International Journal of Bifurcation and Chaos

In this paper we present a panoramic view of numerical simulations associated with nonlinear wave equations which appear in different experimental contexts. Mainly, we deal with scalar wave equations, but also the Maxwell equations in nonlinear media are studied. A basic part of this work is devoted to the construction and verification of numerical schemes on a physical basis. The stochastic perturbations of scalar wave equations are especially analyzed by analytical and numerical approaches. Also, other kinds of perturbations are considered, like nonlocal ones. Finally, a summary of promising experimental results from the numerical simulations of the Maxwell system in a nonlinear media is presented.

The focus of the paper is mainly on the existence of limit cycles of a planar system with third-degree polynomial functions. A previously developed perturbation technique for computing normal forms of differential equations is employed to calculate the focus values of the system near equilibrium points. Detailed studies have been provided for a number of cases with certain restrictions on system parameters, giving rise to a complete classification for the local dynamical behavior of the system. In particular, a sufficient condition is established for the existence of k small amplitude limit cycles in the neighborhood of a high degenerate critical point. The condition is then used to show that the system can have eight and ten small amplitude (local) limit cycles for a set of particular parameter values.

Database Tomography (DT) is a textual database analysis system consisting of two major components: (1) algorithms for extracting multiword phrase frequencies and phrase proximities (physical closeness of the multiword technical phrases) from any type of large textual database, to augment (2) interpretative capabilities of the expert human analyst. DT was used to derive technical intelligence from a Nonlinear Dynamics database derived from the Science Citation Index/Social Science Citation Index (SCI). Phrase frequency analysis by the technical domain experts provided the pervasive technical themes of the Nonlinear Dynamics database, and the phrase proximity analysis provided the relationships among the pervasive technical themes. Bibliometric analysis of the Nonlinear Dynamics literature supplemented the DT results with author/journal/institution publication and citation data.

In this paper a new integrated Chua's circuit is implemented through switched-capacitor techniques. Recent studies on complex systems made of vast arrays or matrices of Chua's cells motivate the need for experimental verifications which could be easily accomplished by an integrated version of the array. Design efforts have been devoted to the reduction of the chip area consumption. The cell layout, realized in CMOS technology, is presented and experimental results confirm the validity of our chip implementation. The implementation presented allows the user to generate a wide gallery of complex behaviors. The system is in fact programmable: both the chaotic attractor generated by the single cell and the frequency range of the chaotic oscillations can be easily set via external control pins.

We study a model equation that mimics convection under rotation in a fluid with temperature-dependent properties (non-Boussinesq (NB)), high Prandtl number and idealized boundary conditions. It is based on a model equation proposed by Segel [1965] by adding rotation terms that lead to a Küppers–Lortz instability [Küppers & Lortz, 1969] and can develop into oscillating hexagons. We perform a weakly nonlinear analysis to find out explicitly the coefficients in the amplitude equation as functions of the rotation rate. These equations describe hexagons and oscillating hexagons quite well, and include the Busse–Heikes (BH) model [Busse & Heikes, 1980] as a particular case. The sideband instabilities as well as short wavelength instabilities of such hexagonal patterns are discussed and the threshold for oscillating hexagons is determined.

We consider the dynamics of impact oscillators with multiple degrees of freedom subject to more than one motion limiting constraint or stop. A mathematical formulation for modeling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom system are considered, along with definitions of the impact map for multiply constrained systems. We consider sticking motions that occur when a single mass in the system becomes stuck to an impact stop, and discuss the computational issues related to computing such solutions. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. Numerical examples of sticking orbits for this system are shown and we discuss identifying the region, S in phase space where these orbits exist. We use bifurcation diagrams to indicate differing regimes of vibro-impacting motion for two different cases; firstly when the stops are both equal and on the same side (i.e. the same sign) and secondly when the stops are unequal and of opposing sign. For these two different constraint configurations we observe qualitatively different dynamical behavior, which is interpreted using impact mappings and two-dimensional parameter space.

