Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems

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doi:10.3934/jgm.2017017

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2017, Volume 9, Issue 4: 439-457. Doi: 10.3934/jgm.2017017

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Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems

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1.
Faculty of Science, University of Ontario Institute of Technology, Oshawa, ONT L1H 7K4, Canada
2.
Department of Mathematics and Statistics, Queen's University, Kingston, ONT K7L 3N6, Canada

* Corresponding author: Pietro-Luciano Buono
* Corresponding author: Pietro-Luciano Buono

Received: May 31, 2010

Revised: March 31, 2017

Published: November 2017

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Abstract

We consider the question of linear stability of a periodic solution $z(t)$ with finite spatio-temporal symmetry group of a reversible-equivariant Hamiltonian system obtained as a minimizer of the action functional. Our main theorem states that $z(t)$ is unstable if a subspace $W$ associated with the boundary conditions of the minimizing problem is a Lagrangian subspace with no focal points on the time interval defined by the boundary conditions and the second variation restricted to the subspace $W$ at the minimizer has positive directions. We show that the conditions of our theorem are always met for a class of minimizing periodic orbits with the standard mechanical reversing symmetry. Comparison theorems for Lagrangian subspaces and the use of time-reversing symmetries are essential tools in constructing stable and unstable subspaces for $z(t)$. In particular, our results are complementary to the recent paper of Hu and Sun Commun. Math. Phys. 290, (2009).

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Mathematics Subject Classification: Primary: 37J25, 37J45; Secondary: 37G40.

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