International Journal of Bifurcation and Chaos

Discrete-time CNN systems are studied in this paper by the application of Chua's local activity principle. These systems are locally active everywhere except for one isolated parameter value. As a result, nonhomogeneous spatiotemporal patterns may be induced by any initial setting of the CNN system when the strength of the system diffusion coupling exceeds a critical threshold. The critical coupling coefficient can be derived from the loaded cell impedance of the CNN system. Three well-known 1D map CNN's (namely, the logistic map CNN, the magnetic vortex pinning map CNN, and the spiral wave reproducing map CNN) are introduced to illustrate the applications of the local activity principle. In addition, we use the cell impedance to demonstrate the period-doubling scenario in the logistic and the magnetic vortex pinning maps.

In this contribution, we show that the incorporation of nonlinear dynamical measures into a multivariate discrimination provides a signal classification system that is robust to additive noise. The signal library was composed of nine groups of signals. Four groups were generated computationally from deterministic systems (van der Pol, Lorenz, Rössler and Hénon). Four groups were generated computationally from different stochastic systems. The ninth group contained inter-decay interval sequences from radioactive cobalt. Two classification criteria (minimum Mahalanobis distance and maximum Bayesian likelihood) were tested. In the absence of additive noise, no errors occurred in a within-library classification. Normally distributed random numbers were added to produce signal to noise ratios of 10, 5 and 0 dB. When the minimum Mahalanobis distance was used as the classification criterion, the corresponding error rates were 2.2%, 4.4% and 20% (Expected Error Rate = 89%). When Bayesian maximum likelihood was the criterion, the error rates were 1.1%, 4.4% and 21% respectively. Using nonlinear measures an effective discrimination can be achieved in cases where spectral measures are known to fail.

Most classification errors occurred at low signal to noise ratios when a stochastic signal was misclassified into a different group of stochastic signals. When the within-library classification exercise is limited to the four groups of deterministic signals, no classification errors occurred with clean data, at SNR = 10 dB, or at SNR = 5 dB. A single classification error (Observed Error Rate = 2.5%, Expected Error Rate = 75%) occurred with both classification criteria at SNR = 0 dB.

Relaxation oscillations appear in processes which involve transitions between two states characterized by fast and slow time scales. When a relaxation oscillator is coupled to an external periodic force its entrainment by the force results in a response which can include multiple periodicities and bistability. The prototype of these behaviors is the harmonically driven van der Pol equation which displays regions in the parameter space of the driving force amplitude where stable orbits of periods 2n ± 1 coexist, flanked by regions of periods 2n + 1 and 2n - 1. The parameter regions of such bistable orbits are derived analytically for the closely related harmonically driven Stoker–Haag piecewise discontinuous equation. The results are valid over most of the control parameter space of the system. Also considered are the reasons for the more complicated dynamics featuring regions of high multiple periodicity which appear like noise between ordered periodic regions. Since this system mimics in detail the less analytically tractable forced van der Pol equation, the results suggest extensions to situations where forced relaxation oscillations are a component of the operating mechanisms.

Stabilization of a class of linear time-delay systems can be achieved by a numerical procedure, called the continuous pole placement method [Michiels et al., 2000]. This method can be seen as an extension of the classical pole placement algorithm for ordinary differential equations to a class of delay differential equations. In [Michiels et al., 2000] it was applied to the stabilization of a linear time-invariant system with an input delay using static state feedback. In this paper we study the limitations of such delayed state feedback laws. More precisely we completely characterize the class of stabilizable plants in the 2D-case. For that purpose we make use of numerical continuation techniques. The use of delayed state feedback in various control applications and the effect of its limitations are briefly discussed.

This work investigates the complexity of one-dimensional cellular neural network mosaic patterns with spatially variant templates on finite and infinite lattices. Various boundary conditions are considered for finite lattices and the exact number of mosaic patterns is computed precisely. The entropy of mosaic patterns with periodic templates can also be calculated for infinite lattices. Furthermore, we show the abundance of mosaic patterns with respect to template periods and, which differ greatly from cases with spatially invariant templates.

As part of an ongoing project on the stability of massively complex electrical power systems, we discuss the global geometric structure of contacts among the basins of attraction of a six-dimensional dynamical system. This system represents a simple model of an electrical power system involving three machines and an infinite bus. Apart from the possible occurrence of attractors representing pathological states, the contacts between the basins have a practical importance, from the point of view of the operation of a real electrical power system. With the aid of a global map of basins, one could hope to design an intervention strategy to boot the power system back into its normal state. Our method involves taking two-dimensional sections of the six-dimensional state space, and then determining the basins directly by numerical simulation from a dense grid of initial conditions. The relations among all the basins are given for a specific numerical example, that is, choosing particular values for the parameters in our model.

