Contours of Liapunov functions

As is well known, the stability of a dynamical system in two dimensions may be demonstrated in a very intuitive fashion from the existence of a suitable positive-definite Liapunov function, providing the contours of this function in a neighborhood of the stable point are Jordan curves. It is shown that the Liapunov function will certainly have this property if the stable point is an isolated stationary point in the sense of the Clarke calculus, but a counterexample is given if this assumption is weakened to the stable point being an isolated local extremum.This work was carried out with the support of the Natural Sciences and Engineering Council of Canada, which is gratefully acknowledged.     

Formulas of Liapunov functions for systems of linear partial differential equations

Formulas of explicit quadratic Liapunov functions for showing asymptotic stability of the system of linear partial differential equations on (0,∞)×Ω, are constructed, where A is an n×n real matrix, u=T(u₁,u₂,…,un), Ω is a bounded domain in Rk with smooth boundary ∂Ω, and Δ denotes the Laplacian operator on Rk with Δu=T(Δu₁,Δu₂,…,Δun). These formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability, traveling wave, and Navier-Stokes equations.     

Formulas for Liapunov Functions for Systems of Linear Difference Equations

Explicit quadratic Liapunov functions that provide necessaryand sufficient conditions for the asymptotic stability of thesystem of linear difference equations x (t + 1) = Ax(t) areconstructed by transforming the original systems to y (t + 1)= Gy(t), where G is a companion matrix associated with the characteristicpolynomial of A. A necessary and sufficient condition for allroots of the characteristic polynomial to lie in the unit circle|z| < 1 on the complex plane is also derived. 2000 MathematicalsSubject Classification 39A11, 93D05.     

基于向量Liapunov函数不连续系统的稳定性研究

基于局部Lipschitz连续且正则(Clarke意义下)的向量Liapunov函数,讨论不连续自治系统的稳定性(Filippov解意义下)．通过定义一类新的向量Liapunov函数的“集值导数”,给出了关于不连续系统的广义比较原理．基于Lipschitz连续且正则的向量Liapunov函数,进一步的给出不连续自治系统的Liapunov稳定性定理．     

Local expansion concepts for detecting transport barriers in dynamical systems

In the last two decades, the mathematical analysis of material transport has received considerable interest in many scientific fields such as ocean dynamics and astrodynamics. In this contribution we focus on the numerical detection and approximation of transport barriers in dynamical systems. Starting from a set-oriented approximation of the dynamics we combine discrete concepts from graph theory with established geometric ideas from dynamical systems theory. We derive the global transport barriers by computing the local expansion properties of the system. For the demonstration of our results we consider two different systems. First we explore a simple flow map inspired by the dynamics of the global ocean. The second example is the planar circular restricted three body problem with Sun and Jupiter as primaries, which allows us to analyze particle transport in the solar system.     

Quasiperiodic functions and dynamical systems in quantum solid state physics

This is a survey article dedicated to the study of topological quantities in theory of normal metals discovered in the works of the authors during the last years. Our results are based on the theory of dynamical systems on Fermi surfaces. The physical foundations of this theory (the so-called Geometric Strong Magnetic Field Limit) were found by the school of I. M. Lifshitz many years ago. Here the new aspects in the topology of quasiperiodic functions are developed.Dedicated to IMPA on the occasion of its 50^th anniversaryThe work of S. P. Novikov is partially supported by the NSF Grant DMS 0072700.     

Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

Tomasz Szarek presented interesting criteria for the existence of invariant measures and asymptotic stability of Markov operators on Polish spaces. Hans Crauel in his book presented the theory of random probabilistic measures on Polish spaces showing that notions of compactness and tightness for such measures are in one-to-one correspondence with such notions for non-random measures on Polish spaces, in addition to the criteria under which the space of random measures is itself a Polish space. This result allowed the transfer of results of Szarek to the case of random dynamical systems in the sense of Arnold. These criteria are interesting because they allow to use the existence of simple deterministic Lyapunov type function together with additional conditions to show the existence of invariant measures and asymptotic stability of random dynamical systems on general Polish spaces.     

On exposed functions in Bernstein spaces

For σ > 0, the Bernstein space {ie427-01} consists of those L ¹(ℝ) functions whose Fourier transforms are supported by [−σ, σ]. Since {ie427-02} is separable and dual to some Banach space, the closed unit ball {ie427-03} of {ie427-04} has sufficiently large sets of both exposed and strongly exposed points: {ie427-05} coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples, and obtain as corollaries relations between the sets of exposed, strongly exposed, weak* exposed, and weak* strongly exposed points of {ie427-06}.     

On a class of dynamical systems with emerging cluster structure

In this paper we discuss and analyze a two-parameter family of systems of quadratic ordinary differential equations of interest in applied sciences, whose dynamics exhibits an emerging cluster structure.     

Nonexpansive selections of metric projections in spaces of continuous functions

A subset A of a metric space X is said to be a nonexpansive proximinal retract (NPR) of X if the metric projection from X to A admits a nonexpansive selection. We study the structure of NPR's in the space C(K) of continuous functions on a compact Hausdorff space K. The main results are a characterization of finite-codimensional and of finite-dimensional NPR subspaces of C(K) and a complete characterization of all NPR subsets of .     

Instability of isolated equilibria of dynamical systems with invariant measure in spaces of odd dimension

We discuss the conjecture asserting that isolated equilibrium states of autonomous systems admitting invariant measures are unstable in spaces of odd dimension. This conjecture is proved for systems for which quasihomogeneous truncations with isolated singularities can be found. We consider a counterexample in the class of systems with infinitely differentiable right-hand sides and zero Maclaurin series at the equilibrium state. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 674–680, May, 1999.     

On common quadratic Lyapunov functions for stable discrete-time LTI systems

This paper deals with the question of the existence of weakand strong common quadratic Lyapunov functions (CQLFs) for stablediscrete-time linear time-invariant (LTI) systems. The mainresult of the paper provides a simple characterization of pairsof such systems for which a weak CQLF of a given form existsbut for which no strong CQLF exists. An application of thisresult to second-order discrete-time LTI systems is presented.     

Cheeger-harmonic functions in metric measure spaces revisited

Let (X,d,μ)$(X,d,\mu )$ be a complete metric measure space, with μ   a locally doubling measure, that supports a local weak L²$L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on (X,d,μ)$(X,d,\mu )$. Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.     

The Banach-Zarecki theorem for functions with values in metric spaces

Using an old result of Luzin about his property , we prove a general version of the Banach-Zarecki theorem (on absolute continuity and Luzin's property ).

     

On stability and stabilization of some discrete dynamical systems

The paper formulates effective and nonimprovable stability conditions for a linear difference system involving 2 integer delays. The used technique combines algorithm of the discrete D‐decomposition method with some procedures of the polynomial theory. Contrary to the related existing results, the derived conditions are fully explicit with respect to both delays, which enables their simple applicability in various scientific and engineering areas. As an illustration, we show their importance in delayed feedback controls of discrete dynamical systems, with a particular emphasis put on stabilization of unstable steady states of the discrete logistic map.