International Journal of Bifurcation and Chaos

In this tutorial we continue our program of clarifying chaos by examining the relationship between chaotic and stochastic processes. To do this, we construct chaotic analogs of stochastic processes, stochastic differential equations, and discuss estimation and prediction models. The conclusion of this section is that from the composition of simple nonlinear periodic dynamical systems arise chaotic dynamical systems, and from the time-series of chaotic solutions of finite-difference and differential equations are formed chaotic processes, the analogs of stochastic processes. Chaotic processes are formed from chaotic dynamical systems in at least two ways. One is by the superposition of a large class of chaotic time-series. The second is through the compression of the time-scale of a chaotic time-series. As stochastic processes that arise from uniform random variables are not constructable, and chaotic processes are constructable, we conclude that chaotic processes are primary and that stochastic processes are idealizations of chaotic processes.

Also, we begin to explore the relationship between the prime numbers and the possible role they may play in the formation of chaos.

The circuit recently proposed by Murali, Lakshmanan and Chua (MLC) is one of the simplest nonautonomous nonlinear electronic circuits which show a variety of dynamical phenomena including various bifurcations, chaos and so on. In this paper we study the spatiotemporal dynamics in one- and two-dimensional arrays of coupled MLC circuits both in the absence as well as in the presence of external periodic forces. In the absence of any external force, the propagation phenomena of traveling wavefront and its failure have been observed from numerical simulations. We have shown that the propagation of the traveling wavefront is due to the loss of stability of the steady states (stationary front) via subcritical bifurcation coupled with the presence of neccessary basin of attraction of the steady states. We also study the effect of weak coupling on the propagation phenomenon in one-dimensional array which results in the blocking of wavefront due to the existence of a stationary front. Further we have observed the spontaneous formation of hexagonal patterns (with penta–hepta defects) due to Turing instability in the two-dimensional array. We show that a transition from hexagonal to rhombic structures occur by the influence of an external periodic force. We also show the transition from hexagons to rolls and hexagons to breathing (space-time periodic oscillations) motion in the presence of external periodic force. We further analyze the spatiotemporal chaotic dynamics of the coupled MLC circuits (in one dimension) under the influence of external periodic forcing. Here we note that the dynamics is critically dependent on the system size. Above a threshold size, a suppression of spatiotemporal chaos occurs, leading to a space-time regular (periodic) pattern eventhough the single MLC circuit itself shows a chaotic behavior. Below this critical size, however, a synchronization of spatiotemporal chaos is observed.

Different strategies for building high-resolution models providing a more detailed description of a limited area of interest as for example, in regional weather forecasts are developed. They are subsequently compared, on the basis of the dynamical behavior generated by the corresponding models. The statistical properties of the relevant fields are analyzed, and predictability experiments are performed on statistical ensembles of close lying trajectories whose mean distance represents the uncertainty in the initial state of the system. The results show that a global, variable-mesh model performs much better than a limited area fine mesh one embedded into a coarser global model.

We modify a combinatorial idea of Peter Lax, to uniformly approximate any volume preserving homeomorphism of the closed unit cube Iⁿ, n≥2, by another one which either: (i) rigidly permutes an infinite family of dyadic cubes of total volume one, or (ii) is chaotic in the sense of Devaney. Our methods apply as well to arbitrary compact manifolds endowed with a nonatomic, locally positive Borel measure.

The twist condition is a necessary condition in integrable Hamiltonian systems and symplectic maps to obtain Poincaré–Birkhoff bifurcations under small perturbations. When this condition does not hold, topological structures other than Poincaré–Birkhoff chains arise in phase space through bifurcations of isolated periodic orbits and reconnections of asymptotic manifolds. In this paper we construct an integrable model Hamiltonian with degeneracies suitable to observe these phenomena close to nontwist resonant tori. The generation of isochronous chains and the stability of their fixed points is determined analytically and a condition for the reconnection is found. Particular examples are given, illustrating the bifurcation and reconnection scenario for several cases of degeneracy.

The parameter renormalization group method proposed in the previous papers is applied to the case where the system is described by two coupled one-dimensional maps to investigate the critical behavior of periodic-doubling. Particular attention is given to the asymmetric case where the maps have different bifurcation parameters. The first-order parameter renormalization equations are found to give qualitatively correct critical behaviors by comparing with numerical simulation. Furthermore, using the parameter reduction technique, it is found that critical behaviors of the coupled maps having a set of parameters on several critical lines are equivalent to those for the corresponding uncoupled maps.

