Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 8.5

Center Manifolds and Bifurcation Theory

Mathematics — Calculus · periodic-solution-floquet-multipliers

Problem

Consider the system x1′=x1+x22−sin⁡t,x2′=−x2.x_1' = x_1 + x_2^2 - \sin t, \quad x_2' = -x_2. (a) Find the solution to the initial value problem x(t,p)x(t, p). (b) Show that there is a unique p∈R2p \in \mathbb{R}^2 such that x(t,p)x(t, p) is periodic. (c) Find the Floquet multipliers of this periodic solution. (d) Show that the periodic solution is hyperbolic and find the local stable and unstable manifolds.

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