International Journal of Bifurcation and Chaos

Over the past fifteen years there have been various attempts to define chaos. In an effort to find a universally acceptable definition we began constructing new examples of chaotic systems in the hope that the salient features of chaos could be captured. Our efforts to date have failed and the examples we have constructed seem to suggest that no such definition exists. However, these examples have proved to be valuable in spite of our inability to hone a universal definition of chaos from them. Consequently, we present this list of examples and their significance. Some interesting conclusions that we can draw from them are: It is possible to construct simple closed form solutions of chaotic one-dimensional maps; sensitive dependence on initial conditions, the most widely used definition of chaos, has many counterexamples; there are invertible chaotic dynamical systems defined by simple differential equations that do not have horseshoes; three important properties that are thought to characterize chaos, continuous power spectral density, exponentially sensitive dependence on initial conditions, and exponential loss of information (Chaitin’s concept of algorithmic complexity), are independent.

Chaos seems to be tied to our notion of rates of divergence of orbits or degradation of information such as is found in systems with positive Lyapunov exponents. The reliance on rates seems to open the door to a pandora’s box of rates, both higher and lower than exponential. The intuitive notion of pseudo-randomness, a practical feature of chaos, is present in examples that do not have positive Lyapunov exponents. And in general, nonlinear polynomial rates of degradation of information are also quite “unpredictable”.

We conclude that it appears that for any given definition of chaos, there may always be some “clearly” chaotic systems which do not fall under that definition, thus making chaos a cousin to Gödel’s undecidability.

Chaotic scattering systems with multiple exit modes typically have fractal phase space boundaries separating the sets of initial conditions (basins) going to the various exits. If the exits number more than two, we show that the system may possess the stronger property that any initial condition which is on the boundary of one exit basin is also simultaneously on the boundary of all the other exit basins. This interesting property is known as the Wada property and basin boundaries having this property are called Wada basin boundaries.

The nonlinear resonance hypothesis of music perception was tested in an experiment comparing a group of musically sophisticated and a group of less sophisticated subjects. The prediction that weakly chaotic music entrains less complex brain wave (EEG) oscillations at the prefrontal cortex was confirmed by using a correlational dimension algorithm. Strongly chaotic (stochastic) and periodic music both stimulated higher brain wave complexity. More sophisticated subjects who prefer classical music showed higher EEG dimensions while less sophisticated subjects responded with a drop in brain wave complexity to rhythmical weakly chaotic music. Subjects ratings of perceived complexity of the musical pieces followed mathematical (objective) structure of the music and did not reflect the changes in brain wave complexity. The results are interpreted in the context of an associated (Hebbian) network theory of nonlinear brain dynamics.

In this paper, it is shown that the number and location of the fixed points of global nonlinear polynomials can be specified in terms of the term clusters and cluster coefficients of the respective models. It is also shown that if the structure (that is, the model basis) of a nonlinear polynomial model of degree ℓ includes all possible terms then such a model will always have ℓ nontrivial fixed points. In modeling problems, where the estimated polynomials are to reproduce fundamental invariants of the original system, this situation will not always be welcome and therefore suggests that the structure of polynomial models should be carefully chosen. Ways of achieving this are briefly mentioned.

We deduce a pde equation governing the evolution in time of the state (i.e., the distribution on the phases) of an ensemble of resonators interacting by means of their own emitting waves. A result due to Rayleigh (concerning the statistical properties of a superposition of a great number of identical waves) is generalized and provides the basic tool of this research. We find a "two mode” equation, whose structure changes for different ranges of the values of the distribution of the phases.

Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. The primary features of the resulting parameter plane bifurcation diagrams are “(Arnold) resonance horns” emanating from zero forcing amplitude. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper, we investigate divisions of these horns into subregions — different subregions corresponding to maps having different numbers of periodic orbits. The existence of subregions having more than the “usual” one pair of attracting and repelling periodic orbits implies the existence of “extra folds” in the corresponding surface of periodic points in the Cartesian product of the phase and parameter spaces. The existence of more than one attracting and one repelling periodic orbit is shown to be generic. For some of the families we create, the resulting parameter plane bifurcation pictures appear in shapes we call “Arnold flames”. Results apply both to circle maps and forced oscillator maps.

