On Takens' Last Problem: tangencies and time averages near heteroclinic networks

TL;DR

This paper constructs a stable family of differential equations with heteroclinic tangencies, demonstrating persistent historic behavior in solutions, thereby affirmatively resolving Takens' Last Problem about systems with positive measure sets of historic behavior.

Contribution

It provides the first example of a stable family of smooth ODEs exhibiting persistent historic behavior near heteroclinic cycles, confirming Takens' Last Problem.

Findings

Existence of a stable family with heteroclinic tangencies

Presence of historic behavior in solutions with positive measure

Explicit example near an SO(2)-equivariant vector field

Abstract

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields $C^2$-close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. We illustrate our results with an explicit example where historic behaviour arises $C^2$-close of a $\textbf{SO(2)}$-equivariant vector field.