International Journal of Bifurcation and Chaos

This paper presents a method to compute the normal forms of differential equations whose Jacobian evaluated at an equilibrium includes a double zero or a triple zero eigenvalue. The method combines normal form theory with center manifold theory to deal with a general n-dimensional system. Explicit formulas are derived and symbolic computer programs have been developed using a symbolic computation language Maple. This enables one to easily compute normal forms and nonlinear transformations up to any order for a given specific problem. The programs can be conveniently executed on a main frame, workstation or a PC machine without any interaction. Mathematical and practical examples are presented to show the applicability of the method.

This paper present an application of the local activity theory [Chua, 1998] to a specific reaction–diffusion cellular nonlinear network (CNN) with cells defined by the model of morphogenesis first proposed in [Gierer & Meinhardt, 1972]. Both the local activity domain and a subset called the "edge of chaos" are identified in the cell parameter space. Within these domains, various cell parameter points were selected to illustrate the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

Methods borrowed from nonequilibrium thermodynamics and statistical physics have been employed in the quantitative analysis of spatiotemporal chaos in a 2D coupled map lattice (CML). Emphasis is made on entropy, entropy variation and entropy production. These quantities manifest peculiar changes in a region where spiral waves emerge. The spiral waves observed in the 2D CML are found to be dissipative objects with an elevated entropy production.

We consider a lattice of coupled van der Pol systems with external periodic forces and Dirichlet boundary conditions. Under the assumption of bounded dissipativeness, we prove that the asymptotic synchronization occurs provided that the coupling coefficients are sufficiently large. Furthermore, we demonstrate from numerical simulations that asymptotic synchronization is affected by the coupling coefficients and the size of lattice.

In this paper, we study an excitable and nonoscillating neuron on the basis of a technique of extending the concept of isochrons from oscillatory to excitable systems. The extended isochrons allow reduction of an excitable system described by possibly high dimensional differential equations to a simpler system. We analytically derive a one-dimensional model of an excitable neuron stimulated by instantaneous pulses with the technique of the extended isochrons and show its similarity to an isochronal map numerically obtained from the FitzHugh–Nagumo model. Response characteristics of the one-dimensional model to periodic impulsive stimulations are also analyzed numerically.

A method for controlling nonlinear dynamics based on avoiding homoclinic intersection is systematically implemented to perform a numerical analysis of the control induced modifications of the steady attractors and bifurcation scenario of a two-well impact system. The work is divided into two parts. This paper (Part I) deals with the analysis of the harmonic (reference) and global optimal excitations, which are both symmetric. The bifurcation diagrams obtained for increasing values of the excitation amplitude show there exist a "basic" attractor and other "complementary" solutions. The range of stability of the principal complementary attractors is numerically established, and the mechanisms leading to their disapperance are identified. The role of classical and nonclassical local bifurcations in determining the system response is emphasized. Chaotic attractors are seen to appear and disappear both by classical period doubling route to chaos and by sudden changes. Subductions, boundary and interior crises are repeatedly observed. By comparison of the system response under different excitations we obtain information on the performances of global control, which furnishes relatively low gain in terms of regularization but succeeds in controlling the whole phase space.

To exploit all potentialities of the optimal control procedure, the analysis initiated in Part I focuses on the system response under one-side control, an excitation which furnishes high gain though, roughly, controlling only one part of the phase space. Many bifurcational and control items related to the unsymmetric and pulsed nature of the excitation are deeply investigated. A nonclassical kind of homoclinic bifurcation is identified and it is discussed how it may lead to major regularity. The system response is very rich, and the main local and global phenomena of the dynamics are analyzed in detail through combined use of bifurcation diagrams and attractor-basin phase portraits. Both the confinement of steady dynamics in the controlled potential well and their successive transition from confined to scattered are studied, and it is discussed how they are obtained through comparison with the case of harmonic excitation. It is shown that the two investigated optimal excitations permit to increase the amplitude level for confined to scattered dynamics and to regularize the steady dynamics, although in a different manner. The analysis shows the effectiveness — in "average" sense — of the proposed method for controlling nonlinear dynamics.

Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.

In this paper we present a theory for control of chaotic systems using sampled data. The output of the chaotic system is sampled at a given sampling rate and the sampled output is used by a feedback subsystem to construct a control signal, which is held constant by a holding subsystem. Hence, during each control iteration, the control input remains unchanged. Theoretical results on the asymptotic stability of the resulting controlled chaotic systems are presented. Numerical experimental results via Chua's circuit are used to verify the theoretical results.

This work is concerned with linear control systems subjected to saturated feedback. A first harmonic approach is used to describe the existence of nonsymmetric periodic orbits in a three-dimensional control system. By taking a high-gain parametrization of the feedback control, the presence of nonsymmetric (first harmonic) periodic orbits is demonstrated for certain values of the parameter. Since it is also shown that nonsymmetric periodic orbits do not exist for small and large values of the parameter, evidences are found of the existence of symmetry breaking bifurcations.

The challenging problems, in the field of control of chaos or of transition to chaos, lie in the domain of infinite-dimensional systems. Access to all variables being impossible in this case and the controlling action being limited to a few collective variables, it will not in general be possible to drive the whole system to the desired behavior. A paradigmatic problem of this type is the control of the transition to turbulence in the boundary layer of fluid motion. By analyzing a boundary layer flow for an ionized fluid near an airfoil, one concludes that active control of the transition amounts to the resolution of an generalized integro-differential eigenvalue problem. To cope with the required response times and phase accuracy, electromagnetic control, whenever possible, seems more appropriate than mechanical control by microactuators.

The present paper proposes a learning control system that automatically stabilizes one-dimensional time-delayed chaotic systems. We give a systematic procedure to design the control system using a few pieces of uncertain information on the chaotic system. Furthermore, this control system can be applied to a technique that moves a stable orbit of nonchaotic systems to coexistent unstable points and stabilizes the moving orbit onto these unstable points. To check the theoretical results, we demonstrate some numerical experiments.

We show that nonlinear, aperiodic oscillatory behavior can be observed in a chloroplast system by measuring the absorption of light. This behavior depends on illumination of the system with light which is absorbed by chlorophyll. The oscillations can be stopped by addition of acetone indicating that chemical reactions are involved. These observations confirm the role of photosynthesis. The effect of stirring and the statistics of the oscillating part of the spectrum suggest that the buoyancy of oxygen bubbles formed during photosynthesis is an important art of the mechanism.

We consider a dissipative mapping derived from a modification of the Chirikov standard mapping. For simplicity, we assume that the dissipative strength is of the order of the square of the perturbing parameter of the conservative model. Under this assumption, we derive an analytical approximation of the solution associated to the dissipative mapping. The equations are explicitly solved up to the order 7 in the perturbing parameter.

Having fixed a frequency ω, a comparison of the associated to the dissipative solutions shows that the two curves coincide for low values of the perturbing parameter, while in most cases they diverge as the breakdown threshold of the invariant curve with rotation number ω is approached.

Bifurcation and chaos in one-cell cellular neural network with delay is investigated. Further controlling of chaotic behavior with suitable additional parameter is studied. Later, the study has been extended by adding one additional cell without delay. Further the dynamics of two coupled homogeneous and hetrogeneous cells with delay is also investigated. The numerical global bifurcaiton analysis for all the systems have also been performed.

A two-dimensional array of diffusively-coupled nonlinear electronic circuits, Chua's cells, operating in a bistable regime, is used to find the shortest path connecting two points of a given image. Images are previously stored in the system by modulating the diffusion coefficients. Different types of images were considered, from black and white pictures (where 0 means no propagation and 1 propagation), to analog figures (with many intermediate states) and, in all cases, the algorithm used for such calculations succeeded to find the shortest path. The full description of the algorithm is here described and applied to nontrivial cases where the shortest path strongly differs from the straight line.