Invariant Sets Of Dynamical Systems Research Articles

On the Minkowski dimension of some invariant sets of dynamical systems

Abstract. It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stability, the closure of the non-singular trajectory is a parametrically defined curve of the fractal type. In this work the fractal dimension of such curves is calculated. In addition, special two-parameter family of functions is studied such that Minkowski dimension of their graphs varies from 1 to 2. The obtained result allows us to implement a regular dynamic system with an isolated hyperbolic point such that the closure of two-dimensional stable manifold of this point may have Minkowski dimension greater than 2. To calculate the graph dimension, the segment of the argument defining the graph is split into two parts. The dimension of the first part of the graph can be estimated from above by direct calculation of the corresponding curve’s length. The upper estimation of the other part’s dimension is provided by means of the area of rectangle containing this curve. The lower estimation of the Minkowski dimension is based on calculating the cardinality of ε -distinguishable set of graph points.

Dimension Analysis via Differential Lyapunov and Dissipation Inequalities⋆

Motivated by complex and diverse dynamics in engineering and nature, this paper considers Hausdorff dimension estimates for compact invariant sets of dynamical systems. Our aim is to develop a framework of dimension analysis as an extension of Lyapunov's stability theory as well as Willems’ dissipativity theory. In particular, we extend the existing result by Smith involving Lyapunov inequalities for Hausdorff dimension estimates by employing state-dependent metrics. The obtained result is then combined with the dissipativity framework for studying interconnected systems. This paper is mainly devoted to mathematics but describes several connections with recent works on the differential framework for contraction analysis.

Pseudo-prolongations in the qualitative theory of dynamical systems

This paper considers the qualitative behaviour of the flow in a neighbourhood of closed invariant sets of dynamical systems. The properties of compactness, invariance, and connectivity of pseudo-prolongations are investigated. A rather deep analysis of the flow in the vicinity of a compact invariant set of asymptotically compact phase spaces is presented. The connection of pseudo-prolongation with the first positive prolongation of T. Ura and the set of weakly elliptic points is refined.

Kleene Algebra to Compute Invariant Sets of Dynamical Systems

In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to provide the definitions and theorems that will allow us to make the link between the theory of invariant sets and the Kleene algebra. This link has never be done before and will allow us to compute rigorously sets that can be defined as a combination of positive invariant sets. Some illustrating examples show the nice properties of the approach.

Provably convergent implementations of the subdivision algorithm for the computation of invariant objects

The subdivision algorithm by Dellnitz and Hohmann for the computation of invariant sets of dynamical systems decomposes the relevant region of the state space into boxes and analyzes the induced box dynamics. Its convergence is proved in an idealized setting, assuming that the exact time evolution of these boxes can be computed. In the present article, we show that slightly modified, directly implementable versions of the original algorithm are convergent under very mild assumptions on the dynamical system. In particular, we demonstrate that neither a fine net of sample points nor very accurate approximations of the precise dynamics are necessary to guarantee convergence of the overall scheme.

Automatic generation of bounds for polynomial systems with application to the Lorenz system

This study covers an analytical approach to calculate positive invariant sets of dynamical systems. Using Lyapunov techniques and quantifier elimination methods, an automatic procedure for determining bounds in the state space as an enclosure of attractors is proposed. The available software tools permit an algorithmizable process, which normally requires a good insight into the systems dynamics and experience. As a result we get an estimation of the attractor, whose conservatism only results from the initial choice of the Lyapunov candidate function. The proposed approach is illustrated on the well-known Lorenz system.

The wild Fox–Artin arc in invariant sets of dynamical systems

ABSTRACTThe modern qualitative theory of dynamical systems is thoroughly intertwined with the fairly young science of topology. Many strange constructions of topology are found sooner or later in dynamics of discrete or continuous dynamical systems. In the present paper, we show that the wild Fox–Artin arc emerges naturally in invariant sets of dynamical systems.

On Solving of a Special Optimization Problem Connected with Determination of Invariant Sets of Dynamical Systems

On Solving of a Special Optimization Problem Connected with Determination of Invariant Sets of Dynamical Systems

Nonlinear phenomena and transient modes in the dynamics of new bioresources control models

This article is dedicated to research of series nonlinear effects in dynamical systems modeling the dynamics of controlled biological systems. The problem of analysis of the results mathematical modeling regularly faced with phenomena which are beyond the scope of interpretation in the proposed area of application. Traditionally, investigation of models focuses on definition of the structure of asymptotically stable of stationary states, closed invariant sets of dynamical systems. It is shown by the examples of this paper, that noninvariant sets and complex transitional regimes have an important influence in the analysis of simulation results.

The computation of convex invariant sets via Newton's method

In this paper we present a novel approach to the computation of convex invariant sets of dynamical systems. Employing a Banach space formalism to describe differences of convex compact subsets of $\mathbb{R}^n$ by directed sets, we are able to formulate the property of a convex, compact set to be invariant as a zero-finding problem in this Banach space. We need either the additional restrictive assumption that the image of sets from a subclass of convex compact sets under the dynamics remains convex, or we have to convexify these images. In both cases we can apply Newton's method in Banach spaces to approximate such invariant sets if an appropriate smoothness of a set-valued map holds. The theoretical foundations for realizing this approach are analyzed, and it is illustrated first by analytical and then by numerical examples.

