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A method of reduction for invariant curves of quasiperiodically forced maps

Amadeu Delshams, Rafael Ortega

Abstract

The existence of translated curves for quasiperiodically forced maps is established, under very mild regularity hypotheses, for rotation numbers of constant type. Among the translated curves, the invariant curves are characterized as the solutions of a scalar bifurcation equation, from which their existence, stability as well as bifurcation can be easily described.

A method of reduction for invariant curves of quasiperiodically forced maps

Abstract

The existence of translated curves for quasiperiodically forced maps is established, under very mild regularity hypotheses, for rotation numbers of constant type. Among the translated curves, the invariant curves are characterized as the solutions of a scalar bifurcation equation, from which their existence, stability as well as bifurcation can be easily described.

Paper Structure

This paper contains 10 sections, 4 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Translated curves
  3. Dependence with respect to parameters
  4. Invariant curves: existence and stability
  5. Final remarks
  6. Case of a rational $\alpha$
  7. On the numbers $\nu_n$
  8. An alternative approach to the existence of invariant curves
  9. The mapping on the torus
  10. Basin of attraction

Key Result

Theorem 1

Assume $\alpha$ is of constant type (Eq:ConsType). Then there exists $\varepsilon^\ast=\varepsilon^\ast(\alpha,k_F^{(3)})>0$ such that for $0\le\varepsilon<\varepsilon^\ast$ and $c\in\mathbb{R}$, then the map $f_\varepsilon$ given in (Eq:FE) has a unique translated curve $r=\psi_c(\theta)$, with $\p Moreover, the map $(c,\theta)\in\mathbb{R}\times\mathbb{T}\longrightarrow \psi_c(\theta)\in\mathbb{

Figures (3)

  • Figure 1: Representation of translated and invariant curves: $\Gamma_0$ is an invariant curve and $\Gamma_1$ a translated one.
  • Figure 2: For the quasiperiodically forced Arnold circle map (\ref{['Eq:qpfAmap']}) with $k=0.8$, $\omega=0$, $\alpha=\frac{\sqrt{5}-1}{2}$, there appear a blue attractor invariant curve and a red repeller one for $\frac{b}{2\pi}=1.1$, and a possible SNA for $\frac{b}{2\pi}=3.1$.
  • Figure 3: Radial action for $F(r,\theta)=(1+\sin^2r)\sin 4\theta$ on $\theta=\frac{\pi}{8},\dots, \frac{15\pi}{8}$. The dynamics is upwards on the set of blue lines, and downwards on the red ones.

Theorems & Definitions (11)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • Remark 7
  • Lemma 8
  • Remark 9
  • Theorem 10
  • ...and 1 more