Vera B. Smirnova

Frequency-domain methods for nonlinear analysis: theory and applications

Classical absolute stability theory dichotomy and stability of equilibria sets cycles, homoclinic and heteroclinic trajectories strange attractors estimates of dimensions.

Non-local methods for pendulum-like feedback systems

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Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chuas circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.

Asymptotic estimates for gradient-like distributed parameter systems with periodic nonlinearities

Many systems, arising in electrical and electronic engineering may be represented as an interconnection of LTI system and a periodic nonlinear block, as exemplified by phase-locked loops (PLL) and more general systems, based on controlled phase synchronization of several periodic processes (“phase synchronization” systems, or PSS). Typically such systems are featured by the gradient-like behavior, i.e. the system has infinite sequence of equilibria points, and any solution converges to one of them (which may be interpreted as the phase locking). This property however says nothing about the transient behavior of the system, whose important qualitative index is the maximal phase error. Before the phase is locked, the error may increase up to several periods, which phenomenon is known as cycle slipping and was introduced by J. Stoker for the model of mathematical pendulum. Since the cycle slipping is considered to be undesired behavior of PLLs, it is important to find efficient estimates for the number of slipped cycles. In the present paper, we address the problem of cycle-slipping for phase synchronization systems with infinite-dimensional linear part. New effective estimates for a number of slipped cycles are obtained by means of Popovs method of “a priori integral indices”, which was originally developed for proving absolute stability of nonlinear systems.

Frequency-domain estimates for transient attributes of discrete phase systems

A multidimensional discrete control system with periodic nonlinearity is investigated. Its two important charac- teristics: the transient time and a number of slipped cycles are considered. By means of Lyapunov direct method and Yakubovich-Kalman theorem certain estimates of these two attributes are obtained. They are formulated as frequency- domain criteria.

Pendulum-Like Systems

In the present chapter we develop the global bifurcation theory of two-dimensional systems with a periodic nonlinearity and the general concept of higher-dimensional pendulum-like systems and their canonical forms, which are of considerable use in studying, by frequency-domain methods, the global behavior of such systems. We show that Lyapunov functions constructed as quadratic forms are not applicable in a standard way for investigating the global behavior of pendulum-like systems. In contrast to this fact we prove global convergence results for a certain class of such systems which use the invariance principle and the theorem of Yakubovich-Kalman.

LOCALIZATION OF ATTRACTORS IN MULTIDIMENSIONAL CHUA'S CIRCUIT EQUATIONS

The well-known double-scroll Chuas attractor occurs in the parameter space where all 3 linearized regions of the dimensionless Chuas equation are unstable. To investigate the surprising existence of a basin of attraction, this paper proves a theorem which can be used to derive a bounded positive invariant set in Chuas circuit.

Phase locking, oscillations and cycle slipping in synchronization systems

Many engineering applications employ nonlinear systems, representable as a feedback interconnection of a linear time-invariant dynamic block and a periodic nonlinearity. Such models naturally describe phase-locked loops (PLLs), which are widely used for synchronization of built-in computer clocks, demodulation and frequency synthesis. Other example include, but are not limited to, dynamics of pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities, often referred to as synchronization systems, are usually featured by the existence of an infinite sequence of equilibria (stable or unstable). The central problem, concerning dynamics of synchronization systems, is the convergence of solutions to equilibria, treated in engineering applications as phase locking. In general, not any solution is convergent (“phase-locked”). This raises a natural question which oscillatory trajectories (such as e.g. periodic solutions) are possible. Even when the solution converges, the transient process can be unsatisfactory due to cycle slippings, leading to demodulation errors. In this paper, we address the mentioned three problems and offer novel criteria for phase locking, estimates for the number of slipped cycles and possible frequencies of periodic oscillations. The methods used in this paper are based on the method of integral quadratic constraints, stemming from Popovs technique of “a priori integral indices”.

Frequency-domain criteria for gradient-like behavior of phase control systems with vector nonlinearities

Asymptotic behavior of control systems with periodic vector nonlinearities and denumerable sets of equibliria is examined. Both continuous systems, described by ordinary differential equations, and discrete systems, described by difference equations, are considered. Certain generalization of periodic Lyapunov-type functions and sequences is offered. By means of generalized Lyapunov-type functions and Yakubovich-Kalman theorem new frequency-domain criteria which guarantee that every solution of the system tends to an equiblirium are obtained.

Frequency estimates for the number of cycle slippings in a phase system with nonlinear vector function

Two classes of phase control systems with vector nonlinearities are considered: systems described by ordinary differential equations and system described by difference equations. They are characterized by the presence of a periodic vector nonlinearity in the mathematical description of the system. The problem of the number of cycle slippings is investigated. For both classes of control systems, frequency estimates of the deviation of each angular coordinate from its initial value are obtained. The estimation technique is based on the direct Lyapunov method with periodic Lyapunov functions. With the use of the Yakubovich-Kalman lemma, all results are formulated in terms of the transfer function of the linear part of the system. The results obtained have the form of frequency inequalities with variable parameters, which satisfy some algebraic inequalities.