Periodic orbits in Hamiltonian systems with involutory symmetries

Reem Alomair, James Montaldi

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    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 aHamiltonian systemwith a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibriumpoint 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 systemwith symmetry R
    Original languageEnglish
    Pages (from-to)1-25
    JournalJournal of Dynamics and Differential Equations
    Early online date8 Feb 2016
    Publication statusPublished - 2016


    • Lyapunov centre theorem
    • time-reversing symmetry
    • nonlinear normal modes


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