A Geometric Approach to Invariant Sets for Dynamical Systems
نویسندگان
چکیده
In this article, we present a geometric framework to study invariant sets of dynamical systems associated with differential equations. This framework is based on properties of invariant sets for an area functional. We obtain existence results for heteroclinic and periodic orbits. We also implement this approach numerically by means of the steepest descent method.
منابع مشابه
Approximation of invariant foliations for stochastic dynamical systems ∗
Invariant foliations are geometric structures for describing and understanding the qualitative behaviors of nonlinear dynamical systems. For stochastic dynamical systems, however, these geometric structures themselves are complicated random sets. Thus it is desirable to have some techniques to approximate random invariant foliations. In this paper, invariant foliations are approximated for dyna...
متن کاملThe Exact Solution of Min-Time Optimal Control Problem in Constrained LTI Systems: A State Transition Matrix Approach
In this paper, the min-time optimal control problem is mainly investigated in the linear time invariant (LTI) continuous-time control system with a constrained input. A high order dynamical LTI system is firstly considered for this purpose. Then the Pontryagin principle and some necessary optimality conditions have been simultaneously used to solve the optimal control problem. These optimality ...
متن کاملEntropy of infinite systems and transformations
The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with ...
متن کاملEntropy and Rotation Sets: a Toymodel Approach
Given a continuous dynamical system f on a compact metric space X and a continuous potential Φ : X → R, the generalized rotation set is the subset of R consisting of all integrals of Φ with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measuretheoretic entropies over all invariant measures whose integrals pr...
متن کاملA local approach to the entropy of countable fuzzy partitions
This paper denes and investigates the ergodic proper-ties of the entropy of a countable partition of a fuzzy dynamical sys-tem at different points of the state space. It ultimately introducesthe local fuzzy entropy of a fuzzy dynamical system and proves itto be an isomorphism invariant.
متن کامل