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Nonintegrability of time-periodic perturbations of analytically integrable systems near homo- and heteroclinic orbits

Kazuyuki Yagasaki

TL;DR

We address the nonintegrability of time-periodic perturbations of analytically integrable systems by embedding the perturbation into a higher-dimensional autonomous framework and applying a generalized Morales-Ramis theory via real-meromorphic integrability. A finite Fourier-series perturbation leads to a Melnikov function $M(\theta;c)$ whose nonconstancy implies nonreal-meromorphic integrability through a reduction to single-frequency components and differential Galois analysis. The proof combines a complexification of the heteroclinic structure, monodromy computations, and Morales-Ramis criteria to show nonintegrability when some Fourier component $\hat{M}_\ell(c)\neq0$. An explicit example for a periodically forced rigid body (quadrotor model) confirms the theory by producing concrete nonzero Melnikov components, illustrating practical nonintegrability criteria for multi-degree-of-freedom, time-periodic perturbations.

Abstract

We consider time-periodic perturbations of analytically integrable systems in the sense of Bogoyavlenskij and study their \emph{real-meromorphic} nonintegrability, 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 \emph{real-meromorphically} integrable near homo- and heteroclinic orbits. We illustrate the theory for rotational motions of a periodically forced rigid body, which provides a mathematical model of a quadrotor helicopter.

Nonintegrability of time-periodic perturbations of analytically integrable systems near homo- and heteroclinic orbits

TL;DR

We address the nonintegrability of time-periodic perturbations of analytically integrable systems by embedding the perturbation into a higher-dimensional autonomous framework and applying a generalized Morales-Ramis theory via real-meromorphic integrability. A finite Fourier-series perturbation leads to a Melnikov function whose nonconstancy implies nonreal-meromorphic integrability through a reduction to single-frequency components and differential Galois analysis. The proof combines a complexification of the heteroclinic structure, monodromy computations, and Morales-Ramis criteria to show nonintegrability when some Fourier component . An explicit example for a periodically forced rigid body (quadrotor model) confirms the theory by producing concrete nonzero Melnikov components, illustrating practical nonintegrability criteria for multi-degree-of-freedom, time-periodic perturbations.

Abstract

We consider time-periodic perturbations of analytically integrable systems in the sense of Bogoyavlenskij and study their \emph{real-meromorphic} nonintegrability, 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 \emph{real-meromorphically} integrable near homo- and heteroclinic orbits. We illustrate the theory for rotational motions of a periodically forced rigid body, which provides a mathematical model of a quadrotor helicopter.
Paper Structure (7 sections, 8 theorems, 55 equations, 4 figures)

This paper contains 7 sections, 8 theorems, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction of the problem to single-frequency systems
  4. Reduction of the VE \ref{['eqn:vehl']}
  5. Proof of Theorem \ref{['thm:main']}
  6. Example
  7. Generalized Morales-Ramis theory

Key Result

Theorem 1.2

Suppose that the Melnikov function $M(\theta;c)$ is not constant under assumptions (A1)-(A8). Then the system eqn:rsys is not real-meromorphically integrable near in $\mathbb{R}^{2N+n+1}$.

Figures (4)

  • Figure 1: Assumptions (A2), (A4) and (A5).
  • Figure 2: Riemann surface $\Gamma=x^\mathrm{h}(U)\cup\tilde{\mathscr{C}}_+\cup\tilde{\mathscr{C}}_-$.
  • Figure 3: Mathematical model for a quadrotor helicopter.
  • Figure 4: Unperturbed orbits of \ref{['eqn:rbody']} with $\varepsilon=0$ on the level set $F_1(\omega)=c$.

Theorems & Definitions (14)

  • Definition 1.1: Bogoyavlenskij
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 4 more