Noise-free stochastic resonance is investigated in two chaotic maps with periodically modulated control parameter close to a boundary crisis: the Hénon map and the kicked spin model. Response of the maps to the periodic signal at the fundamental frequency and its higher harmonics is examined. The systems show noise-free stochastic multiresonance, i.e. multiple maxima of the signal-to-noise ratio at the fundamental frequency as a function of the control parameter. The maxima are directly related to the fractal structure of the attractors and basins of attraction colliding at the crisis point. The signal-to-noise ratios at higher harmonics show more maxima, as well as dips where the signal-to-noise ratio is zero. This opens a way to use noise-free stochastic resonance to probe the fractal structure of colliding sets by a method which can be called "fractal spectroscopy". Using stochastic resonance at higher harmonics can reveal smaller details of the fractal structures, but the interpretation of results becomes more difficult. Quantitative theory based on a model of a colliding fractal attractor and a fractal basin of attraction is derived which agrees with numerical results for the signal-to-noise ratio at the fundamental frequency and at the first two harmonics, quantitatively for the Hénon map, and qualitatively for the kicked spin model. It is also argued that the maps under study belong to a more general class of threshold-crossing stochastic resonators with a modulated control parameter, and qualitative discussion of conditions under which stochastic multiresonance appears in such systems is given.

We give a complete description of the dynamics of the mapping f_ε(z)=z²+(ε/z) for positive real values of ε. We then consider two generalizations: the case of complex ε and the mapping z→zⁿ+(ε/z^m), where ε is positive and real. In both cases we provide a full characterization of the map for a certain set of parameters, and give observations based on numerical evidence for all other parameter values. The dynamics of all maps that we consider bears striking resemblance to that of complex quadratic maps.

We consider in this paper the problem of stability and duration of the synchronization process between two electromechanical devices, both in their regular and chaotic states. Stability boundaries are derived through Floquet theory. The influence of the precision on the synchronization time is also analyzed using numerical simulation of the equations of motion.

In a multiple access spread-spectrum communication system, the bit error performance is usually limited by the mutual interference between users. The use of multi-user detection is an effective method to reduce such interference. In this paper, we consider two linear multi-user detectors, namely, decorrelating detector and minimum mean-square-error detector, applied to a multiple access chaotic-sequence spread-spectrum communication system. An approach to calculate the bit error rate (BER) for such systems is presented. The calculated BERs are compared with those obtained using brute-force computer simulations which give the true system performance.

We present and illustrate a feedback control-based framework that enables microscopic/stochastic simulators to trace their "coarse" bifurcation diagrams, characterizing the dependence of their expected dynamical behavior on parameters. The framework combines the so-called "coarse time stepper" and arc-length continuation ideas from numerical bifurcation theory with linear dynamic feedback control. An augmented dynamical system is formulated, in which the bifurcation parameter evolution is linked with the microscopic simulation dynamics through feedback laws. The augmentation stably steers the system along both stable and unstable portions of the open-loop bifurcation diagram. The framework is illustrated using kinetic Monte Carlo simulations of simple surface reaction schemes that exhibit both coarse regular turning points and coarse Hopf bifurcations.

In this paper, we present the hyperchaos dynamics of a modified canonical Chua's electrical circuit. This circuit, which is capable of realizing the behavior of every member of the Chua's family, consists of just five linear elements (resistors, inductors and capacitors), a negative conductor and a piecewise linear resistor. The route followed is a transition from regular behavior to chaos and then to hyperchaos through border-collision bifurcation, as the system parameter is varied. The hyperchaos dynamics, characterized by two positive Lyapunov exponents, is described by a set of four coupled first-order ordinary differential equations. This has been investigated extensively using laboratory experiments, Pspice simulation and numerical analysis.

