On the stability of projected dynamical systems

A class of projected dynamical systems (PDS), whose stationary points solve the corresponding variational inequality problem (VIP), was recently studied by Dupuis and Nagurney (Ref. 1). This paper initiates the study of the stability of such PDS around their stationary points and thus gives rise to the study of the dynamical stability of VIP solutions. Examples are constructed showing that such a study can be quite distinct from the classical stability study for dynamical systems (DS). We give the definition of a regular solution to a VIP and introduce the concept of a minimal face flow induced by a PDS, which is a standard DS of a lower dimension. We then show that, at the regular solutions of the VIP, the local stability of the PDS is essentially the same as that of its minimal face flow. Hence, we reduce the problem, in this case, to one of the classical stability study of DS, a more developed discipline. In a more direct way, we then establish a series of local and global stability results of the PDS, under various conditions of monotonicity.This research was supported by the National Science Foundation under Grant DMS-9024071 under the Faculty Awards for Women Program. This support is gratefully acknowledged.     

Linear dynamical systems with partial derivatives

Systems with multidimensional time over Lie-algebras are considered. It is proved that a transfer function determines a minimal system up to similarity.     

Partial stability and stabilization of dynamical systems

We prove the theorem on necessary and sufficient conditions of partial instability and the theorem on partial stabilization of nonlinear dynamical systems. We obtain sufficient conditions of controllability for systems linear with respect to control. We also study the problem of control and stabilization of an angular motion of a solid body by rotors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 186–193, February, 1995.     

Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

Let {A₁,…,AK}⊂Cd×d be arbitrary K matrices, where K and d both ?2. For any 0<Δ<∞, we denote by the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R₊ satisfying tj−tj−1?Δ and

     

On the stability of the translation equation and dynamical systems

In this paper we prove the stability of the functional equation F(s,F(t,x))=F(s+t,x) in the class of functions F:R×I→I, which are continuous with respect to each variable, and where I⊂R is a real interval. We also discuss the stability in the sense of Hyers-Ulam of dynamical systems on I. We show some properties of δ-approximate solutions of the translation equation on a real interval.     

Hyperbolicity,stability and monodromy of dynamical systems

We propose a rigorous computational method for proving uniform hyperbolicity of dynamical systems. Besides finding structurally stable parameters, the algorithm can also be applied for the computation of the monodromy of dynamical systems. With this algorithm, we prove that the topology of the 2-dimensional generalization of the Mandelbrot set is totally different from that of the original Mandelbrot set. Furthermore, we show that the monodromy of the complex Hénon map can be used to determine the dynamics of the real Hénon map. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant cones and stability of linear dynamical systems

We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1446–1461, November, 2006.     

Robust stability of uncertain impulsive dynamical systems

This paper studies robust stability of uncertain impulsive dynamical systems. By introducing the concepts of uniformly positive definite matrix functions and Hamilton–Jacobi/Riccati inequalities, several criteria on robust stability, robust asymptotic stability and robust exponential stability are established. An example is also worked through to illustrate our results.     

Asymptotic and total stability for generalized dynamical systems

The existence of solutions to a class of nonlinear problems depending on a parameter is proved using the Gaierkin method The principal difficulty is that when the parameter crosses a critical value the bound edness of the approximate solutions is no longer obvious. Applications are indicated.     

On the stochastic stability of some two-dimensional dynamical systems

We study the stochastic stability of a two-dimensional diffusion process described by a system of two nonlinear Itô stochastic differential equations. The nonlinear drift operator is assumed to be additively separated with respect to the variables, and the diffusion is assumed to be degenerate with respect to only one of the two components of the process. We consider cases in which, in the expression for the nonlinear drift vector, two of the four functions of one variable are linear. We obtain sufficient conditions for the stochastic stability of such systems. As an example, we consider a stochastic system whose deterministic part is equivalent to the classical Lienard equation.     

Stochastic stability in some chaotic dynamical systems

For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of Perron-Frobenius-operators.With 1 Figure     

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.     

Preservation of stability of dynamical systems under homomorphisms

For abstract dynamical systems in the form of systems of motions, we obtain criteria for the preservation of stability properties via an extended notion of a homomorphism of dynamical systems. We obtain corollaries for ordinary differential equations including those with switchings of vector fields. We present examples illustrating the use of these criteria.     

Multivalued maps in stability theory of dynamical systems

Summary Multivalued maps like orbit, limit set, prolongations etc., are an useful tool in Dynamical Systems theory. In this work we develop a calculus for multivalued maps associated with a dynamical system. Then we give general definitions of stability and attraction of a compact set with respect to a multivalued map. On the basis of our calculus, we obtain several characterizations of stability and attraction, which generalise well known classical theorems. Such a general theory is applied to total stability of diffentiable dynamical systems. The equivalence among several approaches to total stability is established.