Invariant Surfaces of Periodic Systems with Conservative Cubic First Approximation

Abstract

Two classes of time-periodic systems of ordinary differential equations with a small parameter ε ≥ 0, those with “fast” and “slow” time, are studied. The corresponding conservative unperturbed systems \({{\dot {x}}_{i}}\) = \( - {{\gamma }_{i}}{{y}_{i}}{{\varepsilon }^{\nu }}\), \({{\dot {y}}_{i}}\) = γ_i(\(x_{i}^{3}\) – \({{\eta }_{i}}{{x}_{i}}\))ε^ν (i = \(\overline {1,n} \), ν = 0, 1) have 1 to 3ⁿ singular points. The following results are obtained in explicit form: (1) conditions on perturbations independent of the parameter under which the initial systems have a certain number of invariant surfaces of dimension n + 1 homeomorphic to the torus for all sufficiently small parameter values; (2) formulas for these surfaces and their asymptotic expansions; (3) a description of families of systems with six invariant surfaces.