A secure transmission application of the State Controlled Cellular Neural Network (SC–CNN)-based circuit is presented. Since the SC–CNN-based circuit has feedback connections between the cells, it is potentially very suitable for realizing a chaotic masking system with feedback algorithm. So, we have constructed a chaotic masking system with feedback using the SC–CNN-based circuit. PSpice simulation experiments verify that the proposed SC–CNN-based secure communication system exhibits a good performance for a wide range of amplitude and spectral characteristics of the information signal.

We study the dynamics of a kink propagating in a Nagumo chain presenting a geometrical bifurcation. In the case of weak couplings, we define analytically and numerically the coupling conditions leading to the pinning of the kink at the bifurcation site. Moreover, real experiments using a nonlinear electrical lattice confirm the theoretical and numerical predictions.

We propose a computational model explaining the pharmacological, behavioral, and neurosurgical treatments of obsessive–compulsive disorder (OCD) in a unified framework. On the basis of the concept that the striatum stores information linked to OCD symptoms, we analyze dynamical changes of a chaotic neural network modeling the striatum. The effect of pharmacological treatment is explained by the increase in refractoriness induced by the neurotransmitter serotonin. The effect of behavior therapy is explained by the time-variant input stimulus weakening neural interconnections according to the covariance rule concerning synaptic plasticity. The effect of neurosurgical treatment is explained by the lesions to the cortical-striatal-thalamic-cortical loop decreasing the feedback of the closed loop. These treatments have a common property that they decrease the stability of stored information in the striatum.

In this paper, the problem of making a nonlinear system chaotic by using state-feedback control is studied, where the feedback controller uses a simple sine function of the system state with only one single component in each dimension. It is proved that the designed control system generates chaos in the sense of Li and Yorke.

In this paper, an online chatting system with an embedded real-time chaotic encryption/decryption method is designed for the Internet. Such system not only provides a real-time communication platform, but also ensures a secure channel for communication. By the use of cipher feedback and the skew tent map, the input text can be real-time encoded and the cipher text is sent via TCP/IP. With the properties of randomness of the map, and its sensitivity on system parameters and the initial conditions, the encrypted transmitting messages are difficult to be eavesdropped. The implemented method is simple and can be easily embedded in any existing systems.

In this paper, the theory of Lyapunov–Andronov is applied to investigate the route to chaos in Rossler system. On the base of a new analytical formula for the first Lyapunov value at the boundary of stability region, we make a detailed bifurcation analysis of this system. From the obtained results the following new conclusions are made: Transition to chaos in the Rossler's system takes place at soft stability loss in the form of a cascade of periodic self-oscillations. Then the occurrence of chaotic self-oscillations in this system takes place under hard stability loss.

The aim of this paper is to derive a sharper lower bound for the spatial entropy of two-dimensional golden mean.

Let f denote a transcendental entire function, and I(f), I₀(f), T(f) and A(f) be denoted as follows:

In this paper we study the relationship between the asymptotic behavior of a numerical simulation by linear multistep method and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of linear multistep method for nonlinear delay differential equation is investigated and we prove that A-stable linear multistep methods are NP-stable. It is shown that a consistent zero-stable linear multistep method does not admit spurious fixed points. The existence of spurious period-two solutions in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period-two solutions.

The dynamical behavior of two coupled parametrically excited van der Pol oscillators is investigated in this paper. A special road to chaos is explored in detail. Period-doubling bifurcation associated with one of the frequencies of the system may be observed, the other frequency of the coupled oscillators plays a role in the evolution. It is found that one of the frequencies of the system contributes to the cascade of period-doubling bifurcations associated with the other frequency, which leads to a generalized modulated chaos.

In this paper we develop a new approach to generally synchronize Genesio–Tesi systems. A suitable scalar signal of the master, Genesio–Tesi system, is transmitted to the slaver, which is derived from an iteration transformation of a Genesio–Tesi system. If a parameter is chosen properly, the stability of synchronization manifold becomes Hurwitz stability. Numerical simulation verifies the effectiveness of this method.