International Journal of Bifurcation and Chaos

This paper deals with cell mapping methodology for global analysis of nonlinear dynamical systems. It serves a mixed set of purposes. It is basically a tutorial paper on cell mapping. But, it also reports on certain new developments in cell mapping and includes a summary of recent publications on the topic. Presented in Sec. 1–3 and 5 are the basic concepts and theory of cell mapping. Two types of cell-to-cell mapping are discussed, namely: simple cell mapping and generalized cell mapping. Once a dynamical system has been cast in the form of a cell mapping, one needs to extract the system behavior from the mapping. In Secs. 4 and 6, computation algorithms for simple cell mapping and generalized cell mapping are discussed in detail. Moreover, a workshop-type example is included to guide the reader if he wishes to gain a working knowledge of the methodology.

The new developments presented in the paper include an algorithm for processing simple cell mappings and a theory of subdomain-to-subdomain global transient analysis of generalized cell mapping. These are reported in Secs. 4–6. Listed in the last section are publications on cell mapping, including applications in many rather diversified areas of dynamics. Hopefully, the breadth of the examples of application will indicate the potential of the cell mapping method.

Numerical methods are often needed if bifurcation phenomena in nonlinear dynamical systems are studied. In this paper the software system CANDYS/QA for numerical qualitative analysis is presented. A wide class of problems is treated: computation of invariant sets (e.g., steady-states and periodic orbits), path-following (continuation) of such sets, and the related bifurcation phenomena. The following bifurcation situations are detected and the corresponding critical points are calculated during path-following: turning, bifurcation, Hopf bifurcation, period-doubling, torus bifurcation points (one-parameter problems) as well as cusp and Takens-Bogdanov points (two-parameter problems). A number of newly developed methods (e.g., for computation of the Poincaré map) as well as algorithms from the literature are described to demonstrate the whole procedure of a qualitative analysis by numerical means. An illustrative example analyzed by CANDYS/QA is included.

Theory and examples of attractors with basins which are of positive measure, but contain no open sets, are developed; such basins are called riddled. A theorem is established which states that riddled basins are detected by normal Lyapunov exponents. Several examples, both mathematically rigorous and numerical, motivated by applications in the literature, are presented.

Let fµ(s)=µs(1−s) be the family of logistic maps with parameter µ, 1≤µ≤4. We present a study of the second-order difference equation x_n+1=fµ([1−∈]xn+∈x_n−1), 0≤∈≤1, which reduces to the well-known logistic equation as ∈=0.

In this paper, the existence of strange attractor of the Lozi mapping in the parameter region 0<b<1, 1+b<a< 2−b/2 is proved. This result coincides with numerical results. By means of the theory of transversal heteroclinic cycle, the structure of strange attractor is discussed. We conjecture that it consists entirely of unstable manifolds of infinite hyperbolic periodic points.

An infinite class of integer valued matrices of unbounded size and of arbitrary dimensions is presented here as a generalization of the so-called excess-matrix associated with a particular cellular automaton. These matrices are constructed through a recursive arithmetical procedure, and show an intricate fractal-like distribution of integers. The elements in these matrices have a number theoretical interpretation which is directly related to a proper number-base representation of the coordinates of the element. These excess-matrices exhibit numerical and symbolical self-similarities under agglomerating and decimating coarse-graining operations, and under so-called disguising transformations.

We investigate thermally induced optical nonlinearity of a glass doped with semiconductor quantum dots. With the feedback of a Fabry-Perot resonator this glass shows dispersive optical bistability. The reflected light signal is coupled to a hybrid ring cavity with a round trip time much longer than that of the Fabry-Perot resonator and even longer than the thermal relaxation time of the glass. The self-oscillations occurring for certain input parameters are regular and the appearance of different modes as a function of the light intensity coupled to the resonator is observed. Due to the broad hysteresis and the high inclination of both branches of the bistable loop we observe large numbers of ascending and descending steps in the output intensity, i.e., long oscillation periods. A simple model of adiabatic simulations by an iterated map shows that the mode structure follows a Farey-tree construction.

The dynamics of homogeneously broadened ring lasers supporting two transverse modes is numerically studied. The regimes with periodic and chaotic intensity pulsations have been obtained for lasers having both fast and slow-response relaxation of the population inversion. A phenomenon of metachaos has been found and statistical properties of metastable chaotic states and their dependence on system parameters have been studied.

This paper presents a classical approach to controlling the Lorenz equations, a well known chaotic system. It is shown that proportional-plus-integral control using an easily measurable state variable gives both good stability and tracking properties well into the normally chaotic region. Here the input signal to the Lorenz equations is the applied heat via the Rayleigh number. This paper demonstrates that widely used classical control approaches can stabilize a chaotic system and have it track input signals in its usually chaotic regime.

In this paper we present two generalizations of the equations governing Chua’s circuit. In order to obtain the first generalization we simplify Chua’s equations by replacing the piecewise-linear term with a signum function. The resulting simplified system produces a double scroll similar to the one observed experimentally in Chua’s circuit. What is significant about this simplified system is that it can be reduced to what we shall call a two-dimensional single scroll, and from the two-dimensional single scroll we are able to derive a one-dimensional map. This entire derivation is carried out analytically, in contrast to the one-dimensional map analysis that has been carried out for the Lorenz equations which is based on axioms.

