Stochastic analysis of dynamical systems with delayed control forces

Reduction of structural vibration in actively controlled dynamical system is usually performed by means of convenient control forces dependent of the dynamic response. In this paper the existent studies will be extended to dynamical systems subjected to non-normal delta-correlated random process with delayed control forces. Taylor series expansion of the control forces has been introduced and the statistics of the dynamical response have been obtained by means of the extended Itô differential rule. Numerical application provided shows the capabilities of the proposed method to analyze stochastic dynamic systems with delayed actions under delta-correlated process contrasting statistics of response with estimates from Monte-Carlo (MC) simulation.     

非自治动力系统拓扑压变分原理的一点注记

用类似于非自治熵的变分原理的方法,证明了非自治拓扑压的变分原理的一个不等式,推广了非自治熵的变分原理,丰富了非自治变分原理的内容.     

Reversible dynamical systems: Dissipation-induced destabilization and follower forces

This paper uses results from the theory of reversible dynamical systems to examine the nonlinear stability of undamped follower force systems. The well-known dissipation-induced destabilization found in these systems is also examined and it is indicated how the postdestabilization dynamics can be determined using normal forms and an eigenvalue splitting criterion.     

Chaos induced by weak A-coupled-expansion of non-autonomous discrete dynamical systems

This paper focuses on chaos induced by weak A-coupled-expansion of non-autonomous discrete systems in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, separately. A new concept of weak A-coupled-expansion for non-autonomous discrete systems, whose condition is weaker than that of A-coupled-expansion, is introduced, and several new criteria of chaos induced by weak A-coupled-expansion of non-autonomous discrete systems are established. By applying some close relationships between chaotic dynamical behaviours of the original system and its induced systems, two criteria of chaos are established. One example is provided for illustration.     

Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

Hyperspace dynamical system (E²,f²) induced by a given dynamical system (E,f) has been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on E² employed in these studies is non-metrizable when E is not compact metrizable, e.g., E=Rn. Consequently, metric related dynamical concepts of (E²,f²) such as sensitivity on initial conditions and metric-based entropy, could not even be defined. Moreover, a condition on (E²,f²) equivalent to the transitivity of (E,f) has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When E is HLCSC, the hit-or-miss topology on E² is again HLCSC, thus metrizable. In this paper, the concepts of co-compact mixing, co-compact weak mixing and co-compact transitivity are introduced for dynamical systems. For any HLCSC system (E,f), these three conditions on (E,f) are respectively equivalent to mixing, weak mixing and transitivity on (E²,f²) (hit-or-miss topology equipped). Other noticeable properties of co-compact mixing, co-compact weak mixing and co-compact transitivity such as invariants for topological conjugacy, as well as their relations to mixing, weak mixing and transitivity, are also explored.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

Remarks on global controllability for the Burgers equation with two control forces

In this paper we deal with the viscous Burgers equation. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. In a first result, we prove that the global exact null controllability does not hold for small time. In a second one, we prove that the exact controllability result does not hold even for large time.     

Linear dynamical systems with integer constraints

We study properties of dynamic discrete-time linear systemswith the input or the state constrained to have integer components.The reachability of input-constrained systems is investigated,and consistency conditions for state-constrained systems aregiven.     

Linear dynamical systems with partial derivatives

Systems with multidimensional time over Lie-algebras are considered. It is proved that a transfer function determines a minimal system up to similarity.     

Impulsive surfaces on dynamical systems

This work is devoted to the construction of impulsive sets in \({\mathbb{R}^n}\). In the literature, there are many examples of impulsive dynamical systems whose impulsive sets are chosen in an abstract way, and in this paper we present sufficient conditions to characterize impulsive sets in \({\mathbb{R}^n}\) which satisfy some “tube conditions” and ensure a good behavior of the flow. Moreover, we present some examples to illustrate the theoretical results.     

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.     

Remarks and examples concerning the weak basis theorem

This work was undertaken with the support of the AvH.