Analytical solutions of a particular Hill's differential system

Consider a second order differential linear periodic equation. This equation is recast as a first-order homogeneous Hill’s system. For this system we obtain analytical solutions in explicit form. The first solution is a periodic function. The second solution is a sum of two functions; the first is a continuous periodic function, but the second is an oscillating function with monotone linear increasing amplitude. We give a formula to directly compute the slope of this increase, without knowing the second numerical solution. The periodic term of second solution may be computed directly. The coefficients of fundamental matrix of the system are analytical functions.

REMARKS ON SKEW PRODUCTS OVER IRRATIONAL ROTATIONS

We prove several results of the orbit behavior of skew product diffeomorphisms generated by quasi-periodic differential systems. The first diffeomorphism is derived from a periodic differential equation on the circle by means of a construction proposed by Z. Opial to get a scalar quasi-periodic equation with all its solutions bounded but without an almost periodic solution. We consider both possible cases for the irrational rotation number, transitive and singular (intransitive). The main result for a transitive case is that the related skew product diffeomorphism has a foliation into invariant curves with pure irrational rotation on each curve (being the same for each curve). For intransitive case, we get invariant sets of two types: a collection of continuous invariant curves and invariant sets being dimensionally inhomogeneous ones.Section 3 is devoted to perturbations of a skew product diffeomorphism over an irrational rotation being initially foliated into invariant curves. We prove an analog of Poincaré–Pontryagin theorem which sets conditions when a perturbation of a one-degree-of-freedom Hamiltonian system (given in an annulus and written down in action-angle variables) has limit cycles. Our theorem provides sufficient conditions when a perturbation of a foliated skew product diffeomorphism has isolated invariant curves (asymptotically stable or unstable).In Sec. 4 we present some results on the geometry of skew product diffeomorphisms derived by a quasi-periodic Riccati equation.