The problems and methods of control of chaos, which in the last decade was the subject of intensive studies, were reviewed. The three historically earliest and most actively developing directions of research such as the open-loop control based on periodic system excitation, the method of Poincare map linearization (OGY method), and the method of time-delayed feedback (Pyragas method) were discussed in detail. The basic results obtained within the framework of the traditional linear, nonlinear, and adaptive control, as well as the neural network systems and fuzzy systems were presented. The open problems concerned mostly with support of the methods were formulated. The second part of the review will be devoted to the most interesting applications.

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Control of Chaos: Methods and Applications. I. Methods

Interval state observers provide an estimate on the set of admissible values of the state vector at each instant of time. Ideally, the size of the evaluated set is proportional to the model uncertainty, thus interval observers generate the state estimates with estimation error bounds, similarly to Kalman filters, but in the deterministic framework. Main tools and techniques for design of interval observers are reviewed in this tutorial for continuous-time, discrete-time and time-delayed systems.

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Design of interval observers for uncertain dynamical systems

We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.

/pdf/convex-computation-of-the-maximum-controlled-invariant-set-3bwd6s0nyr.pdf

Convex computation of the maximum controlled invariant set for polynomial control systems

https://www-m6.ma.tum.de/~botkin/PLA2006.pdf

Localization of compact invariant sets of the Lorenz system

Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

Recently, a number of problems that can be called localization problems have been considered in qualitative theory of differential equations [1–15]. Their statements slightly vary, but geometrically one seeks subsets with some properties in the phase space. (For example, these subsets should contain certain solutions of a system of differential equations.) Such problems include the construction of positively invariant sets containing some separatrices of the Lorenz system [1]; analysis of the asymptotic behavior of solutions of the Lorenz system; finding sets that contain an attractor of the Lorenz system [2–5, 14]; finding sets containing all periodic trajectories [6–13] or separatrices and other trajectories [10, 11]. Such sets are naturally said to be localizing, and it is obviously of interest to consider methods and results that permit one to find the sharpest possible localizing sets for any structure in the phase space. The present paper focuses attention on the localization of invariant compact sets of a system of differential equations in the phase space of the system, that is, on the problem of finding subsets of the phase space that contain all invariant compact sets of the system. Invariant compact sets include equilibria, periodic trajectories, separatrices, limit cycles, invariant tori, and other sets as well as their finite unions. These sets and their properties largely determine the structure of the phase space and the qualitative behavior of solutions of a system of differential equations. In Section 1, we introduce the notation and present the main results obtained in [6–8] for the problem of localization of periodic trajectories. Finding localizing sets for a specific autonomous system of differential equations with the use of the method in [6–8] requires finding the least upper and greatest lower bounds of a function on some set. The latter problem has a special form, and its properties are discussed in Section 2. In Section 3, we show that the localizing sets constructed in [6–8] actually contain all invariant compact sets; we state a theorem that justifies our method for finding localizing sets for invariant compact sets of the system. As a corollary, we obtain a sufficient condition for the absence of invariant compact sets in a subset of the phase space; this condition can be viewed as an analog of the well-known Bendixson theorem for two-dimensional systems. In Section 4, we construct localizing sets for invariant compact sets of the Lorenz system, and in Section 5, we discuss the efficiency of various approaches to localization of the attractor. Section 6 gives an iterative procedure for the localization of invariant compact sets and discusses results obtained by this procedure.

Localization of Invariant Compact Sets of Dynamical Systems

A method for estimating domains with limit cycles and finding surfaces with the traces of all cycles is proposed. Corresponding estimations of domains with cycles for Lorenz and Rossler systems are indicated.

Estimations of domains with cycles

http://www-m6.ma.tum.de/~botkin/CNSNS2013nopages.pdf

On the global dynamics of one cancer tumour growth model

We generalize the localization method for invariant compact sets of an autonomous dynamical system to the case of a nonautonomous system of differential equations. By using this method, we solve the localization problem for the Vallis third-order dynamical system governing some processes in atmosphere dynamics over the Pacific Ocean. For this system, we construct a one-parameter family of localizing sets bounded by second-order surfaces and find the intersection of all sets of the family.

Alexander P. Krishchenko

Papers

Localization of compact invariant sets of the Lorenz system

Localization of Invariant Compact Sets of Dynamical Systems

Estimations of domains with cycles

On the global dynamics of one cancer tumour growth model

Localization of invariant compact sets of nonautonomous systems