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Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations

Shuntaro Tomizawa

TL;DR

The paper analyzes 3D $C^{r}$ diffeomorphisms with a quadratic focus–saddle homoclinic tangency under the boundary condition $|\lambda\gamma|=1$. By introducing a proper three-parameter unfolding, the authors trigger a Hopf bifurcation on the tangency curve and show the existence of a homoclinic point to the bifurcating cycle, yielding a nonhyperbolic Hopf–tangent structure. Through careful normal form reductions, invariant cone fields, and Lyapunov coefficient calculations, they produce a generic Hopf point $Q_k$ with a negative Lyapunov coefficient in a three-parameter family and demonstrate a Hopf–homoclinic cycle. Perturbations within this framework then produce a coindex-one heterodimensional cycle, showing that $f$ can be $C^{r}$-approximated by a map with robust saddle-type heterodimensional dynamics under the expanding condition (EC). This result extends prior work on homoclinic tangencies by incorporating the critical boundary case and linking Hopf bifurcations to heterodimensional phenomena in dimension three.

Abstract

We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|λγ| = 1$. For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus $|λγ|$, we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map $f$ can be $C^{r}$-approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case.

Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations

TL;DR

The paper analyzes 3D diffeomorphisms with a quadratic focus–saddle homoclinic tangency under the boundary condition . By introducing a proper three-parameter unfolding, the authors trigger a Hopf bifurcation on the tangency curve and show the existence of a homoclinic point to the bifurcating cycle, yielding a nonhyperbolic Hopf–tangent structure. Through careful normal form reductions, invariant cone fields, and Lyapunov coefficient calculations, they produce a generic Hopf point with a negative Lyapunov coefficient in a three-parameter family and demonstrate a Hopf–homoclinic cycle. Perturbations within this framework then produce a coindex-one heterodimensional cycle, showing that can be -approximated by a map with robust saddle-type heterodimensional dynamics under the expanding condition (EC). This result extends prior work on homoclinic tangencies by incorporating the critical boundary case and linking Hopf bifurcations to heterodimensional phenomena in dimension three.

Abstract

We analyze three-dimensional diffeomorphisms () exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy . For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus , we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map can be -approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case.
Paper Structure (33 sections, 21 theorems, 354 equations, 4 figures, 3 tables)

This paper contains 33 sections, 21 theorems, 354 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Previous work and our approach
  3. Plan of proof of Theorem \ref{['thm-main']}
  4. Idea of proof of Theorem \ref{['thm-sec']}
  5. Three-parameter family of diffeomorphisms
  6. Proper Unfolding
  7. Our result in a three-parameter family
  8. Existence of non-hyperbolic periodic points
  9. Iterated local map and global map
  10. First-return map
  11. Representation of the local map and the global map
  12. Normal form for the first-return map
  13. Invariant cone fields
  14. Non-hyperbolic periodic points
  15. Verifying the periodic point is a generic Hopf point
  16. ...and 18 more sections

Key Result

Theorem A

For the above three-dimensional $C^r$, $r \geq 1$, diffeomorphism $f$ with $|\lambda^* \gamma^*| = 1$, there exists a $C^r$ diffeomorphism $g$ arbitrarily $C^r$-close to $f$ such that $g$ has a heterodimensional cycle involving two hyperbolic periodic orbits $L_1$ and $L_2$ of saddles satisfying Moreover, if the pair $(f, \Gamma)$ satisfies the expanding condition (EC), then the $g$ can be chosen

Figures (4)

  • Figure 1.1: (a) The phase portrait of $f$ in the Focus-Saddle (2, 1) class when $\dim M_\mathrm{ph} = 3$. (b) The phase portrait of $g$ having a heterodimensional cycle involving $O^*(g)$ and $Q$. The new hyperbolic periodic point $Q$ arises near the orbit of the homoclinic tangency. The right picture indicates the cycle in a topological view.
  • Figure 1.2: The $xy$-plane. The main region is $R = \{0 < x < 1, y > 1\}$.
  • Figure 1.3: The phase portrait of $g$ when $\dim M_\mathrm{ph} = 3$ and $\mathrm{LC}(Q) > 0$. In this setting, $\widetilde{W}^s(Q)$ contains $W^c_\mathrm{loc}(Q)$ and $\widetilde{W}^u(Q)$ is one-dimensional manifold.
  • Figure A.1: The transformation of $C_2$ under $f$. After being linearly stretched, it is further modified by nonlinear transformations such as rotation and bending, resulting in the configuration shown in the rightmost diagram.

Theorems & Definitions (61)

  • Theorem A: Main theorem
  • Remark 1.2
  • Definition 1.3: Generic Hopf point
  • Definition 1.4: Hopf-homoclinic cycle
  • Theorem B: Secondary theorem
  • Theorem 1.5: Three-dimensional version of Theorem 1.1 in T2019
  • Remark 2.1
  • Definition 2.2: Proper unfolding
  • Remark 2.3
  • Remark 2.4
  • ...and 51 more