International Journal of Bifurcation and Chaos

A general approach of the automated algorithms for derivation of amplitude equations for dynamical and distributed systems is presented. A brief description of the Mathematica functions designed using this approach is given. Application of the functions is illustrated by several examples dealing with dynamical and distributed systems.

A system of nonautonomous differential equations having Chua's piecewise-linearity is studied. A brief discussion about equilibrium points and their stability is given. It is also extended to obtain a system showing "multispiral" strange attractors, and some of the fundamental routes to "multispiral chaos" and bifurcation phenomena are demonstrated with various examples. The same work is done for other systems of autonomous or nonautonomous differential equations. This is achieved by modifying Chua's piecewise-linearity in order to have additional segments. The evolution of the dynamics and a mechanism for the development of multispiral strange attractors are discussed.

In a programmable (multistage) cellular neural network (CNN) structure, the CPU is a CNN universal chip which supports massively parallel computations on patterns and images, including videos. In this paper, we decompose the structure of a class of simultaneous recurrent networks (SRN) into a CNN program and run it on a von Neumann-like stored program CNN structure. To train the SRN, we map the back-propagation-through-time (BTT) learning algorithm into a sequence of CNN subroutines to achieve real-time performance via a CNN universal chip. By computing in parallel, the CNN universal chip can be programmed to implement in real time the BTT learning algorithm, which has a very high time complexity. An estimate of the time complexity of the BTT learning algorithm based on the CNN universal chip is presented. For small-scale problems, our simulation results show that a CNN implementation of the BTT learning algorithm for a two-dimensional SRN is at least 10,000 times faster than that based on state-of-the-art sequential workstations. For the few large-scale problems which we have so far simulated, the CNN implemented BTT learning algorithm maintained virtually the same time complexity with a learning time of a few seconds, while those implemented on state-of-the-art sequential workstations dramatically increased their time complexity, often requiring several days of running time. Several examples are presented to demonstrate how efficiently a CNN universal chip can speed up the learning algorithm for both off-line and on-line applications.

The human voice source generates complex signals including subharmonics and toroidal oscillations. Essential features of voice production are covered by two-mass models where each vocal fold is represented by two oscillators. We present a related analog circuit which allows online analysis of the generated complex signals. Using narrow-band spectrograms various bifurcations due to left–right asymmetry are monitored. Time series, spectra, return maps, and response measurements provide further detailed information about phase-locking and toroidal oscillations. Finally, possible clinical applications are discussed.

A frequency domain approach to the bifurcation analysis of time-delay SISO control systems with saturation is proposed. Generic bifurcations to be expected for these systems are emphasized. The main goal is to give a classification of system behavior modes that are relevant for stability studies, reaching a global perspective of the parameter space. The classification is obtained through the bifurcation analysis, which is implemented by frequency domain methods that are well known to the control systems engineers. Some specific cases are analyzed with this methodology. The theoretical results are in agreement with numerical simulations.

In this paper we study the dynamics of a charged particle in a constant external magnetic field and the field due to a polarized electromagnetic wave. We observe a new phenomenon, the chaotic gun effect, which appears in the case of a sufficiently large amplitude of the wave. We show that the longitudinal component of the momentum undergoes oscillations with a chaotic modulation of the amplitude which increases suddenly to a value which remains constant between two consecutive phase Larmor circles. The novel effect may be modeled by a time-regressive system associated with an Ikeda map depending on a monotonically time-varying parameter.

In the paper we propose a method of constructing cryptosystems, utilizing a nonpredictability property of discrete chaotic systems. We point out the requirements for such systems to ensure their security. The presented algorithms of encryption and decryption are based on multiple iteration of a certain dynamical chaotic system coming from gas dynamics models. A plaintext message specifies a part of the initial condition of the system (a particle's initial position). A secret key specifies the remaining part of initial condition (the particle's initial angle) as well as a sequence of discrete choices of the pre-images in the encryption procedure. We also discuss problems connected with the practical realization of such chaotic cryptosystems. Finally we demonstrate numerical experiments illustrating the basic properties of the proposed cryptosystem.

