[Paper Review] Perturbations of Mathieu equations with parametric excitation of large period

Mário Bessa|arXiv (Cornell University)|Jun 25, 2010

Advanced Differential Equations and Dynamical Systems10 references3 citations

TL;DR

This paper studies perturbations of Mathieu equations with large-period parametric excitation, proving that arbitrarily close to any such system, there exists a Hamiltonian damped Mathieu equation with periodic coefficients where all but finitely many closed orbits exhibit unstable solutions. The instability arises via small $C^0$ perturbations of the periodic coefficients, leveraging cocycle theory and traceless matrix dynamics on $sl(2,\mathbb{R})$. The key result establishes generic instability in the presence of long closed orbits.

ABSTRACT

We consider a linear differential system of Mathieu equations with periodic coefficients over periodic closed orbits and we prove that, arbitrarily close to this system, there is a linear differential system of Hamiltonian damped Mathieu equations with periodic coefficients over periodic closed orbits such that, all but a finite number of closed periodic coefficients, have unstable solutions. The perturbations will be peformed in the periodic coefficients.

Motivation & Objective

• To investigate the dynamical behavior of solutions to linear differential systems derived from Mathieu equations with periodic coefficients.
• To understand how small perturbations in the periodic coefficients affect the stability of solutions, particularly in the presence of long closed orbits.
• To demonstrate that, arbitrarily close to any Mathieu equation with periodic coefficients, there exists a damped Hamiltonian system with unstable solutions for almost all closed periodic orbits.
• To establish a general mechanism for inducing instability through $C^0$ perturbations of the coefficient functions in linear skew-product flows.

Proposed method

• Represent the Mathieu equation as a linear skew-product flow on a compact manifold using the cocycle formalism.
• Model the system as a linear differential system $(M, \varphi^t, A)$ with $A: M \to \mathfrak{gl}(2,\mathbb{R})$, and consider perturbations within the $C^0$ topology.
• Use the Lie algebra structure of $sl(2,\mathbb{R})$ to ensure Hamiltonian (traceless) dynamics and preserve area-preservation in the fundamental solution.
• Construct explicit perturbations via small directional homotheties and rotations in the monodromy matrix, using Lemma 2.1 to ensure $C^0$-closeness to the original system.
• Apply a result from [3, Lemma 6.6] to show that for sufficiently long orbits ($\pi(p) \geq T$), perturbations can be designed to make the monodromy matrix have real eigenvalues (hyperbolic or parabolic).
• Define perturbed coefficients $\Gamma_\eta(t)$ and $\Psi_\eta(t)$ via a transformation involving $g(t)$, $\eta$, and matrix components, ensuring $\Gamma_\eta \to 0$ and $\Psi_\eta \to \psi(t)$ as $\eta \to 0$.

Experimental results

Research questions

• RQ1 Can small $C^0$ perturbations of the periodic coefficients in a Mathieu equation induce instability in the solutions for most closed periodic orbits?
• RQ2 Is it possible to construct a damped Hamiltonian Mathieu system arbitrarily close to a given Mathieu system such that all but finitely many periodic orbits have unstable solutions?
• RQ3 What is the role of orbit length in enabling the creation of instability through perturbations of the monodromy matrix?
• RQ4 How can perturbations be localized to the orbit while preserving the $C^0$-closeness to the original system and ensuring real eigenvalues in the monodromy map?
• RQ5 Under what conditions can a system with complex eigenvalues (elliptic) be perturbed to yield real eigenvalues (hyperbolic or parabolic), thus inducing instability?

Key findings

• For any $\delta > 0$, there exists $T > 0$ such that for all closed orbits with period $\pi(p) \geq T$, a $C^0$-perturbation $B$ of the original system exists with $B \in \mathcal{U}(A, \delta)$ and $\Phi_B^{\pi(p)}$ having only real eigenvalues.
• The perturbed system $B$ is a Hamiltonian damped Mathieu equation with $\Gamma_\eta(t)$ close to zero and $\Psi_\eta(t)$ close to $\psi(t)$, ensuring $C^0$-closeness to the original system.
• All but finitely many closed periodic orbits of the perturbed system have unstable solutions, as evidenced by real eigenvalues of the monodromy matrix with modulus not equal to one.
• The perturbation is localized: $B = A$ outside a small neighborhood of the periodic orbit, preserving the original dynamics away from the orbit.
• The construction relies on decomposing the monodromy into $k$-fold products of $C_j$ maps and applying small rotations $R_{\theta_j}$ with $|\theta_j| < \delta$ to force real eigenvalues.
• The perturbed system satisfies $\ddot{y} + \Gamma_\eta(t)\dot{y} + (\omega^2 + \epsilon \Psi_\eta(t))y = 0$, with $\Gamma_\eta(t) \to 0$ and $\Psi_\eta(t) \to \psi(t)$ as $\eta \to 0$, confirming the perturbation is small and physical.