2015/07/06 by R. A. Alomair, Alomair, Reem, James Montaldi +1
Materials Science · Physics and Astronomy · #37C27 #37J15 #Dynamical Systems (math.DS) #FOS: Mathematics #Magnetism in coordination complexes #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems
paper · doi:10.48550/arxiv.1507.01392
openalex publication_date 2015/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 symmetries. In both classes, involutions reverse the sign of the Hamiltonian function. 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 there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian. 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.