Devaney [1991] and Devaney and Durkin [1991] exhibited the chaotic burst in the dynamics of certain critically finite entire transcendental functions such as λexp z and iλ cos z. The Julia set of critically finite entire function E_λ(z)=λ e^z for 0<λ<(1/e) is a nowhere dense subset entirely contained in the right half-plane, while it explodes and equals to the extended complex plane for λ>(1/e), a phenomena referred to as chaotic burst in the dynamics of functions in one parameter family ℰ≡{E_λ: E_λ(z)=λ exp z, λ>0}. In the present work, a class of noncritically finite entire functions is introduced and the dynamics of functions in one parameter family generated from each function g(z) in the class is investigated. It is proved that there exists a real number such that bifurcation in the dynamics of functions occurs at . Further, it is established that chaotic burst occurs in the dynamics of noncritically finite functions in one parameter family at the parameter value . Finally, certain interesting examples of the family , viz. , where is the well-known modified Bessel function of zero order and , where with fixed k=1, 2,… and are provided.

The response of a chaotic Chua's electronic circuit and that of an atom show interesting common features. Starting from the comparison of the qualitative behavior of bifurcation diagrams and electronic spectroscopic series of atoms we associate order–chaos transition with those of an electron passing from a quantized to a continous energy spectrum. The resulting atomic model suggests an analogy between the synchronization of two Chua's atoms and chemical covalent bonding.

A model of coupled nonlinear oscillators is discussed, wherein Langevin-type bistable systems are combined with self-sustained oscillators. An external harmonic signal is coupled in a subthreshold manner into the bistable systems at the initial stage of the signal chain. Signal transfer through the oscillators is studied under the influence of noise. Different noise contributions, including spatially-incoherent and spatially-coherent noise sources are considered. Results reveal a stochastic resonance kind of behavior at different stages of the signal transfer, specifically the harmonic signal is transduced through the whole system of coupled oscillators. The combined action of spatially-incoherent and spatially-coherent noise exhibits constructive as well as destructive influences on signal amplification.

A new criterion is given for choosing the coupling constant in a system of coupled chaotic oscillators to guarantee their synchronization. The criterion is derived from a new observer design methodology based on Lyapunov stability theory. As an example and application, we prove the conjecture that synchronization of two chaotic Chua circuits can be achieved with the second state as the coupling variable provided that the coupling constant is suitably chosen according to the new criterion.

In this Letter a method for synchronizing hyperchaotic systems described by ordinary differential equations (ODEs) as well as functional differential equations (FDEs) is illustrated. The technique, which exploits the concept of observer from system theory, is at first applied for synchronizing an example of high-order circuit. A remarkable feature is that hyperchaos synchronization is achieved via a scalar signal. Successively, the tool is applied for synchronizing an example of high-dimensional chaos generator, represented by a cell equation in Cellular Neural Networks with delay. Finally, the advantages of the proposed approach are discussed in detail.

A method of secure communication based on the inversion of a chaotic dynamical system is analyzed. The approach does not require chaotic synchronization of two or several oscillators to be used. The influence of noise in the communication channel and parameter mismatch on the quality of decoded signal is discussed. The feasibility of the approach is demonstrated by the examples of nonautonomous pendulum equation and autonomous Rössler oscillator.

Discrete planar maps such as x_n+1=f(x_n)+μ g(x_n, μ), x_n∈ℝ², n∈ℤ, μ∈ℝ, are studied under the assumption that the unperturbed map x_n+1=f(x_n) has a critical homoclinic orbit to a hyperbolic fixed point. We give either necessary and sufficient conditions for a bifurcation from zero to two homoclinic orbits as the small parameter μ crosses zero. These conditions are stated in terms of geometrical objects such as critical lines, stable and unstable manifolds.

The effect of considering previous results (history) in the spatial formulation of the Prisoner's Dilemma is assessed. It is concluded that history protects homogeneous populations against the irruption of mutant behavior.

A novel method of binary data description using cellular automata is proposed. As an actual example, several trials are made to describe vocal and musical sound data digitized in a standard data format. The reproduced sounds are evaluated by "actual listening", by calculating "the signal to noise ratio" with respect to the original sound data, and by comparing "the Fourier power spectra" with those of the original sounds. The results show that this method is quite effective and provides a new means of data compression applicable in any field of digital data recording or transferring.

The goal of this letter is to report a novel class of dynamical behaviors observed from a generalized cellular automata CNN [Chua, 1998] with piecewise-linear (PWL) cells. Starting from an almost homogeneous initial condition, self-making (autopoietic in the sense of [Varela et al., 1974]) patterns, reminiscent of simple living systems, emerge as a result of the nonlinear coupling among cells. Similar to patterns of organization characterizing living systems, our patterns display features such as growth, maturity and death. The discovery of such patterns was made possible via mutations in several piecewise-linear CNN cell realizations of the "Game of Life" [Conway, 1982].