Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

Abstract

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales-Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable.We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homocliic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Figures (5)

• Figure 1: Assumptions (A1) and (A2).
• Figure 2: Riemann surface $\Gamma=x^\mathrm{h}(U)\cup W_+^\mathrm{s}\cup W_-^\mathrm{u}$.
• Figure 3: Phase portrait of \ref{['eqn:ex1']} with $\varepsilon=0$.
• Figure 4: Phase portrait of \ref{['eqn:ex2']} with $\varepsilon=0$.
• Figure 5: Phase portrait of \ref{['eqn:ex2']} with $\varepsilon=0$.

Theorems & Definitions (23)

• Definition 1.1: Bogoyavlenskij
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• proof