American Institute of Mathematical Sciences

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August  2006, 15(3): 747-757. doi: 10.3934/dcds.2006.15.747

Collisionless orbits of singular and nonsingular dynamical systems

Daniel Franco 1, and  J. R. L. Webb 2,

1. 

Departamento de Matemática Aplicada, ETSI Industriales, UNED, Apartado de Correos 60149, Madrid, 28080
2. 

Department of Mathematics, University of Glasgow, Glasgow G12 8QW

Received  October 2005 Revised  December 2005 Published  April 2006

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: weak singularity., fixed point index, Periodic solutions.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34B16, 47N2.
Citation: Daniel Franco, J. R. L. Webb. Collisionless orbits of singular and nonsingular dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 747-757. doi: 10.3934/dcds.2006.15.747
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