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Unbounded dynamics for vector fields

Eran Igra

TL;DR

The paper develops a general topological-dynamical framework to relate local fixed-point behavior with unbounded global dynamics in three-dimensional flows. By employing Poincaré compactification and detailed geometric analysis of level sets $H=\\{F_i=0\\}$ and their tangency loci, it proves the existence of unbounded one-dimensional invariant manifolds from fixed points to $\infty$ and an unknot-type connecting structure in $S^3$. It then demonstrates the framework on concrete systems: Belousov-Zhabotinsky and Genesio-Tesi exhibit unbounded invariant manifolds toward infinity, while the Michelson system exhibits infinite instances of homoclinic/heteroclinic topologies as parameters vary. The results illuminate how local spectral data and infinity-oriented topology drive global, unbounded dynamics, offering tools for analyzing chaotic and forcing phenomena in 3D flows and their applications in chemical and mechanical systems.

Abstract

Consider a three-dimensional vector field $F$ which generates a finite number of fixed points - what can we say on its unbounded dynamics? In this paper we tackle this question, and prove sufficient conditions for $F$ to have fixed points with unbounded invariant manifolds. Following that, we use these results to study the dynamics of the Genesio-Tesi system, the Belousov-Zhabotinsky reaction, and the Michelson system.

Unbounded dynamics for vector fields

TL;DR

The paper develops a general topological-dynamical framework to relate local fixed-point behavior with unbounded global dynamics in three-dimensional flows. By employing Poincaré compactification and detailed geometric analysis of level sets and their tangency loci, it proves the existence of unbounded one-dimensional invariant manifolds from fixed points to and an unknot-type connecting structure in . It then demonstrates the framework on concrete systems: Belousov-Zhabotinsky and Genesio-Tesi exhibit unbounded invariant manifolds toward infinity, while the Michelson system exhibits infinite instances of homoclinic/heteroclinic topologies as parameters vary. The results illuminate how local spectral data and infinity-oriented topology drive global, unbounded dynamics, offering tools for analyzing chaotic and forcing phenomena in 3D flows and their applications in chemical and mechanical systems.

Abstract

Consider a three-dimensional vector field which generates a finite number of fixed points - what can we say on its unbounded dynamics? In this paper we tackle this question, and prove sufficient conditions for to have fixed points with unbounded invariant manifolds. Following that, we use these results to study the dynamics of the Genesio-Tesi system, the Belousov-Zhabotinsky reaction, and the Michelson system.

Paper Structure

This paper contains 10 sections, 9 theorems, 10 equations, 17 figures.

Table of Contents

  1. Introduction
  2. The general theorem:
  3. Stage $I$ - basic qualitative analysis of $F$
  4. Stage $II$ - proving the existence of $\Gamma_1$ under idealized assumptions.
  5. Stage $III$ - removing the idealized assumptions and concluding the general existence of $\Gamma_1$.
  6. Stage $IV$ - proving the existence of $\Gamma_2$ and concluding the proof.
  7. The applications:
  8. Unbounded dynamics in the Belousov-Zhabotinsky reaction and the Genesio-Tesi system.
  9. Homoclinic nooses and Heteroclinic knots in the Michelson System.
  10. Discussion - weakening the assumptions of Th.\ref{['infinitytheorem']}

Key Result

Theorem 1

Let $\dot{s}=F(s)$, $s=(x,y,z),F=(F_1,F_2,F_3)$ be a $C^k$, $k>0$ vector field satisfying the following: Then, there exist two fixed points $x_1,x_2$ (not necessarily distinct) s.t. each fixed point generates a respective one-dimensional invariant manifold, $\Gamma_1$ and $\Gamma_2$, connecting it to $\infty$ (see the illustration in Fig.conn). Moreover, there exists some curve $\gamma\subseteq\m

Figures (17)

  • Figure 1: The scenario where $F$ has two fixed points with unbounded dynamics (up) and the scenario where it has precisely one (down).
  • Figure 2: A $0$-index fixed point.
  • Figure 3: A sketch of $H=H_+\cup H_-,l_1,l_2,x_1$ and $x_2$ in Case $A$ - along with a red flow line tangent to $l_2$ - along with the direction of the vector field $F$ on $H$. By definition, $l$ is the straight line corresponding to $l_1\cup l_2\cup l_3$.
  • Figure 4: Case $A$ - the plane $H_1$ is transverse to $H$, and the forward trajectory of every initial condition in $l_1$ eventually hits either $H_-$ or $H'=H_1\cap\{F_1(s)>0\}$ transversely (where $H'$ is the shadowed half-plane). The green curve denotes the points where initial conditions from $l_1$ hit $H_-\cup H'$. On $H'$ the vector field points into $\{x>c_1\}$.
  • Figure 5: The idealized assumptions - the set $V$ connects with $Q_1$ in a green curve in some neighborhood of $\infty$.
  • ...and 12 more figures

Theorems & Definitions (21)

  • Theorem
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • ...and 11 more