Considering an n–dimensional map T, a necessary and sufficient condition for the existence of a cusp point on a fold bifurcation curve in a parameter plane of T is proposed. In the case of a nondegenerated cusp point, a necessary and sufficient condition for the stability of the associated nonhyperbolic fixed point is established. These conditions, obtained using the classical methods (center manifold, normal form), are expressed in an explicit algebraic well adapted for numerical computing.

The rapid bifurcation described by Kriegsmann [1987] is shown to be a generic bifurcation for planar symmetric piecewise-linear systems. The bifurcation can be responsible for the abrupt appearance of stable periodic oscillations. Although it has some similarities with the Hopf bifurcation for smooth systems, since the stability change of an equilibrium involves the appearance of one limit cycle, the dependence of the limit cycle amplitude on the bifurcation parameter is different from the Hopf's case. To characterize this bifurcation, accurate estimates for the amplitude and period of the bifurcating limit cycle are given.

The analysis is just illustrated with the application of the theoretical results to the Wien bridge oscillator. Comparisons with experimental data and Kriegsmann's analysis are also included.

We consider cellular neural networks with symmetric space-variant feedback template. The complete stability is proved via detailed analysis on the energy function. The proof is presented for the two-dimensional case with Dirichlet boundary condition. It can be extended to other dimensions with minor adjustments. Modifications to the cases of Neumann and periodic boundary conditions are also mentioned.

Checkerboard spiral waves in a 2D CML under consideration are generic solutions in a narrow parameter layer located adjacent to the saddle-node bifurcation boundary in the spatiotemporal chaos region. The spiral wave self-organization is in close agreement with the Haken's slaving principle. Scaling correspondence between sizes of the spiral waves and the diffusion length that is a natural unit of length in the CML is established.

Intermittency of checkerboard spiral waves and traveling clusterings originating from sudden shrinking of the strange attractor of the 2D CML in the neighborhood of the saddle-node bifurcation boundary is found. A power-law probability density for lifetimes in the spiral wave (laminar) phase is observed, while in the checkerboard clusterings (bursting) phase the above quantity exhibits an exponential decay. This difference can be interpreted through the self-organized behavior of the spiral waves, and the passive relaxation of the disordered checkerboard clusterings.

The linearized stability and Hopf bifurcation of the delay Lienard equations are studied. The linearized asymptotic stability for the null solution of the equations with two parameters is analyzed. The Hopf bifurcation and the stability of periodic solutions are investigated by constructing the center manifold and using the normal form method. Several examples for typical nonlinear delay Lienard equations are given to show the coincidence between the theoretical analysis and the numerical results.

A large array of dynamical systems can be broadly classified as weakly nonlinear, where a common feature is the presence of dual time-scale dynamics. Often it is desirable to isolate the dynamics at one time scale (typically the slower time scale), and this paper presents a generic phase-sensitive detection method that successfully isolates the slow-time dynamics, stripping off the less-relevant and often dominant fast-time dynamics. We shall also show for the first time, under certain phase considerations, the equivalence of the present method and the traditional Poincaré section of the original full response. The method is first presented in theory and then applied to data from a numerical simulation of the forced spherical pendulum followed by data from a similar spherical pendulum experiment. The method is successful provided certain conditions apply for phase stationarity.

It has been shown that chaotic signal can be obtained from cascade coupled phase-locked loop systems in a broad region of parameter variation. In this paper we demonstrate by computer simulation that the cascade coupled PLLs may be a useful tool for the solution of the problem of synchronization of two PLL systems in the regime of chaotic oscillations (with stabilized average frequency). We give an example of using cascade coupled PLLs for data communication by means of the chaotic signal.

A numerical investigation of the stability of invariant librational tori is presented. The method has been developed for a model describing the spin-orbit coupling in Celestial Mechanics. Periodic orbits approaching the librational torus are computed by means of Newton's method. According to Greene's criterion, their stability is strictly related to the survival of invariant tori.

We consider librational tori around the main spin-orbit resonances (1:1, 3:2). Their existence provides the stability of the resonances, due to the confinement properties in the three-dimensional phase space associated to our model. The results are consistent with the actual observations of the eccentricity and of the oblateness parameter. A different behavior of the Moon and Mercury around the main resonances is evidenced, providing interesting suggestions about the different probabilities of capture in a resonance.