International Journal of Bifurcation and Chaos

We present here an informal, encyclopedic glossary of terms and formulae covering the new geometrical concepts of nonlinear dynamics and chaos. It is arranged as an alphabetical dictionary, which can also be read as a connected introduction to the subject by following the go to instructions starting with the entry dynamical system and ending with archetypal equations.

In this review we survey methods for detecting nonlinearities in stationary time series. These methods are based on the estimation of so-called correlation integrals. These correlation integrals provide a way of analyzing time series and reveal aspects which are often complementary to the information one obtains from power spectra and autocorrelations. So we also focus our attention on the meaning and the estimation of the correlation integrals.

The polynomials P_d,λ(z)≔λ[1+(z/d)]^dconverge uniformly on compact sets to E_λ(z)≔λe^z. What this convergence means for the dynamics of these functions when iterated was first studied in Devaney et al. [preprint]. Here we show the convergence of the corresponding Julia sets in the Hausdorff metric for two cases: (1) for λ such that E_λ has an attracting periodic orbit, in which case its Julia set is a Cantor set of curves, and (2) for λ such that the Julia set of E_λ is the whole plane . Finally, we give the key ideas of the algorithms designed to illustrate this convergence.

The parametrized dynamics of a standard nonlinear model neuron with self-interaction is discussed. For units with a self-excitatory connection a hysteresis effect is observed, and the underlying mechanism is identified as that of a cusp catastrophe. This is true for discrete as well as for continuous dynamics. For the discrete dynamics of self-inhibiting units there appear period-doubling bifurcations from stationary states to stable period-2 orbits.

The occurrence of chaos in recurrent neural networks is supposed to depend on the architecture and on the synaptic coupling strength. It is studied here for a randomly diluted architecture. We produce a bifurcation parameter independent of the connectivity that allows a sustained activity and the occurrence of chaos when reaching a critical value. Even for weak connectivity and small size, we find numerical results in accordance with the theoretical ones previously established for fully connected infinite sized networks. Moreover the route towards chaos is numerically checked to be a quasiperiodic one, whatever the type of the first bifurcation is. In the discussion, we connect these results to some recent theoretical results about highly diluted networks. Hints are provided for further investigations to elicit the role of chaotic dynamics in the cognitive processes of the brain.

Real cellular automata (RCA) are time-discrete dynamical systems on ℝ^N. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝ^N, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates.

We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters.

Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

We present the bifurcation analysis of Chua’s circuit equations with a smooth nonlinearity, described by a cubic polynomial. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory. We emphasize how a combination of these approaches actually works in practice. We compare the dynamics of Chua’s circuit equations with piecewise-linear and with smooth nonlinearity. The dynamics of these two variants are similar, but we also present some differences. We conjecture that this similarity is due to the central role of homoclinicity in this model. We describe different ways in which the type of a homoclinic bifurcation influences the behavior of branches of periodic orbits. We present an overview of codimension 1 bifurcation diagrams for principal periodic orbits near homoclinicity for three-dimensional systems, both in the generic case and in the case of odd symmetry. Most of these diagrams actually occurs in the model. We found several homoclinic bifurcations of codimension 2, related to the so called resonant conditions. We study one of these bifurcations, a double neutral saddle loop.

An efficient method of numerical and qualitative analysis of homoclinic trajectories in Hamiltonian systems (or systems possessing a first integral of the motion) with two degrees of freedom is suggested. It allows us to reduce the problem of finding homoclinic loops as intersections of two-dimensional stable and unstable manifolds of an equilibrium point in the four-dimensional phase space to the problem of finding intersections of one-dimensional manifolds (hereafter referred to as touch-curves) on a certain two-dimensional boundary surface. The application of this algorithm in a number of problems leading to Hamiltonian dynamical systems with two degrees of freedom to the analysis of bifurcations of homoclinic loops occurring under the variation of structural parameters, is illustrated.

