally given by components of the complement of the stable and unstable manifolds. The shift map defining the dynamics is a factor of a subshift of finite type, and is obtained from a graph related to the tangle. The entropy of this shift map is a lower bound for the topological entropy of the planar diffeomorphism. We give examples arising from the Hénon family.

The discrete BVP oscillator obtained through the Euler method is investigated, and also first proved that there exist chaotic phenomena in the sense of Marotto's definition of chaos and two-period cycles. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including the ten-periodic orbit, a cascade of period-doubling bifurcation, quasiperiodic orbits and the chaotic orbits in Marotto's chaos and intermitten's chaos. The computations of Lyapunov exponents confirm the existence of dynamical behaviors.

We study the influence of spatiotemporal noise on the propagation of square waves in an electrical dissipative chain of triggers. By numerical simulation, we show that noise plays an active role in improving signal transmission. Using the Signal to Noise Ratio at each cell, we estimate the propagation length. It appears that there is an optimum amount of noise that maximizes this length. This specific case of stochastic resonance shows that noise enhances propagation.

We show that coupled Kerr and Duffing oscillators with small nonlinearities and strong external pumping can generate chaotic and hyperchaotic beats. The appearance of chaos within beats depends strongly on the type of interactions between the nonlinear oscillators. To indicate chaotic behavior of the system we make use of the Lyapunov exponents. The structure of chaotic beats can be qualitatively different — the envelope function can be smooth if the system is undamped or can give the impression of noise structure in the presence of strong damping and nonlinear interactions between the individual oscillators. The systems considered can be used, in practice, as generators of chaotic beats with chaotically modulated envelopes and frequencies.

Shape memory and pseudoelastic effects are thermomechanical phenomena associated with martensitic phase transformations, presented by shape memory alloys. The dynamical analysis of intelligent systems that use shape memory actuators involves a multi-degree of freedom system. This contribution concerns with the chaotic response of shape memory systems. Two different systems are considered: a single and a two-degree of freedom oscillator. Equations of motion are formulated assuming a polynomial constitutive model to describe the restitution force of oscillators. Since equations of motion of the two-degree of freedom oscillator are associated with a five-dimensional system, the analysis is performed considering two oscillators, both with single-degree of freedom, connected by a spring-dashpot system. With this assumption, it is possible to analyze the transmissibility of motion between two oscillators. Results show some relation between the transmissibility of order, chaos and hyperchaos with temperature.

This letter reports the finding of a new chaotic attractor in a simple three-dimensional autonomous system, which connects the Lorenz attractor and Chen's attractor and represents the transition from one to the other.

It was observed in earlier studies, that the mean field of globally coupled maps evolving under synchronous updating rules violated the law of large numbers, and this remarkable result generated widespread research interest. In this work we demonstrate that incorporating increasing degrees of asynchronicity in the updating rules rapidly restores the statistical behavior of the mean field. This is clear from the decay of the mean square deviation of the mean field with respect to lattice size N, for varying degrees of asynchronicity, which shows 1/N behavior upto very large N even when the updating is far from fully asynchronous. This is also evidenced through increasing 1/f² behavior regimes in the power spectrum of the mean field under increasing asynchronicity.