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2015.110: Periodic orbits in Hamiltonian systems with involutory symmetries

2015.110: Reem Alomair and James Montaldi (2016) Periodic orbits in Hamiltonian systems with involutory symmetries. J. Dynamics and Differential Equations, to appear.

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DOI: 10.1007/s10884-016-9519-5


We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetry. In both classes, the involution reverses the sign of the Hamiltonian function, and the system is in 1:-1 resonance. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R, and furthermore that there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian (correcting a minor error in their paper). In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R.

Item Type:Article
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Version 2, post-refereeing (24 Jan 2016)

Uncontrolled Keywords:symmetry, time-reversing symmetry, Liapunov centre theorem, nonlinear normal modes
Subjects:MSC 2000 > 37 Dynamical systems and ergodic theory
MIMS number:2015.110
Deposited By:Dr James Montaldi
Deposited On:05 December 2015

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