Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.     

The Conley index for discontinuous vector fields

In this paper, we define the Conley index for a region of discontinuity D of a piecewise C ^k discontinuous vector field Z on an n-dimensional compact Riemannian smooth orientable manifold and prove it to be a homotopy invariant. This invariance is obtained by regularization of the discontinuous vector field. We use an adapted form of Lyapunov graph continuation to produce, in a few examples, a regularization of the discontinuous vector field with the property that the dynamics in a regularized neighborhood of D has the same Conley index as .      

Conley type index applied to Hamiltonian inclusions

In the paper we develop the theory of a cohomological index of the Conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional Hilbert space. An application to the existence of periodic orbits to asymptotically linear Hamiltonian inclusions is presented.     

On the Conley index in the invariant manifolds

Let ? be a flow on a manifold M and assume that N⊂M is an invariant manifold. The aim of this note is to compare the Conley indices of an isolated invariant set S⊂N with respect to the flow ? and the flow ? restricted to N.     

Detection of renewal system factors via the Conley index

Let be an isolating neighborhood for a map . If we can decompose into the disjoint union of compact sets and , then we can relate the dynamics on the maximal invariant set to the shift on two symbols by noting which component of each iterate of a point lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to . In essence, we measure the difference between the Conley index of and the sum of the indices of and .

     

The Conley index for fast-slow systems II: Multidimensional slow variable

We use the Conley index theory to develop a general method to prove existence of periodic and heteroclinic orbits in a singularly perturbed system of ODEs. This is a continuation of the authors' earlier work [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470] which is now extended to systems with multidimensional slow variables. The key new idea is the observation that the Conley index in fast-slow systems has a cohomological product structure. The factors in this product are the slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast-slow systems with one-dimensional slow flow [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast-slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427-470].     

The Conley index over a base

We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in , and the problem of continuing two periodic orbits which are not homotopic as loops.

     

A semi-algebraic approach for asymptotic stability analysis

This paper deals with the problem of computing Lyapunov functions for asymptotic stability analysis of autonomous polynomial systems of differential equations. We propose a new semi-algebraic approach by making advantage of the local property of the Lyapunov function as well as its derivative. This is done by first constructing a semi-algebraic system and then solving this semi-algebraic system in an adaptive way. Experiment results show that our semi-algebraic approach is more efficient in practice, especially for low-order systems.     

Algebraic transition matrices in the Conley index theory

We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.

     

Lower bounds on entropy via the Conley index with application to time series

A proof of a localized version of the proven entropy conjecture for C^∞ smooth maps is given. This allows for computational methods for bounding topological entropy through properties of the Conley index. Chaos can then be determined by a non-global index calculation robust under possibly large (and noisy) perturbations. In addition, a proof of a Wazewski's Principle for time series analysis is given which allows for lifting of entropy to the observed dynamical system under certain conditions.     

Unstable manifold,Conley index and fixed points of flows

We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.     

On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that is negative definite. In this paper we prove that if is bounded then the condition that is negative definite can be weakened and replaced by that and is negative definite.     

Guaranteed asymptotic stability for a class of uncertain linear dynamical systems

We consider a class of linear dynamical systems with bounded, Lebesgue-measurable uncertainties in the system and input matrices as well as in the input itself. A state feedback control is derived, which guarantees global, uniform asymptotic stability of the zero state; this control is continuous, except at the zero state.This paper is based in part on research supported by the National Science Foundation.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index

We introduce a new approach to the algorithmic computation of the Conley index for continuous maps. We use the technique of splitting an index pair into two layers which is inspired by the work of Mrozek, Reineck and Srzednicki [M. Mrozek, J.F. Reineck, R. Srzednicki, The Conley index over a base, Trans. Amer. Math. Soc. 352 (2000) 4171–4194]. The main advantage of our construction over the approach based directly on the one introduced by Mischaikow, Mrozek and Pilarczyk [K. Mischaikow, M. Mrozek, P. Pilarczyk, Graph approach to the computation of the homology of continuous maps, Found. Comput. Math. 5 (2005) 199–229] is that our cubical sets have the excision property. Moreover, our solution has some advantages in comparison to the approach recently proposed by Mrozek [M. Mrozek, Index pair algorithms, Found. Comput. Math. 6 (2006) 457–493].     

On explicit conditions for the asymptotic stability of linear higher order difference equations

We derive some explicit sufficient conditions for the asymptotic stability of the zero solution in a general linear higher order difference equation, and compare our estimations with other related results in the literature. Our main result also applies to some nonlinear perturbations satisfying a kind of sublinearity condition.     

Phase boundary solutions to model kinetic equations via the Conley index theory. Part I

In this paper, we consider phase boundary solutions to a four-velocity kinetic model of a kinetic equation governing the motion of van der Waals fluids. These solutions connect such equilibrium states, which are saddle critical points of the suitable dynamic system. Solutions of this type can be interpreted as dynamic phase transition. The mathematical apparatus is that of the Coney index theory.