Publication detail

On the Position of Chaotic Trajectories

DIBLÍK, J. CALAMAI, A. FRANCA, M. POSPÍŠIL, M.

Original Title

On the Position of Chaotic Trajectories

English Title

On the Position of Chaotic Trajectories

Type

journal article in Web of Science

Language

en

Original Abstract

The main purpose of this paper is to locate trajectories of a perturbed system, which is known to behave chaotically. The unperturbed system is assumed to have the origin as a hyperbolic fixed point, and to admit a trajectory homoclinic to the origin. This homocline is assumed to lie in a prescribed region having the origin in its border. Using a Mel’nikov type approach, we introduce natural conditions ensuring that all the chaotic trajectories of the perturbed system, given by classical results, lie in the same region. The applicability of our results is illustrated in two examples. In the first one, we find positive radial solutions for a class of P.D.E.’s, obtaining new results in the case of critical equations ruled by Laplacian with Hardy potentials. In the other one, we show that under certain conditions one of two weakly coupled pendula moves in one direction only.

English abstract

The main purpose of this paper is to locate trajectories of a perturbed system, which is known to behave chaotically. The unperturbed system is assumed to have the origin as a hyperbolic fixed point, and to admit a trajectory homoclinic to the origin. This homocline is assumed to lie in a prescribed region having the origin in its border. Using a Mel’nikov type approach, we introduce natural conditions ensuring that all the chaotic trajectories of the perturbed system, given by classical results, lie in the same region. The applicability of our results is illustrated in two examples. In the first one, we find positive radial solutions for a class of P.D.E.’s, obtaining new results in the case of critical equations ruled by Laplacian with Hardy potentials. In the other one, we show that under certain conditions one of two weakly coupled pendula moves in one direction only.

Keywords

Chaotic behaviour; Hardy potential;· Bernoulli shift; Mel’nikov integral

Released

01.12.2017

Publisher

Springer

ISBN

1040-7294

Periodical

Journal of Dynamics and Differential Equations

Year of study

29

Number

4

State

US

Pages from

1423

Pages to

1458

Pages count

36

URL

Documents

BibTex


@article{BUT142523,
  author="Alessandro {Calamai} and Josef {Diblík} and Matteo {Franca} and Michal {Pospíšil}",
  title="On the Position of Chaotic Trajectories",
  annote="The main purpose of this paper is to locate trajectories of a perturbed system, which is known to behave chaotically. The unperturbed system is assumed to have the origin as a hyperbolic fixed point, and to admit a trajectory homoclinic to the origin. This homocline is assumed to lie in a prescribed region having the origin in its border. Using a Mel’nikov
type approach, we introduce natural conditions ensuring that all the chaotic trajectories of the perturbed system, given by classical results, lie in the same region. The applicability of our results is illustrated in two examples. In the first one, we find positive radial solutions for a class of P.D.E.’s, obtaining new results in the case of critical equations ruled by Laplacian
with Hardy potentials. In the other one, we show that under certain conditions one of two weakly coupled pendula moves in one direction only.",
  address="Springer",
  chapter="142523",
  doi="10.1007/s10884-016-9520-z",
  howpublished="print",
  institution="Springer",
  number="4",
  volume="29",
  year="2017",
  month="december",
  pages="1423--1458",
  publisher="Springer",
  type="journal article in Web of Science"
}