Lyapunov families of periodic motions in a reversible system (Q2761255)

The author studies the smooth reversible system \(\dot u = Av+U(u,v)\), \(\dot v = Bu+V(u,v)\), where \(u\in \mathbb R^l\), \(v\in \mathbb R^n\), \(l\geq n\), \(U(0,0)= V(0,0)=0\), \(U(u,-v)=-U(u,v)\), \(V(u,-v)=U(u,v)\), and \(A\) and \(B\) are constant matrices. The system is assumed to possess a set of fixed points \(M = \{(u,v): v=0\}\). The author establishes conditions for the existence of \((l-n)\)-parametric manifold of equilibrium state of the set \(M\) and finds a one-parametric Lyapunov family of periodic motions. The bifurcations of periodic solutions is analyzed when a pair of roots of the characteristic equation passes through zero. As an example, the author considers the motion of heavy homogeneous ellipsoid on an absolutely rough plane.