On the birth of minimal sets for perturbed reversible vector fields
A four dimensional autonomous system with the symmetry \((x,y,z,w)\mapsto (x,-y,z-w)\) is called reversible. It is shown that the system \[ \begin{aligned} \dot{x}&=-py-y(ax^2+ay^2+bz^2+bw^2)\\ \dot{y}&= px+x(ax^2+ay^2+bz^2+bw^2)\\ \dot{z}&=-qw-w(cx^2+cy^2+dz^2+dw^2)\\ \dot{w}&=qz+z(cx^2+cy^2+dz^2+dw^2),\end{aligned} \] with \(a,b,c,d\in\mathbb{R}\) and \(p,q\in\mathbb{N}\) coprime, is a normal form up to order three for reversible systems around an elliptic equilibrium point. Using the Lyapunov-Schmidt reduction, the authors find sufficient conditions for (1) to possess a one-parameter family of periodic orbits. There is also proved the existence of one, two or three tori filled with periodic orbits for the class of cubic systems (i.e. \(f_i\), \(i=\overline{1,4}\) are cubic polynomials) \[ \begin{aligned}\dot{x}&=-py-\varepsilon y(ax^2+ay^2+bz^2+bw^2)+\varepsilon f_1(x,y,z,w)\\ \dot{y}&=px+\varepsilon x(ax^2+ay^2+bz^2+bw^2)+\varepsilon f_2(x,y,z,w)\\ \dot{z}&=-qw-\varepsilon w(cx^2+cy^2+dz^2+dw^2)+\varepsilon f_3(x,y,z,w)\\ \dot{w}&=qz+\varepsilon z(cx^2+cy^2+dz^2+dw^2)+\varepsilon f_4(x,y,z,w).\end{aligned} \]
scientific article; zbMATH DE number 5988154