International Journal of Bifurcation and Chaos

Based on a simple circuit model of a tunneling phase logic (TPL) element that is driven by a sinusoidal voltage source and biased by a DC voltage source, we present simulations of operations in cellular nonlinear networks (CNN) that could potentially be used to perform general computations in 2D arrays of simple, locally connected nanoscale devices. Some examples are presented to demonstrate the image computation capability of TPL–CNN. In particular, we use a simple 2D TPL–CNN structure to perform edge detection, image enhancement and image segmentation. Some cellular automata (CA)-like behaviors of our 2D TPL-CNN are also presented.

In this paper, we study the relationship between the standard cellular neural network (CNN) and the resonant tunneling diode (RTD)-based CNN. We investigate the functional and advanced capabilities of a new generation of CNNs that exploit the multiplicity of steady states. We also include in the analysis higher order CNNs. Furthermore, some methods for designing RTD-based CNNs with multiple steady states are presented.

In this paper we present an overview of recently published work on the Faraday disk self-exciting homopolar dynamos. We also extend the analysis of two such coupled self-exciting disk dynamos with linear series motors [Moroz et al., 1998a; Moroz et al., 1998b] to the situation where one or both dynamo subunits incorporate nonlinear motors. We examine the differences in some of the bifurcation transition sequences between the various cases and consider whether the nonlinear quenching of oscillatory solutions can occur [Hide, 1998].

Many problems of engineering and applied science result in the consideration of piecewise-smooth dynamical systems. Examples are relay and pulse-width control systems, impact oscillators, power converters, and various electronic circuits with piecewise-smooth characteristics. The subject of investigation in the present paper is the dynamical model of a constant voltage converter which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations. A specific type of phenomena that arise in the dynamics of piecewise-smooth systems are the so-called border-collision bifurcations. The paper contains a detailed analysis of this type of bifurcational transition in the dynamics of the voltage converter, in particular, the merging and subsequent disappearance of cycles of different types, change of solution type, and period-doubling, -tripling, -quadrupling and -quintupling. We show that a denumerable set of unstable cycles can arise together with stable cycles at border-collision bifurcations. The characteristic peculiarities of border-collision bifurcational transitions in piecewise-smooth systems are described and we provide a comparison with some recent results.

There are some reports about sawtooth oscillations on various systems. The authors constructed the sawtooth oscillator circuit exploiting a Uni-Junction Transistor (UJT), and investigated various nonautonomous-like oscillations induced by an external sinusoidal input. Different from the cases of ordinary differential equation (ODE) systems driven with some periodic oscillation, this input periodically alternates only the threshold of the mode transition. Many systems in various fields of electrical circuits, biology and so on utilize threshold variation or modulation between operational modes, and our system is a very simple representative. Though our circuit behaves as a piecewise-continuous ODE system, we will suggest one-dimensional map depending on the phase of the external sinusoidal input. And we will study the background about the existence of various rational locking mode in the analytic method, numerical investigations and circuit experiments, identifying our system as a discrete dynamical system. This paper reveals not only the existing regions of the respective modes, but also reports about respective itineraries of periodic solutions geometrically. Readers will see that the mode distribution constitutes a Cantor set structure.

One of the applications of two degrees of freedom swing equation system is the transient stability problem of an electrical power system. Using the Liapunov function method, there are many reports on the sufficient conditions of normal operations. However the basin structure of the stable equilibrium point, which corresponds to the normal operating state, is not entirely known. In this paper we will approach this problem by investigating the structure of the separatrix of the corresponding conservative system. This separatrix decomposes the phase space into regions of bound motions and divergent motions. This boundary concerns the invariant manifolds of the closed orbits on the center-manifolds of the saddle-center equilibrium points, which appear under the conservative condition. By numerical simulations, we will confirm in this report that there are transverse homoclinic intersections on these manifolds, whose cross-sectional view is different from a familiar homoclinic structure due to the asymmetric feature of the potential. Such circumstances are not observed, for example, on the well-known Hénon–Heiles system whose saddle-center equilibrium point has a trivial homoclinic loop.

Chaotic attractors containing Shilnikov's saddle-focus homoclinic orbits have been observed in many physical systems. Past and current researches of this type of Shilnikov homoclinic phenomena have focused on the orbit and nearby structures only. In this paper we will look at the role such orbits play in a type of attractor, which arises from one-dimensional return maps at the singular limits of some singularly perturbed systems. Results on symbolic dynamics, natural measures, and Lyapunov exponents are obtained for a sequence of a one-parameter caricature family of such attractors.

It is well known that the dynamics of the Arnol'd circle map is phase locked in regions of the parameter space called Arnol'd tongues. If the map is invertible, the only possible dynamics is either quasiperiodic motion, or phase-locked behavior with a unique attracting periodic orbit. Under the influence of quasiperiodic forcing the dynamics of the map changes dramatically. Inside the Arnol'd tongues open regions of multistability exist, and the parameter dependency of the dynamics becomes rather complex. This paper discusses the bifurcation structure inside the Arnol'd tongue with zero rotation number and includes a study of nonsmooth bifurcations that occur for large nonlinearity in the region with strange nonchaotic attractors.