The paper introduces a class of third-order nonsymmetric Cellular Neural Networks (CNNs), and shows through computer simulations that they undergo a cascade of period doubling bifurcations which leads to the birth of a large-size complex attractor. A major point is that these bifurcations and complex dynamics happen in a small neighborhood of a particular CNN with a symmetric interconnection matrix.

We observe purely spatial intermittency accompanied by temporally periodic behavior in an inhomogeneous lattice of coupled logistic maps where the inhomogenity appears in the form of different values of the map parameters at distinct sites. Linear analysis shows that the spatial intermittency appears in the neighborhood of tangent period doubling bifurcation points. The intermittency near the bifurcation points is associated with a power-law distribution for the laminar lengths. The scaling exponent ζ for the laminar length distribution is obtained.

The generalized Hénon maps (GHM) are discrete-time systems with given finite dimension, which show chaotic and hyperchaotic behavior for certain parameter values and initial conditions. A study of these maps is given where particularly higher-dimensional cases are considered.

The ideas underpinning a technique of using nonlinear approximations to control chaotic dynamical systems are shown to be useful as a basis for the development of effective methods for targeting these systems. The inverses of the maps under consideration are not required. The methods are applied to the Hénon and Ikeda maps and also to an Ikeda-like map.

Projective synchronization (PS), in which the state vectors synchronize up to a scaling factor, is usually observable only in partially linear systems. We show that PS could, by means of control, be extended to general classes of chaotic systems with nonpartial linearity. Performance of PS may also be manipulated by controlling the scaling factor to any desired value. In numerical experiments, we illustrate the applications to a Rössler system and a Chua's circuit. The feasibility of the control for high dimensional systems is demonstrated in a hyperchaotic system.

This paper presents a new control approach for steering trajectories of three-dimensional nonlinear chaotic systems towards stable stationary states or time-periodic orbits. The proposed method mainly consists in a sliding mode-based control design that is extended by an explicit consideration of system energy as basis for both controller design and system stabilization. The control objective is then to regulate the energy with respect to a shaped nominal representation implicitly related to system trajectories. In this paper, we establish some theoretical results to introduce the control design approach referred to as Energy based Sliding Mode Control (ESMC for short). Then, some capabilities of the proposed approach are illustrated through examples related to the chaotic circuit of Chua.

In this Letter, we study the popular parametric variation chaos control and state-feedback methodologies in chaos control, and point out for the first time that they are actually equivalent in the sense that there exist diffeomorphisms that can convert one to the other for most smooth chaotic systems. Detailed conversions are worked out for typical discrete chaotic maps (logistic, Hénon) and continuous flows (Rösller, Lorenz) for illustration. This unifies the two seemingly different approaches from the physics and the engineering communities on chaos control. This new perspective reveals some new potential applications such as chaos synchronization and normal form analysis from a unified mathematical point of view.

This letter investigates a new chaotic system and its role as a joint function between two complex chaotic systems, the Lorenz and the Chen systems, using a simple variable constant controller. With the gradual tuning of the controller, the controlled system evolves from the canonical Lorenz attractor to the Chen attractor through the new transition chaotic attractor. This evolving procedure reveals the forming mechanisms of all similar and closely related chaotic systems, and demonstrates that a simple control technique can be very useful in generating and analyzing some complex chaotic dynamical phenomena.

Chen's attractor was recently reported and proved to be topologically different from the Lorenz butterfly attractor. In this paper, Chen's attractor is realized with an electronic circuit. Although the circuitry is straight-forward, it is the first design reported. In addition, the synchronization scheme between two Chen's attractors is also considered and realized.

An improved realization of mixed-mode chaotic circuit which has both autonomous and nonautonomous chaotic dynamics is proposed. Central to this study is inductorless realization of mixed-mode chaotic circuit using FTFN-based inductance simulator. FTFN-based topology used in this realization enables the simulation of ideal floating and grounded inductance. This modification provides an alternative solution to the integration problem of not only mixed-mode chaotic circuit but also other chaotic circuits in the literature using CMOS VLSI technologies. In addition to this major improvement, CFOA-based nonlinear resistor was used in the new realization of mixed-mode chaotic circuit. The usage of CFOA-based nonlinear resistor in the circuit's structure reduces the component count and provides buffered and isolated output.

In this paper, a sliding mode controller is applied to control the cubic Chua's circuit system. The sliding surface of this paper used is one dimension higher than the traditional surface and guarantees its passage through the initial states of the controlled system. Therefore, using the characteristic of this sliding mode we aim to design a controller that can meet the desired specification and use less control energy by comparing with the result in the current existing literature. The results show that the proposed controller can steer Chua's circuit system to the desired state without the chattering phenomenon and abrupt state change.