Computation of polytopic invariants for polynomial dynamical systems using linear programming

This paper deals with the computation of polytopic invariant sets for polynomial dynamical systems. An invariant set of a dynamical system is a subset of the state space such that if the state of the system belongs to the set at a given instant, it will remain in the set forever in the future. Polytopic invariants for polynomial systems can be verified by solving a set of optimization problems involving multivariate polynomials on bounded polytopes. Using the blossoming principle together with properties of multi-affine functions on rectangles and Lagrangian duality, we show that certified lower bounds of the optimal values of such optimization problems can be computed effectively using linear programs. This allows us to propose a method based on linear programming for verifying polytopic invariant sets of polynomial dynamical systems. Additionally, using sensitivity analysis of linear programs, one can iteratively compute a polytopic invariant set. Finally, we show using a set of examples borrowed from biological applications, that our approach is effective in practice.

Dynamics of inner mappings

We study properties of invariant sets of dynamical systems generated by inner mappings. We prove that if x is a nonwandering point of a finitely multiple inner mapping, then not only its positive trajectory O+(x) consists of nonwandering points but also the negative trajectory O−(x) contains at least one partial semitrajectory consisting of nonwandering points.

Symbolic dynamics for nonhyperbolic systems

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.

Topological Approach to Rigorous Numerics of Chaotic Dynamical Systems with Strong Expansion of Error Bounds

A new algorithm for obtaining rigorous results concerning the existence of chaotic invariant sets of dynamical systems generated by non-autonomous, time-periodic differential equations is presented. Unlike all other algorithms the presented algorithm does not require the numerical integration of the solutions and as a consequence it is insensitive to the rapid error growth in the case of long integration. The result is based on a new theoretical approach to the computation of the homology of the Poincare map. A concrete numerical example concerning a time-periodic differential equation in the complex plane is provided.

Stability of invariant measures

We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and it also differs from topologies induced by the Riesz Representation Theorem. It turns out that the constructed topology is a solution of a limit case of a $p$-optimal transport problem, for $p=\infty$.

Global Optimization using a Dynamical Systems Approach

We develop new algorithms for global optimization by combining well known branch and bound methods with multilevel subdivision techniques for the computation of invariant sets of dynamical systems. The basic idea is to view iteration schemes for local optimization problems – e.g. Newton’s method or conjugate gradient methods – as dynamical systems and to compute set coverings of their fixed points. The combination with bounding techniques allow for the computation of coverings of the global optima only. We show convergence of the new algorithms and present a particular implementation.

On the Structure of a Neighborhood of Stable Compact Invariant Sets

The analysis of the phase portrait of trajectories in a neighborhood of stable invariant sets of dynamical systems is one of the main problems in qualitative theory of stability of motion. This problem has been under study starting from Lyapunov’s work [1]; in particular, for abstract dynamical systems, it was considered in [2–8], where a wide bibliography on the topic can be found. However, only the case of asymptotic stability has been sufficiently well analyzed (see [1–3, 9]). The so-far existing practice of scientific research shows that, in the case of nonasymptotic stability, the qualitative picture of trajectories near compact invariant sets is very complicated and immensely variegated, which explains why the problem of qualitatively distinguishing all possible situations has not been satisfactorily solved yet. In the present paper, we suggest an approach to the study of the structure of a neighborhood of compact invariant sets for locally compact dynamical systems and use it to give some classification of types of orbitally nonasymptotically stable invariant sets. The analysis carried out below permits one to single out five basic classes of first-level sets and 121 subclasses of second-level sets. Related illustrative examples of dynamical systems are given. Let the triple (X, , π) denote the dynamical system defined on a locally compact metric space (X, d) with phase mapping π : X × → X (π : (x, t) → xt) [2, 3]. For each point x ∈ X, the mapping x : t → xt is referred to as the motion passing through x, and the sets

Localization of Invariant Sets of Dynamical Systems

Invariant sets of dynamical systems are considered. The localization inside some compact set and isolating neighborhoods are constructed. Bibliography: 6 titles. Illustrations: 3 figures.

Invariant sets of mechanical systems

Problems of the existence and stability of invariant sets of mechanical systems are discussed (in particular, steady motions and relative equilibria). The existence problem of stable steady motions (zero-dimensional invariant sets) was first investigated in [1]. The Routh theory [1–15] gives stability conditions of steady motions of conservative mechanical systems with first integrals as well as a construction method for such steady motions. This method was generalized for dissipative systems with first integrals in [12–15]. Recently, the Routh theory was modified for the existence and stability problems of invariant sets of dynamical systems with a nonincreasing energy function and first integrals [7,14] (in particular, for conservative and dissipative mechanical systems with symmetry [16–19]).

SPECTRA OF DIMENSIONS FOR POINCAR´ E RECURRENCES FOR SPECIAL FLOWS

We prove the variational principle for dimensions for Poincar´e recurrences, in the case of invariant sets of dynamical systems with continuous time. To achieve this goal we show that these dimensions can be expressed as roots of a non–homogeneous Bowen equation.