R. Genesio

Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems

Abstract The paper considers the problem of determining the conditions under which a nonlinear dynamical system can give rise to a chaotic behaviour. On the basis of the harmonic balance principle, which is widely used in the frequency analysis of nonlinear control systems, two practical methods are presented for predicting the existence and the location of chaotic motions. This is formulated as a function of the system parameters, when the system structure is fixed by rather general input-output or state equation models. Several examples of application are presented to show the rather straightforward computations involved in the proposed methods, the kind of results which can be obtained and, due to the heuristic approach to the problem, their corresponding approximation.

Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics☆

The harmonic balance method is applied to the analysis of period-doubling bifurcations in a general class of nonlinear feedback systems. Compact conditions for the prediction and stability analysis of period-doubling bifurcations are obtained. Specializations of these conditions for systems in which the nonlinear subsystem is static are given. The implications of the results to control design are illustrated through the development of a design procedure for solving a representative control problem. The objectives of the control design can include delay of a given period-doubling bifurcation as well as stability objectives. The results are illustrated with a detailed example.

Dynamical phases of the Hindmarsh-Rose neuronal model: Studies of the transition from bursting to spiking chaos

The dynamical phases of the Hindmarsh-Rose neuronal model are analyzed in detail by varying the external current I. For increasing current values, the model exhibits a peculiar cascade of nonchaotic and chaotic period-adding bifurcations leading the system from the silent regime to a chaotic state dominated by bursting events. At higher I-values, this phase is substituted by a regime of continuous chaotic spiking and finally via an inverse period doubling cascade the system returns to silence. The analysis is focused on the transition between the two chaotic phases displayed by the model: one dominated by spiking dynamics and the other by bursts. At the transition an abrupt shrinking of the attractor size associated with a sharp peak in the maximal Lyapunov exponent is observable. However, the transition appears to be continuous and smoothed out over a finite current interval, where bursts and spikes coexist. The beginning of the transition (from the bursting side) is signaled from a structural modification in the interspike interval return map. This change in the map shape is associated with the disappearance of the family of solutions responsible for the onset of the bursting chaos. The successive passage from bursting to spiking chaos is associated with a progressive pruning of unstable long-lasting bursts.

A Frequency Method for Predicting Limit Cycle Bifurcations

The paper studies the bifurcations of limit cycles in a rather general class of nonlinear dynamic systems. Relying on the classical harmonic balance approach as applied in control engineering neat frequency conditions for such bifurcations are derived. These results, approximate in nature, make clear the structural mechanism of the considered phenomena and can be applied to predict the occurrence of bifurcations as a function of system parameters. The application to several examples of different complexity shows the simplicity and accuracy of the proposed method for solving complicated problems of nonlinear dynamics.

Dead-beat chaos synchronization in discrete-time systems

The problem of synchronizing discrete-time chaotic systems is investigated. A new appealing property, the dead-beat synchronization, or exact synchronization in finite time, is presented, and conditions for its accomplishment in a simple important class of nonlinear maps are given. An original secure communication scheme which effectively exploits this property is also introduced. >

A frequency approach for analyzing and controlling chaos in nonlinear circuits

The paper presents a frequency domain approach for studying the chaotic dynamics of an important class of nonlinear circuits. By formulating an elementary model of chaos and using the harmonic balance principle, techniques for the analysis and the stabilization to a periodic solution of complex systems are developed. They result in engineering tools which are simple and practical, although not rigorous in principle, providing a qualitative view of the global dynamics under study. Some applications concerning the recent unfolded Chuas circuit are considered. >

On the stability domain estimation via a quadratic Lyapunov function: convexity and optimality properties for polynomial systems

The problem of estimating the stability domain of the origin of an n-order polynomial system is considered. Exploiting the structure of this class of systems it is shown that, for a given quadratic Lyapunov function, an estimate of the stability domain can be obtained by solving a suitable convex optimization problem. This estimate is shown to be optimal for an important subclass including both quadratic and cubic systems, and its accuracy in the general polynomial case is discussed via several examples.

Stabilizing periodic orbits of forced systems via generalized Pyragas controllers

This brief deals with the problem of designing linear time-invariant feedback controllers to stabilize unstable periodic orbits for a class of sinusoidally forced nonlinear systems. Exploiting the classical circle criterion, a sufficient condition for the stabilization of unstable periodic solutions is derived for a class of controllers that generalizes the time delayed one proposed by Pyragas. Based on this rendition, a technique for designing optimal controllers is proposed. The validity of the technique and the performance of the designed controllers are illustrated via one example concerning the forced Duffing oscillator.

DISTORTION CONTROL OF CHAOTIC SYSTEMS: THE CHUA'S CIRCUIT

The paper presents a new approach to the control of chaotic systems for the stabilization of a periodic orbit. The problem formulation requires preserving a number of original system characteristics and making use of a low energy control. The proposed method follows a frequency harmonic balance technique employed in the approximate analysis of complex nonlinear phenomena. The design procedure is applied to the well-known Chuas circuit showing the main characteristics of the approach: it gives reasonably accurate results, partly intuitive in nature, by means of quite simple computations.

On the dynamics of chaotic spiking-bursting transition in the Hindmarsh–Rose neuron

The paper considers the neuron model of Hindmarsh-Rose and studies in detail the system dynamics which controls the transition between the spiking and bursting regimes. In particular, such a passage occurs in a chaotic region and different explanations have been given in the literature to represent the process, generally based on a slow-fast decomposition of the neuron model. This paper proposes a novel view of the chaotic spiking-bursting transition exploiting the whole system dynamics and putting in evidence the essential role played in the phenomenon by the manifolds of the equilibrium point. An analytical approximation is developed for the related crucial elements and a subsequent numerical analysis signifies the properness of the suggested conjecture.