Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Collisionless orbits of singular and nonsingular dynamical systems

Pages: 747 - 757, Volume 15, Issue 3, July 2006      doi:10.3934/dcds.2006.15.747

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Daniel Franco - Departamento de Matemática Aplicada, ETSI Industriales, UNED, Apartado de Correos 60149, Madrid, 28080, Spain (email)
J. R. L. Webb - Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom (email)

Abstract: In this paper we study the existence of periodic solutions for some nonlinear systems of differential equations. We assume the nonlinearity satisfies suitable properties in one direction and we consider both the singular and the nonsingular case. As an application we present the existence of collisionless orbits for singular Lagrangian systems where the singular potential can have an attractive or repulsive behaviour near the singularity and we do not need to consider so-called strong force conditions. Our method is based on fixed point index theory for completely continuous operators, involving a new type of cone. In contrast with previous work using this type of technique, the nonlinearity neither needs to be positive nor to have a constant sign behaviour. The results improve recent work even in the scalar case.

Keywords:  Periodic solutions, fixed point index, weak singularity.
Mathematics Subject Classification:  Primary: 34C25; Secondary: 34B16, 47N20.

Received: October 2005;      Revised: December 2005;      Available Online: April 2006.