International Journal of Bifurcation and Chaos

Finding a natural meeting ground between the highly developed complexity theory of computer science — with its historical roots in logic and the discrete mathematics of the integers — and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis — with its rich history and longstanding traditions in the continuous mathematics of analysis — presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution.

Let Ω be a bounded planar domain, its boundary ∂Ω a closed curve of fractal dimension d_f ∈ (1, 2). In Piacquadio [1992] we approximated such ∂Ω by closed ∂Ω^p curves, p ∈ ℕ, p → ∞. In this paper we deal with the pathological case d_f = 2 — i.e. the so-called "plane filling" ∂Ω. Such ∂Ω are both frequent and important in physics. We study a variety of planar tilings, corresponding to crystalline and quasicrystalline configurations, by means of approximating curves ∂Ω^p to different plane filling ∂Ω. We give some applications to elliptic partial differential equations (P.D.E.) and to the study of crystalline and quasicrystalline matter structure.

We construct solutions g of the functional equation g^m-1 (g(x)) = f(x), m ∈ N*, m > 1, f is a given map. The set of definition of f is strictly included in the set of definitions of these solutions.

We study the behavior of the solutions in a neighborhood of a closed contour formed by two heteroclinic connections to two equilibrium points of saddle-focus type. We consider both the three-dimensional case, as in the well-known Chua's circuit, as well as the infinite-dimensional case.

A general model of the neuron-like medium with a selfconsistent excitation threshold is considered. The necessary and sufficient conditions of stability of static collective-activity structures in this model are found. The structure bifurcations which are possible by additional inertial inhibition introduced by a local excitation threshold are analyzed using a linear approach.

The correspondence between a general model of the neuron-like medium with a self-consistent activation threshold and a broad class of active two-component ("activator-inhibitor") diffusion systems is established. The transformation of static collective-activity structures into dynamical (propagating and pulsating) structures is investigated for two types of active systems with positive and negative diffusion.

The mutual information and the redundancy are used in the time series analysis as tools to determine the best possible time-lag for successful phase space reconstruction. The known methods exhibit difficulties and inaccuracy when applied to noisy and small data sets. In this paper, we will introduce two methods to make a more reliable determination of the redundancy possible. Firstly, we will modify the Θ-function used in the correlation intergrals and, secondly, we will use the "Neural Gas" algorithm to calculate reference vectors and local radii, which enter the redundancy calculations. A clear differentiation between the redundancy and the mutual information on information theoretical background provides the path along which the new methods can be introduced and applied to the Mackey–Glass and Lorenz systems.

Investigation of non-Feigenbaum types of period-doubling universality is undertaken for a single Chua's circuit and for two systems with a unidirectional coupling. Some codimension-2 critical situations are found numerically that were known earlier for bimodal 1D maps. However, the simplest of them (tricritical) does not survive in a strict sense when the exact dynamical equations are used instead of the 1D map approximation. In coupled systems double Feigenbaum's point and bicritical behavior are found and studied. Scaling properties that are the same as in two logistic maps with a unidirectional coupling are illustrated.

We provide several pieces of evidence for possible chaotic dynamics in the irregular behavior of normal speech signals of the Japanese vowel /a/. First, principal component analysis demonstrates that a simple geometric structure underlying the complex speech signal is well reconstructed in a three-dimensional delay-coordinate space. Observations of the reconstructed speech trajectory at multiple cross sections also display speech dynamics with stretching, folding and compressing. Second, Lyapunov spectrum analysis indicates sensitive dependence on initial conditions with a positive Lyapunov exponent for the speech signals of several different speakers. Third, nonlinear modeling analysis with an artificial neural network shows that the nonlinear dynamics of the vowel sound is well reproduced by a deterministic dynamical model.

We will develop new methods to determine the topology of the basin boundary in a class of three-dimensional dynamical systems. One approach is to approximate the basin boundary by backward integration. Unfortunately, there are dynamical systems where it is hard to approximate the basin boundary by a numerical backward integration algorithm. We will introduce topological methods which will provide new information about the structure of the basin boundary. The topological invariants which we will use can be numerically computed.

In this paper, conditions for effective local control of nonlinear, chaotic dynamical systems are analyzed. The aim is to determine the size of a neighbourhood of a fixed point in which the system remains stable when subjected to a linear control law. The stability margins are obtained using ideas from functional analysis, and ways of implementing the obtained results in practice are proposed. An implementation of obtained conditions is illustrated on a real application.

The effects of both hard and soft nonlinearities are examined in the frequency domain. Softening the hard nonlinearity in Chua's diode has a similar effect to low pass filtering or reducing the level of high frequency noise components.

We define the topological winding number for unimodal maps that share the essential properties of that of winding numbers for forced oscillators exhibiting period-doubling cascades. It is demonstrated how this number can be computed for any of the periodic orbits in the first period-doubling cascade. The limiting winding number at the accumulation point of the first period-doubling cascade is also derived. It is shown that the limiting value for the winding number ω_∞ can be computed as the Farey sum of any two neighbouring topological winding numbers in the period-doubling cascade. The derivations are all based on symbolic dynamics and simple combinatorics.

In this letter, we study the use of adaptive controllers to maintain the synchronization of two Chua's oscillators when the channel and circuit parameters are time-varying. We present both computer simulation results and physical experimental results to verify the operation of the designs.

This paper shows an abstract computational system with interesting properties concerning the emulation of a learning process by time evolution, using an unsupervised neural net, called "Autoreflexive Monoconnected Neural Net". We show how this particular unsupervised neural network can be used to simulate chaotic activities related to learning processes, which have been observed in many physiological experiments. We demonstrate how this chaos activity is distributed inside the net system, leading to the generation of synchronous oscillations with different amplitudes and frequencies, consistent with the coherent and rhythmic activity of large numbers of neurons in the brain. These findings suggest that such local oscillations are important for information processing. The results suggest that the learning process is composed of different phases, which are characterized by interspersed chaotic and periodic behaviors. Small perturbations by changing net parameters can be used to control chaotic activity, and to stabilize unstable periodic orbits, offering a new approach for the learning process control.

In this letter we show that the monotone synchronization is possible for chaotic systems and can be easily obtained using the method of continuous chaos control. We also discuss possible applications of this type of synchronization for secure communications.