International Journal of Bifurcation and Chaos

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

This paper presents recent results concerning possibilities of complex behavior in extremely simple second-order digital filter structures. Two types of such behavior have been confirmed by simulation experiments and rigorous mathematical analysis: 1) chaotic behavior defined here as the existence of aperiodic, bounded trajectories displaying sensitive dependence to the initial states. Such trajectories often form very complex, self-similar patterns in the state-space. 2) an abundance of oscillatory solutions and final state sensitivity with respect to system parameters. This type of behavior can be identified by an extremely complicated structure of the parameter space (existence of Arnold tongues) and fine structure of changes of dynamic behavior when varying filter parameters (devil’s staircase). The first type of behavior has been discovered in filter sections employing two’s complement adder overflow characteristics and confirmed via symbolic dynamics technique. Properties of the second type have been found in extensive numerical experiments carried out for filter sections employing saturation arithmetic and in which a full confirmation via the mathematical analysis of an associated one-dimensional model of the system has been made.

We suggest a mathematical concept of self-stochasticity that may underlie a possible scenario of spacetime chaos in ideal medium (e.g. ideal turbulence). The self-stochasticity implies that the deterministic system which describes evolution of deterministic vector fields has trajectories whose ω-limit sets lie in the space of random vector fields. We consider a class of systems where almost all trajectories exhibit this property and, hence, the attractor consists of random fields.

Chaotically advected fluids are both a visual demonstration of stretching and folding leading to chaos and a prototypical example of a multiplicative process with weakly correlated steps. Complementary aspects of the process are studied by means of stretching calculations for different flows under both globally and partially chaotic conditions. Stretching is examined in two different ways: (i) as stretching plots, focusing primarily on stretching at small scales and on the comparison of the spatial distribution of stretching with dye structures and with unstable manifolds, and (ii) as time evolving probability density functions, analyzed using scaling techniques that renormalize the distributions by means of their moments. The first approach leads to the conclusion that the manifold structure generates the striation pattern observed in dye deformation experiments, and that both manifolds and dye patterns agree with stretching plots even at small scales. The second approach demonstrates that the multiplicative nature of stretching generates universal statistics which are reflected in self-similar scaling distributions.

We investigate numerically the effect of a spatiotemporal forcing on travelling waves produced in a model reaction-diffusion system operating under periodic boundary conditions. It is shown that, for suitable parameter values, this external forcing can stabilize the progressive wave regime and impose its directionality through wave inversion. When the field fails to reverse waves travelling in an opposite direction, chaotic-like spatiotemporal patterns may be observed.

The paper proposes a practical engineering approach for predicting chaotic dynamics in an important class of nonlinear systems. The aim of this approach is to provide a heuristic method of analysis which can give reasonably accurate answers but is far simpler to apply than other more rigorous methods based on nonlinear dynamics. Our approach is founded on the harmonic balance principle and uses standard describing function techniques well known to design engineers. Our method consists of a synergism of two independent techniques, each one constituting a possible mechanism for chaos. These two techniques are combined into a single algorithm which is highly efficient computationally, taking only a fraction of the time normally required by other more exact procedures.

In order to present and illustrate our algorithm clearly, and in order to compare its predictions with readily available results obtained by rigorous methods, we have chosen Chua’s circuit as a vehicle to demonstrate the effectiveness of this approach. Chua’s circuit was chosen not only for the huge amount of results already published concerning the dynamics of this system, but also because it represents a real physical system easily built in the laboratory, and whose simple mathematical model has proven to be realistic and mathematically tractable.

Although our algorithm does not guarantee its prediction is fully reliable, any more than the widely used describing function method does, its significance is based entirely on the empirical evidence that it yields qualitatively correct, though not exact, results for all of the chaotic phenomena that we have investigated in Chua’s circuit, as well as in many other chaotic systems. We hope therefore that our chaos prediction algorithm will find practical uses among engineers and scientists not familiar with more specialized mathematical approaches, in search of a simple and practical, although not rigorous, tool for analyzing systems with complex dynamics.

This paper examines the role of damping on both the steady state and basin behavior of a periodically driven damped oscillator with the ability to escape from a potential well. We examine the effect of damping on both the qualitative and quantitative resonance response of the system. Particular attention is paid to how the damping scales the main steady state bifurcations; saddle-nodes, period-doubling flips, cascades to chaos, boundary crises, etc. We also investigate how the damping level effects the main homoclinic and heteroclinic basin bifurcations that may result in a rapid erosion and stratification of the basin of attraction and hence a loss of engineering integrity of the system.

This paper presents the 1:1 resonant horn bifurcation phenomena for the forced van der Pol—Duffing equation. It is shown that the transition to chaos in the case of small dissipation evolves in two parallel processes: a sequence of period-doubling bifurcations and the birth, growth and merging of homoclinic structures.

Ship capsizing is a highly nonlinear dynamic phenomenon where global system behavior is dominant. However the industry standards for analysis are limited to linear dynamics or nonlinear statics. Until recently, most nonlinear dynamic analysis relied upon perturbation methods which are severely restricted both with respect to the relative size of the nonlinearity and the region of consideration in the phase space (i.e., they are usually restricted to a small local region about a single equilibrium), or on numerical studies of idealized system models.