After we have carried out our analysis for this simplified version of Chua’s equations, we use these equations as a guide to the construction of the first generalization to be presented in this paper. We call this a type-I generalization of Chua’s equations. The generalization consists in using a two-dimensional autonomous flow as a component in a three-dimensional autonomous flow in such a way that the resulting equations will have double scroll attractors similar to those observed experimentally in Chua’s circuit.

The value of this generalization is that: (1) it provides a building block approach to the construction of chaotic circuits from simpler two-dimensional components which are not chaotic by themselves. In so doing it provides an insight into how chaotic systems can be built up from simple nonchaotic parts; (2) it illustrates a precise relationship between three-dimensional flows and one-dimensional maps. Of particular significance in this regard is a recent paper of Misiurewicz [1993], which analytically connects the two-dimensional single scroll to the class of unimodal maps, thus providing a framework within which a theory linking unimodal maps to three-dimensional flows may be possible.

The second generalization is suggested by considering three-dimensional flows whose only nonlinearities are sigmoid, sgn, or piecewise-linear functions. Clearly, such flows are a generalization of the Chua equations. We call these equations type-II generalization Chua equations. The significance of this direction of investigation is that attractors similar to the Lorenz and Rössler attractors can be produced from type-II generalized Chua equations in a building block approach using only piecewise-linear vector fields. As a result we have a method of producing the Lorenz and Rössler dynamics in a circuit without the use of multipliers. This suggests that the type-II generalized Chua equations are in some sense fundamental in that the dynamics of the three most important autonomous three-dimensional differential equations producing chaos are seen as variations of a single class of equations whose nonlinearities are generalizations of the Chua diode.

Bifurcations in two-parameter families containing a dynamical system having a heteroclinic orbit connecting type-a saddle equilibrium points are investigated. The existence of a countable set of bifurcation curves corresponding to saddle connections is discussed.

Pattern formation and switching between self-organized states are often associated with instabilities in open, nonequilibrium systems. We describe an experiment which shows that systematically changing a control parameter induces qualitative changes in sensorimotor coordination and brain activity, as registered by a 37-SQUID (Superconducting Quantum Interference Device) array. Near the instability point, predicted features of nonequilibrium phase transitions (critical slowing down, fluctuation enhancement) are observed in both the psychophysical data and the brain signals obtained from single SQUID sensors. Further analysis reveals that activity from the entire array displays spatial patterns evolving in time. Such spatiotemporal patterns are characterized by the dynamics of only a few coherent spatial modes.

Large discrete arrays of chaotic attractors, coupled by diffusion, organize into asynchronous periodic spiral waves, synchronous periodic bands, turbulent fields or synchronous chaos as a function of coupling strength and array size. Self-organization of periodic spirals in both two and three dimensional arrays of nonexcitable systems appears to require the early establishment of an antipodal phase relationship between the few cells that will form the vortex. Cells within or close to the vortex maintain low z amplitude, near limit cycle trajectories, with stable, well-defined phase relationships. In periodic banding structures, initial antipodal phase seeds evolve to isochrons that form nested periodic trajectories. The likelihood that biological systems are fundamentally oscillatory and chaotic is discussed.

By using analytical and numerical methods the authors study one of the basic models of mathematical physics—the so-called complex Ginzburg-Landau equation

The transmission of digital signals by means of chaotic synchronization is demonstrated, numerically as well as experimentally, via Chua’s circuit.

The appearance of a hole in the two-dimensional probability density of a bistable system excited by colored noise is investigated. To analyze the phenomenon, a cumulant analysis is used. In the Gaussian approximation, it was shown that the hole formation corresponded to the bifurcation “saddle-node-repeller” of the state of equilibrium in the system of cumulant equations.

An experimental model of a simple pendulum, harmonically shaken, displays chaotic dynamics. Moreover, in strongly excited chaotic regimes the time series of total angular displacement, which is rarely examined, wanders unboundedly, displaying a power spectrum which falls off as 1/f^α over several decades. This behavior corresponds to deterministic diffusion, which has been found in simulations of nonlinear maps with periodic translational symmetry. The displacement time series obtained by sampling the pendulum displacement once per cycle is self-affine and quantitatively similar to Brownian motion.

This paper deals with the role of neural-network based prediction for the modeling of nonlinear dynamical systems. We show experimentally that the backpropagation learning rule to train neural networks and the prediction error, so widely utilized in teaching and comparing nonlinear predictors, do not consistently indicate that the neural network based model has indeed captured the dynamics of the system that produced the time series. Frequently, but not always, the neural network when used as an autonomous system in a feedback configuration was able to generate a time series that has dynamical invariants similar to the original time series. We show that the estimation of the dynamical invariants (correlation dimension, largest Lyapunov exponent) of the predicted and original time series are an appropriate tool to validate the predictive model.

In this work, an example of the difficulties that may arise when exploring dynamical systems with a digital computer, is presented. It is the case of a linear piecewise function with a period −3 orbit. Computational simulations move us to believe that this orbit is an attracting one, but a theoretical analysis demonstrates that what the computer shows is not, in this case, the real behavior of the system.

Singularities with two multipliers, S₁=−S₂=1, organized in cascades by period doubling, are encountered in m-dimensional diffeomorphisms, m>2. It is shown that endormorphisms with m=2 give rise to singularities equivalent to those of diffeomorphisms with m>2.

We present a new three-dimensional autonomous chaotic dynamical system that appears to have a closer relationship to turbulence than the Lorenz system. We have developed this system using the new technique of dynamical synthesis.

Experimental evidence showing the existence of signal amplification via perturbation of periodic and chaotic orbits in an electronic circuit is presented.