平均跟踪与伪轨跟踪

证明了紧致度量空间中有伪轨跟踪的distal同胚不具有平均跟踪性,并给出了有平均跟踪性的同胚是极小的一个充分条件。     

Generic points for dynamical systems with average shadowing

We study the Besicovitch pseudometric \(D_B\) for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to \(D_B\). It is proved that the weak specification property implies the average asymptotic shadowing property and the latter property does not imply the former one nor the almost specification property. Furthermore an example of a proximal system with the average shadowing property is constructed. It is proved that to every invariant measure \(\mu \) of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for \(\mu \). A simple consequence of the theory presented is that every invariant measure has a generic point in a system with the asymptotic average shadowing property.     

Weak shadowing for discrete dynamical systems on nonsmooth manifolds

In J. Math. Anal. Appl. 189 (1995) 409-423, Corless and Pilyugin proved that weak shadowing is a C⁰ generic property in the space of discrete dynamical systems on a compact smooth manifold M. In our paper we give another proof of this theorem which does not assume that M has a differential structure. Moreover, our method also works for systems on some compact metric spaces that are not manifolds, such as a Hilbert cube (or generally, a countably infinite Cartesian product of manifolds with boundary) and a Cantor set.     

Tubes in dynamical systems

Important concepts concerning tubes in dynamical systems are defined in details. In the cases of a homoclinic tube and a heteroclinically tubular cycle in autonomous systems, existence of tubular chaos is established. The main goal of this article is to stress the importance of tubes in high dimensional dynamical systems.     

Dynamical systems with Lipschitz inverse shadowing properties

In this paper, the notion of the Lipschitz inverse shadowing property with respect to two classes of d-methods that generate pseudotrajectories of dynamical systems is introduced. It is shown that if a diffeomorphism of a Euclidean space has the Lipschitz inverse shadowing property on the trajectory of an individual point, then the Mañé analytic strong transversality condition must be satisfied at this point. This result is used in the proof of the main theorem: a diffeomorphism of a smooth closed manifold that has the Lipschitz inverse shadowing property is structurally stable.     

Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property

We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let be a compact metric space.

First we show the following. If is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map with , then is topologically mixing. If and are commuting expansive onto continuous maps with POTP and if is topologically transitive with period , then for some dividing , , where the , , are the basic sets of with such that all have period , and the dynamical systems are a factor of each other, and in particular they are conjugate if is a homeomorphism.

Then we prove an extension of a basic result in symbolic dynamics. Using this and many techniques in symbolic dynamics, we prove the following. If is a topologically transitive, positively expansive onto continuous map having POTP, and is a positively expansive onto continuous map with , then has POTP. If is a topologically transitive, expansive homeomorphism having POTP, and is a positively expansive onto continuous map with , then has POTP and is constant-to-one.

Further we define `essentially LR endomorphisms' for systems of expansive onto continuous maps of compact metric spaces, and prove that if is an expansive homeomorphism with canonical coordinates and is an essentially LR automorphism of , then has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.

     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

Quasi-random dynamical systems

This paper is the author's abstract of his dissertation for the degree Doctor of Physiomathematical Science. The dissertation was defended on March 7, 1969, in the Study of the Soviet Mechanical-Mathematical Faculty of the M. V. Lomonosov Moscow State University. Official opponents: Academician A. N. Kolmogorov, Dr. of Physicomathematical Sciences D. V. Anosov, and Dr. of Physicomathematical Sciences Professor V. B. Lidskii.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 489–498, October, 1969.     

具有渐近平均跟踪性质的系统

简记渐近平均跟踪性质为AASP.对于紧致度量空间上的连续映射f,证明了:(1)f有AASP当且仅当其逆极限空间上的移位映射有AASP;(2)若f有AASP且是等度连续的,则f是极小同胚.此外,讨论了AASP的拓扑共轭不变性.     

Hopf bifurcations in dynamical systems

The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

     

Coarse-grained entropy in dynamical systems

Let M be the phase space of a physical system. Consider the dynamics, determined by the invertible map T: M → M, preserving the measure µ on M. Let ν be another measure on M, dν = ρdµ. Gibbs introduced the quantity s(ρ) = ?∝ρ log ρdµ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (fine-grained) entropy the so called coarse-grained entropy. First we obtain a formula for the difference between the coarse-grained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information. Then we consider the behavior of the coarse-grained entropy as a function of time. The dynamics transforms ν in the following way: ν → ν _n , dν _n = ρ ○ T ^?n dµ. Hence, we obtain the sequence of densities ρ _n = ρ ○ T ^?n and the corresponding values of the Gibbs and the coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map T. Finally, we give numerical calculation of the coarse-grained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements.