Invariant tori of Duffing-type equations
A class of Duffing-type equations \[ {d^2x\over dt^2}+x^{2n+1} +\sum^{2 n}_{j=0} p_j(t)x^j= 0,\;n\geq 1,\tag{1} \] where \(p_j(t)\) \((j=0,1,\dots,2n)\) are 1-periodic functions, is considered. It is known that equation (1) possesses invariant tori in the \((x,\dot x,t)\)-space arbitrary far from the \(t\)-axis when \(p_j(t)\) are sufficiently smooth. The existence of invariant tori means that all solutions to (1) are bounded and there exist quasiperiodic solutions to (1). The problem of differentiability of the functions \(p_j(t)\) is discussed. Recently it was proved that the functions \(p_j(t)\) are 1-periodic \(C^{(5+\varepsilon)}\) functions \((\varepsilon >0)\). Here, the required degree of differentiability of the \(p_j(t)\)'s decreased to \(C^{(1)}\) for \(p_j(t)\), \(j=0,1,\dots,n\), and to \(C^{(2)}\) for \(p_j(t)\), \(j=n+1,\dots,2n\). The following theorem is proved. Assume that \(p_j(t)\), \(j=0,1,\dots,n\), are 1-periodic \(C^{(1)}\) functions and \(p_j(t)\), \(j=n+1,\dots,2n\), are 1-periodic \(C^{(2)}\) functions, then the Poincaré map \(P:(x,\dot x)_{t=0} \mapsto(x,\dot x)_{t= 1}\) of equation (1) possesses countable many invariant curves whose radii tend to infinity. The boundedness and the existence of quasiperiodic solutions to (1) are obtained as simple corollaries of the theorem.
scientific article; zbMATH DE number 1212872