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

We study the behavior of nonlinear bistable systems as the parameters are slowly perturbed periodically. Using averaged equations, typical features of the transition from regular to chaotic dynamics are determined in an externally driven Duffing oscillator and in a system consisting of a van der Pol and a linear oscillator coupled resonantly. We find that general features of the transition to chaos in these models are in many respects similar. In both systems considered, chaotic regions, arising due to similar mechanisms, are situated inside or near the regions of bistability.

Recent experimental studies indicate that energy can be transferred from high- to low-frequency modes in structures with weak nonlinearity. In each of these experiments, a high-frequency mode was driven near its natural frequency but the response included a large contribution due to the first mode of the structure. In this paper, an analytical study of the response of a two-degree-of-freedom nonlinear system with widely spaced modes to a simple-harmonic excitation near the natural frequency of its high-frequency mode is presented. This system serves as a paradigm for the interaction of high- and low-frequency modes.

In a parameter plane, crossroad areas and spring areas are two typical organizations of fold and flip bifurcation curves centred at a fold cusp point. Till now only spring areas in a “symmetrical” configuration have been described. This letter introduces another type of spring area for which such a “symmetry” does not exist. It is called a dissymmetrical spring area. When a third parameter is varied, qualitative modifications of the parameter plane are considered, and an example of a two-dimensional diffeomorphism is given.

The dynamics and geometry of multi-armed spirals in the light-sensitive ruthenium-catalyzed Belousov-Zhabotinsky reaction were investigated experimentally. The patterns were generated by a novel technique based on laser illumination, which allows the formation, control and annihilation of spiral waves. The rotation of n-armed spirals (n≤6) is stabilized by an unexcitable disk, created by a green laser beam. After switching off the laser, the central disk region becomes excitable again and individual cores are formed for each spiral arm. A symmetric collision structure appears periodically in the central region. Spiral arms are not attached to specific cores, but continuously passed on to the adjacent one. The geometry of multi-armed spirals is analyzed in polar coordinates and single arms are compared with a simple Archimedian shape.

We study numerically the behavior of a scroll wave in a three-dimensional excitable medium with stepwise heterogeneity, using a FitzHugh Nagumo type model. We find that if the refractory periods in the two homogeneous subregions are sufficiently different, the scroll breaks into two scrolls rotating independently in each part of the medium. Eventually, the faster scroll eliminates the slower one leading to a stationary process, in which the scroll wave surviving in the region of faster recovery acts as a source for planar waves in the region of slower recovery.

We use the equations of motion for a particle moving in one dimension under the action of forces which vary cubically with displacement to discuss differential equations associated with double-periodic Jacobian elliptic functions. We show that analytic solutions for this dynamical system can be given in terms of Jacobian elliptic functions. These periodic functions are natural extensions of the well-known trigonometric functions.

Observation of different sequences of bifurcations in an ion-beam plasma system excited by an external sinusoidal signal is reported. Under the influence of an external periodic oscillation applied to the anode of the source chamber of a double plasma device, periodic windows are observed with the variation of theamplitude of the external signal when the frequency of the applied signal is at the edge of the total suppressed position of the internal oscillation. On the other hand, period subtracting phenomena are observed with the variation of the amplitude of the applied signal, when the frequency of the applied signal is lowered but fixed within the suppression range of the internal oscillation and with low modulation depth.

We describe an effective method for controlling chaos by coupling a main chaotic system to a new but simple system with easily changeable parameters. The method is applied to Duffing’s oscillator (numerical and analytical study) and to Chua’s circuit (experimental study). The effectiveness of our method in controlling noisy systems, as well as an illustrative application in a mechanical and an electrical system, are discussed.

In this letter we demonstrate experimentally how Chua’s circuits can be used to implement a secure communication system. This system is compared to another system proposed earlier. This system has the advantage of transmitting a spread spectrum signal and is also more secure in the sense that better parameter matching is required in order to recover the signal. Furthermore, there is less of a problem with loss of synchronization.

We propose a simple method for the synchronization and the control of chaos. A general qualitative description of the method is presented.