Alternating Lyapunov functions in the theory of linear extensions of dynamical systems on a torus

We consider a series of problems connected with the application of quadratic-form Lyapunov functions to the investigation of the properties of regularity of linear extensions of dynamic systems on a torus. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 488–500, April, 2007.     

Linear extensions of dynamical systems on a torus and Green's functions

We found new structures of linear extensions of dynamical systems on a torus which have a unique Green's function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1219–1224, 1990.     

On local perturbations of linear extensions of dynamical systems on a torus

We investigate the problem of preservation of regularity of linear extensions of dynamic systems on a torus under perturbations.     

Linear extensions of dynamical systems on a torus that possess Green-Samoilenko functions

By using Lyapunov functions with alternating signs, we study problems of regularity and weak regularity for some linear extensions of dynamical systems on a torus. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 178–188, February, 1998.     

Sets of linear expansions of dynamical systems on a torus for a fixed Lyapunov function

We consider sets of linear expansions of dynamical systems on a torus with general alternating Lyapunov function. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1707–1713, December, 2007.     

On some classes of regular linear extensions of dynamical systems on a torus

We select some classes of linear extensions of dynamical systems on a torus for which weak regularity implies regularity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1479–1485, November, 1994.This work was supported by the Ukrainian State Committee on Science and Technology.     

Conditions for exponential stability and dichotomy of pulse linear extensions of dynamical systems on a torus

Conditions for exponential stability and dichotomy of pulse linear extensions of dynamical systems on a torus are investigated. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 451–453, March, 1998.     

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.     

Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.      

Separation of variables in linear extensions of dynamical systems on tori

We study the problem of separation of variables in linear extensions of dynamical systems on tori.     

On equicontinuous factors of linear extensions of minimal dynamical systems

The concept of the equicontinuous factor of the linear extension of a minimal transformation group is introduced and investigated. It is shown that a subset of motions, bounded and distal with respect to the extension, forms a maximal equicontinuous subsplitting of the linear extension. As a consequence, any distal linear extension has a nontrivial equicontinuous invariant subsplitting. The linear extensions without exponential dichotomy possess similar subsplittings if the Favard condition is satisfied. The same statement holds for linear extensions with the property of recurrent motions additivity provided that at least one nonzero motion of this sort exists.Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 45, No. 2, pp. 233–238, February, 1993.     

On common linear copositive Lyapunov functions for pairs of stable positive linear systems

We study the common linear copositive Lyapunov functions of positive linear systems. Firstly, we present a theorem on pairs of second order positive linear systems, and give another proof of this theorem by means of properties of geometry. Based on the process of the proof, we extended the results to a finite number of second order positive linear systems. Then we extend this result to third order systems. Finally, for higher order systems, we give some results on common linear copositive Lyapunov functions.     

Markov extensions for multi-dimensional dynamical systems

By a result of F. Hofbauer [11], piecewise monotonic maps of the interval can be identified with topological Markov chains with respect to measures with large entropy. We generalize this to arbitrary piecewise invertible dynamical systems under the following assumption: the total entropy of the system should be greater than the topological entropy of the boundary of some reasonable partition separating almost all orbits. We get a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy. We illustrate our results by quoting an application to a class of multi-dimensional, non-linear, non-expansive smooth dynamical systems. Part of this work was done at Université Paris-Sud, dép. de mathématiques, Orsay.     

On some qualitative properties of monotone linear extensions of dynamical systems

We study monotone linear extensions of dynamical systems. The problem of the existence of invariant manifolds and exponential separation is investigated for linear extensions that preserve the order structure. We also study the relationship between the monotonicity of linear extensions and the existence (weak regularity, quasiregularity) of bounded solutions of inhomogeneous linear extensions.     

Construction of Lyapunov functions for nonlinear planar systems by linear programming

Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions (Giesl and Hafstein, 2010 [7]). Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in Marinosson (2002) [17] and further improved in Hafstein (2007) [10]. The algorithm constructs a linear programming problem for the system at hand, and any feasible solution to this problem parameterizes a Lyapunov function for the system. We prove that the algorithm always succeeds in constructing a Lyapunov function if the system possesses an exponentially stable equilibrium. The size of the region of the Lyapunov function is only limited by the region of attraction of the equilibrium and it includes the equilibrium.     

A criterion for the existence of the unique invariant torus of a linear extension of dynamical systems

Under the assumption that a linear homogeneous system defined on the direct product of a torus and the Euclidean space is exponentially dichotomous on semiaxes, we obtain a necessary and sufficient condition for the existence of the unique invariant torus of the corresponding inhomogeneous linear system. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 3–13, January, 2007.     

Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.