The chaotic dynamics of a nonlinear buckled elastic beam are shown to be controllable in a periodic orbit using a digital occasional proportional feedback control circuit. The method is similar to that developed by Hunt [1991] and does not require a detailed knowledge of the underlying two-dimensional map. The control is effected by perturbing the potential wells synchronously with the forcing function. Experiments show that the system possesses both periodic orbits and strange attractors different than those of the uncontrolled system.

A new concept is developed to facilitate transitions between the two stable states of a bistable dynamical system. The transitions take place without loss of stability or hysteresis. The underlying theoretical concept is to couple an additional dynamical variable to the original system in a strictly linear fashion. Explicit formulae are given which determine the couplings. The method is applied to the double well oscillator. In its bifurcation diagrams possible paths from one stationary point to the other are indicated.

We show that two coupled chaotic systems initially operating on two different simultaneously co-existing attractors can be synchronized. Synchronization is achieved as one of the systems switches its evolution to the attractor of the other one. The final attractor of the synchronized state strongly depends on the precise position of trajectories on their attractors at the moment when coupling is introduced. Our system is the first reported example of locally-intermingled basins of attraction which occur in physical systems.

This paper will address the problem of unmasking a new chaotic communication scheme using synchronizing circuits, where the Lorenz system is modulated by the message and the x-coordinate of the modulated system is added to the message and transmitted to the receiver. The receiver is driven into perfect synchrony with the transmitter even in the presence of the message, and since the message becomes part of the dynamics it provides very little distortion to the phase space of the dynamical system. However, this paper will demonstrate that it is still possible to extract a sinusoidal message from the transmitted signal. It will also be shown that it is possible to extract the sinusoidal signal solely from the x-coordinate, without secondarily adding back the message sinusoid before transmission. The message extraction is also shown to work for simple frequency-modulated and phase-modulated message signals. The modulated communication scheme does effectively nullify a multi-step unmasking technique which had been somewhat successful when applied to chaotic communication schemes which employed additive message signals.

In this paper we consider the estimation of the correlation dimension from a scalar chaotic time series using delay coordinates. Past work has shown that there appears to be a reconstruction space for which the correlation integral has the longest scaling region. We give a firmer foundation to this idea by developing a theory that estimates the dimension of this “optimal” reconstruction space in terms of dynamical quantities such as the largest Lyapunov exponent.

Compounding is a statistical notion. Essentially, it comprises of regarding the parameters in a particular statistical distribution as random variables with a prescribed distribution. The compound distribution then acquires the parameters of the compounding distribution as its own. As deterministic chaos, in spite of being deterministic, appears like a statistical phenomenon, the notion of compounding can be extended to chaotic systems. It is shown with illustrations that a chaotic system can be compounded by another chaotic system, giving rise to compound chaos which is, in general, “chaoticer”. The concept can also be used to make a periodic system chaotic, thus opening possibilities of “chaoticization”. Examples of compound chaos and chaoticization are given using Lorenz and Rössler systems, including their attractors and limit cycles as “compoundee” and/or “compounder” systems. The conclusions are based on quantitative studies of Lyapunov exponents and correlation dimensions.

We introduce homotopy transformations of vector fields as an efficient way to design families of chaotic attractors or transients with specific geometrical and dynamical properties. Those properties are inherited from different progenitor attractors, which allows a morphing between the progenitors. The method can provide a simple tool for a systematic design of attractors of desired dynamical characteristics. It is based on the robustness of invariant manifolds of the homotopic vector fields with respect to parametric perturbations. We analytically construct a convex subfamily of Chua attractors which illustrates this approach. Connections to control and synchronisation schemes of chaotic systems are shown. We also discuss applications in the area of sound synthesis.

The flow of an incompressible viscous fluid between parallel plates becomes unstable when the plates are tumbled. As the tumbling rate increases, the flow restabilizes. This phenomenon is elucidated by path-following techniques. The solution of the Navier-Stokes equations is approximated by spectral techniques. The linear stability of these solutions is studied.