On the control of stability of periodic orbits of completely integrable systems

Abstract

We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.

Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics

“…The aim of this section is to apply the main results of the above section in order to provide an answer to the following problem: given a conservative n−dimensional dynamical system (i.e., a dynamical system which admits a (k + p)−dimensional vector type first integral, where k + p < n; for a brief introduction see, e.g., [2], [3]) and an invariant set S (given as the level set of a k−dimensional first integral defined by some k−dimensional projection of the original (k + p)−dimensional first integral), construct a curve of dynamical systems starting from the original system, such that each system on this curve is still conservative (admitting the p−dimensional first integral which together with the k−dimensional first integral, forms the original (k + p)−dimensional first integral), keeps invariant the set S ∩ Mrk (where Mrk is the open set consisting of the points where the rank of the (k +p)−dimensional first integral is maximal), and moreover, the intersection of S ∩ Mrk with each level set (corresponding to regular values) of the p−dimensional first integral, is an attracting set of each system on the curve (excepting the original system).…”

Asymptotic stabilization with phase of periodic orbits of three-dimensional Hamiltonian systems

“…The purpose of this section is to improve, for locally generic three-dimensional Hamiltonian dynamical systems, the stabilization technique introduced in [9], and also to provide a new stabilization result. More precisely, we show that one of the main hypothesis required by the stabilization technique from [9], is always satisfied in the case of locally generic three-dimensional Hamiltonian systems.…”

“…The purpose of this section is to improve, for locally generic three-dimensional Hamiltonian dynamical systems, the stabilization technique introduced in [9], and also to provide a new stabilization result. More precisely, we show that one of the main hypothesis required by the stabilization technique from [9], is always satisfied in the case of locally generic three-dimensional Hamiltonian systems. Moreover, we will show that the elimination of this hypothesis leads to a new stabilization method, which provides phase asymptotic stability of a certain periodic orbit, even in the case when we are not able to find a parameterization of the orbit (which is the case in many concrete dynamical systems).…”

“…In all cases, the perturbed families of dynamical systems are parameterized by arbitrary strictly positive smooth real functions. This method is an improvement of the n-dimensional stabilization technique introduced in [9], for the particular case of locally generic three-dimensional Hamiltonian systems. In contrast with the general case analyzed in [9], where in order to apply the stabilization method, one needs to know a parameterization of the periodic orbit to be stabilized, here, the only requirement regarding the periodic orbit is the knowledge of the values of the Hamiltonian and the Casimir at the orbit.…”

“…This method is an improvement of the n-dimensional stabilization technique introduced in [9], for the particular case of locally generic three-dimensional Hamiltonian systems. In contrast with the general case analyzed in [9], where in order to apply the stabilization method, one needs to know a parameterization of the periodic orbit to be stabilized, here, the only requirement regarding the periodic orbit is the knowledge of the values of the Hamiltonian and the Casimir at the orbit. Moreover, for the class of locally generic three-dimensional Hamiltonian systems, we prove that one of the main hypotheses from [9] is always fulfilled.…”