We present a new probabilistic symmetric key encryption scheme based on the chaotic dynamics of properly designed chaotic systems. This technique exploits the concept of virtual attractors, which are not real attractors of the underlying chaotic dynamics but are created and maintained artificially. Each virtual attractor represents a symbol of the alphabet used to encode messages. The state space is partitioned over the virtual attractors creating clusters of states. The enciphering process randomizes over the set of states mapped to a virtual attractor in order to construct the ciphertext for the transmited symbol. The receiver can reconstruct perfectly this virtual state space, given the possession of the same chaotic system of difference equations with parameters tuned perfectly to those of the transmitter. Therefore, from the ciphertext chunk corresponding to a state, the virtual attractor can be derived from the details of the virtual state space. The knowledge of the virtual attractor leads to the recovery of the transmitted symbol.

We demonstrate that the new algorithm is secure, reliable and very fast. It uses discrete time chaotic recurrent systems and is simple, flexible and modular. These systems can be constructed easily dynamically from an alphanumeric encryption key. The cryptographic security of the algorithm is evaluated with combinatorial arguments.

This paper investigates the driving synchronization of spatiotemporal chaos. Using this method, synchronization could be achieved efficiently. By improving the driving sequence generation method, the spatial periodicity in the original system is overcome. Large quantities of chaotic sequences are available, which possess the excellent properties expected in Code Division Multiple Access (CDMA) communications. These chaotic sequences can be used as the spreading sequences in place of the traditional pseudo-noise sequences. We also evaluate the performance of the chaotic CDMA communication systems and compute the average bit error rate caused by the multi-user interference and the noise in the channel. The theoretical and simulation results show the excellent performance of the scheme.

We present two modifications of the delayed feedback control technique for controlling chaotic dynamical systems. In these methods, control force is applied only when trajectories enter neighborhoods of the targets. So three shortcomings in the delayed feedback technique, the nonsmallness of control force, impossibility of targeting unstable periodic orbits and birth of undesirable stable orbits, are improved. The delay coordinate embedding technique is also used for specifying the target orbits and determining whether trajectories enter their neighborhood. We demonstrate the effectiveness of the two approaches for an experimental system, a feedback controlled pendulum.

In the method of secure communication suggested by Abarbanel and Linsay, a binary message is encoded by phase modulation of an unstable periodic orbit (UPO) of a chaotic system. It is demonstrated that the encoded message can be decoded from the transmitted signal alone without the knowledge of chaotic system, its orbit, and encoding scheme. The decoding method is based on the fact that errors in single step prediction for models which reconstruct the state space using the local linear approximation are relatively large for the points on UPO in the transmitted signal. This enables us to extract the UPO from the transmitted signal. The identification of UPO is then used to recover the original message.

We study the collective behavior of a 2D network of nonidentical oscillators in the chaotic regime and find that ordered phase can emerge in the collective behavior, due to mutual coupling. The resulting phase cluster is a frequency-locking. The configuration of pattern depends on time and can be divided into several phase clusters when coupling is over some threshold. The frequency-locked spatiotemporal patterns are determined by the mismatched parameter and the coupling strength, and its regularity can be measured by an average nearest-neighbor distance.

In the conventional integration of a continuous dynamical system, the interaction between the model and fixed-step algorithms may produce important numerical effects over the resulting discrete representation. Our results indicate that there are remarkably simple scaling laws connecting the relevant parameters of the system to that value of integration step capable of overflowing the calculations. Moreover, we have identified a new type of chaotic numerical instability, which appears as the step size approaches some critical value. This effect is accurately described by means of nonanalytical power laws characteristic of phase transition phenomena. Finally, it is shown that simple nonlocal replacements in the discrete constructions significantly reduce or eliminate some of these numerical instabilities. These discretization effects were tested in several nonlinear dynamical systems of physical importance.

A new qualitative method is presented for studying the bifurcation problems of nonlinear systems with time-dependent parameters. The concept of stability on finite time intervals is introduced and some related stability theorems are established in order to analyze time-dependent bifurcation problems. As some applications of the new method, three different types of delayed bifurcation transitions and jump phenomena of time-dependent Duffing–van der Pol's equation are investigated qualitatively. The bifurcation transition values are predicted by constructing the V-functions or by the linear stability. The sensitivity of motion to the initial values and parameters is also discussed.

We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.

As dynamical models, cellular automata sometimes provide compelling alternatives to differential equations. In addition to rapid simulations, their stylized dynamics may elucidate the essence of the underlying physics. In this Letter, we demonstrate the efficacy with which cellular automata can model spatiotemporal nonlinear dynamics. We explicitly construct and carefully test an automaton that faithfully simulates an array of slowly torqued, heavily damped pendulums. This example is representative of a variety of systems having two time scales (slow driving followed by fast readjustment) that naturally discretize in the singular limit where the ratio of the time scales vanishes.