Symmetric periodic solutions of symmetric Hamiltonians in 1 : 1 resonance

Abstract

The aim of this work is to prove analytically the existence of symmetric periodic solutions of the family of Hamiltonian systems with Hamiltonian function Hfalse(q1,q2,p1,p2false)=12false(q12+p12false)+12false(q22+p22false)+a0.1emq14+b0.1emq12q22+c0.1emq24$$ H\left({q}_1,{q}_2,{p}_1,{p}_2\right)=\frac{1}{2}\left({q}_1^2+{p}_1^2\right)+\frac{1}{2}\left({q}_2^2+{p}_2^2\right)+a\kern0.1em {q}_1^4+b\kern0.1em {q}_1^2{q}_2^2+c\kern0.1em {q}_2^4 $$ with three real parameters a,b$$ a,b $$, and c$$ c $$. Moreover, we characterize the stability of these periodic solutions as a function of the parameters. Also, we find a first‐order analytical approach of these symmetric periodic solutions. We emphasize that these families of periodic solutions are different from those that exist in the literature.