Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.     

Localization of invariant compact sets of families of discrete-time systems

The functional method of localization of invariant compact sets developed for continuous- and discrete-time dynamical systems is extended to families of discrete-time systems. Positively invariant compact sets are considered. As an example, the method is applied to the Hénon system with uncertain parameters.     

Construction of Lyapunov functions by the method of localization of invariant compact sets

We suggest a new method for constructing Lyapunov functions for autonomous systems of differential equations. The method is based on the construction of a family of sets whose boundaries have the properties typical of the level surfaces of Lyapunov functions. These sets are found by the method of localization of invariant compact sets. For the resulting Lyapunov function and its derivative, we find analytical expressions via the localizing functions occurring in the specification of the above-mentioned sets. An example of a system with a degenerate equilibrium is considered.     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

Functional method for the localization of invariant compact sets in discrete systems

We suggest a method for the localization of invariant compact sets in discrete autonomous systems. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for invariant compact sets of the discrete Henon system.     

The invariant manifolds of systems with first integrals

Several additional possibilities of the Routh–Lyapunov method for isolating and analysing the stationarity sets of dynamical systems admitting of smooth first integrals are discussed. A procedure is proposed for isolating these sets together with the first integrals corresponding to the vector fields for these sets. This procedure is based on solving the stationarity equations of the family of first integrals of the problem in part of the variables and parameters occurring in this family. The effectiveness of this approach is demonstrated for two problems in the dynamics of a rigid body.     

Localization of invariant compact sets in uncertain discrete systems

A functional method for the localization of invariant compact sets in discrete autonomous systems is generalized to discrete systems with uncertainty. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for positively invariant compact sets of the discrete Henon system with uncertainty.     

Expanding maps on Cantor sets and analytic continuation of zeta functions

In this paper, we study a class of Ruelle dynamical zeta functions related to uniformly expanding maps on Cantor sets. We show that under a non-local integrability condition, the zeta function enjoys a non-vanishing analytic continuation in a strip on the left of the line of absolute convergence. Applying these results to Fuchsian Schottky groups and Julia sets yields precise asymptotics of the number of closed geodesics for convex co-compact surfaces and the distribution of periodic points for a family of Cantor-like Julia sets.     

Indecomposable involutive set-theoretic solutions of the Yang–Baxter equation

We describe the indecomposable involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation as dynamical extensions of non-degenerate left cycle sets. Moreover we characterize the indecomposable dynamical extensions and we produce several examples. As an application we construct a family of finite indecomposable solutions whose structure groups have not the unique product property.     

Invariant critical sets of conserved quantities

For a dynamical system we will construct various invariant sets starting from its conserved quantities. We will give conditions under which certain solutions of a nonlinear system are also solutions for a simpler dynamical system, for example when they are solutions for a linear dynamical system. We will apply these results to the example of Toda lattice.     

Uniform exponential attractors for second order non-autonomous lattice dynamical systems

In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, α-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.     

Bounds for a new chaotic system and its application in chaos synchronization

This paper has investigated the localization problem of compact invariant sets of a new chaotic system with the help of the iteration theorem and the first order extremum theorem. If there are more iterations, then the estimation for the bound of the system will be more accurate, because the shape of the chaotic attractor is irregular. We establish that all compact invariant sets of this system are located in the intersection of a ball with two frusta and we also compute its parameters. It is a great advantage that we can attain a smaller bound of the chaotic attractor compared with the classical method. One numerical example illustrating a localization of a chaotic attractor is presented as well.     

A note on multistep methods and attracting sets of dynamical systems

Summary We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.This work was supported by the U.S. Department of Energy Contract DE-A503-76 ER72012     

Self-similar sets satisfying the common point property

A self-similar set is a fixed point of iterated function system (IFS) whose maps are similarities. We say that a self-similar set satisfies the common point property if the intersection of images of the attractor under the maps of the IFS is a singleton and this point has a common pre-image, under the maps of the IFS, and the pre-image is in the attractor.Self-similar sets satisfying the common point property were introduced in Sirvent (2008) in the context of space-filling curves. In the present article we study some basic topological and dynamical properties of self-similar sets satisfying the common point property. We show examples of this family of sets.We consider attractors of a sub-IFS, an IFS formed from the original IFS by removing some maps. We put conditions on this attractors for having the common point property, when the original IFS have this property.     

Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system

The problem of estimating trajectory tubes of a nonlinear control dynamical system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for set-valued states of dynamical systems under uncertainty.     

Computing reachable sets for uncertain nonlinear monotone systems

We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work Ramdani (2008) [22], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the signs of off-diagonal elements of system’s Jacobian matrix, a hybrid automaton can be obtained, which yields component-wise bounds for the reachable sets. One shortcoming of the method is induced by the need to use whole sets for addressing mode switching. In this paper, we improve this method and show that for the broad class of monotone dynamical systems, component-wise bounds can be obtained for the reachable set in a separate manner. As a consequence, mode switching no longer needs to use whole solution sets. We give examples which show the potentials of the new approach.     

Spectral decomposition theorem in equicontinuous nonautonomous discrete dynamical systems

In this paper, we define the notion of weak chain recurrence and study properties of weak chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. Our main result is the Smale’s spectral decomposition theorem in an equicontinuous nonautonomous discrete dynamical system.     

A computational method for a class of non-standard time optimal control problems involving multiple time horizons

In this paper, we consider a class of non-standard time optimal control problems involving a dynamical system consisting of multiple subsystems evolving over different time horizons. Different subsystems are required to reach their respective target sets at different termination times. The goal is to minimize the maximum of these termination times. By introducing a discrete variable to represent the system termination ordering, we reformulate this problem as a discrete optimization problem. A discrete filled function method is developed to solve this discrete optimization problem. For illustration, a numerical example is solved.