In this work, recently developed global analysis techniques (e.g., those found in Guckenheimer and Holmes [1986], and Wiggins [1988, 1990]) are used to study transient rolling motions of a small ship which is subjected to a periodic wave excitation. This analysis is based on determining criteria which can predict the qualitative nature of the invariant manifolds which represent the boundary between safe and unsafe initial conditions, and how these depend on system parameters for a specific ship model. Of particular interest is the transition which this boundary makes from regular to fractal, implying a loss in predictability of the ship’s eventual state. In this paper, actual ship data is used in the development of the model and the effects of various ship and wave parameters on this transition are investigated. Finally, lobe dynamics are used to demonstrate how unpredictable capsizing can occur.

The model most often used by ecologists to describe interactions between predator and prey populations is analyzed in this paper with reference to the case of periodically varying parameters. A complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique. The results perfectly agree with the local theory of periodically forced Hopf bifurcation. The two classical routes to chaos, i.e., cascade of period doublings and torus destruction, are numerically detected.

In a homoclinic bifurcation to infinity, a limit cycle blows up to infinite amplitude and disappears. Based on linearizations at infinity, we argue that this bifurcation occurs in models of the Kondratieff economic long wave. A formal proof of the existence of stable and unstable manifolds at infinity is given. A numerical method is proposed and applied to a new model in which regulation of consumer goods production is taken into account. The economic implications of the analysis are discussed.

This paper aims at understanding the statistical features of nucleic acid sequences from the knowledge of the dynamical process that produces them. Two studies are carried out: first, mutual information function of the limiting sequences generated by simple sequence manipulation dynamics with replications and mutations are calculated numerically (sometimes analytically). It is shown that elongation and replication can easily produce long-range correlations. These long range correlations could be destroyed in various degrees by mutation in different sequence manipulation models. Second, mutual information functions for several human nucleic acids sequences are determined. It is observed that intron sequences (noncoding sequences) tend to have longer correlation lengths than exon sequences (protein-coding sequences).

A new method for the identification of true and spurious Lyapunov exponents computed from time series is presented. It is based on the observation that the true Lyapunov exponents change their signs upon time reversal whereas the spurious exponents do not. Furthermore by comparison of the spectra of the original data and the reversed time series suitable values for the free parameters of the algorithm used for computing the Lyapunov exponents (e.g., the number of nearest neighbors) are determined. As an example for this general approach an algorithm using local nonlinear approximations of the flow map in embedding space by radial basis functions is presented. For noisy data a regularization method is applied in order to get smooth approximating functions. Numerical examples based on data from the Hénon map, a four-dimensional analog of the Hénon map, a quasiperiodic time series, the Lorenz model, and Duffing’s equation are given.

Let X be a bounded subset of Rⁿ and let A be the Lebesgue measure on X. Let {X:τ₁,…, τ_N} be an iterated function system (IFS) with attractor S. We associate probabilities p₁,…, p_N with τ₁,…, τ_N, respectively. Let M(X) be the space of Borel probability measures on X, and let M: M(X)→M(X) be the Markov operator associated with the IFS and its probabilities given by:

Recent progress in understanding the dynamics and bifurcation of two coupled Josephson Junctions is numerically presented. The dependencies of the in-phase (IP) states and the anti-phase (AP) states on the parameters are shown when the coupling is symmetric. The connecting branches between the IP and AP are computed, and are found to be saddles which form the basin boundaries between the IP and AP states. A framework is presented based on a two oscillator model with asymmetric coupling which can predict the dynamics of higher dimensional systems. The behavior of out-of-phase (OP) states of n-junctions is examined using this asymmetric model when the oscillators are organized into two distinct groups.

In this letter we introduce a new concept, the control manifold, which simultaneously points us toward a general algorithm relating the existence of horseshoes in a Poincare map to the parameters of an ODE, while also providing a mechanism for understanding how to control nonlinear phenomena and therefore utilize these phenomena in a disciplined and predictable manner in the design of nonlinear circuits, systems, and neural networks.

In this letter we illustrate three methods of using nonlinear devices as sensors. We show that the sensory features of these devices is a result of sensitive dependence on parameters which we show is equivalent to sensitive dependence on initial conditions. As a result, we conjecture that sensitive dependence on initial conditions is nature’s sensory device in cases where remarkable feats of sensory perception are seen.

In this letter controlling of chaos in Bonhoeffer-van der Pol oscillator is investigated using various different control algorithms. Specifically, using an adaptive control algorithm we analyse the resetting of the system dynamics from a chaotic attractor to a limit cycle attractor. We demonstrate that, starting from a chaotic regime, it is possible to bring the system to a regular regime by means of a small parametric perturbation as well as by the addition of a weak external force. It is then shown that one can eliminate chaotic evolution by stabilizing an unstable periodic orbit by making small adjustments to an external parameter. Finally, we show the suppression of chaos by adding appropriate external noise.

Experimental results of friction in a mechanical system are reported. Strange properties of the friction force-velocity dependence are shown, where this relationship does not follow the typical assumption of the force proportional to the sign of the relative velocity.

An algorithm is presented for computing the relative rotations of orbits in a suspension of Smale horseshoes consisting of n stripes in spiral configuration. Bifurcational precedence relations in the creation of such horseshoes may thus be more readily determined. An application to a driven oscillator is discussed.