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

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 $\text{SO(2)}$-equivariant vector field.

References in corpus (2)

• Takens' last problem and existence of non-trivial wandering domains
• Dense heteroclinic tangencies near a Bykov cycle

Cited by in corpus (15)

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• Persistent Strange attractors in 3D Polymatrix Replicators
• Boundary crisis for degenerate singular cycles
• Abundance of observable Lyapunov irregular sets
• Lyapunov exponents for random maps
• Ergodic Formalism for topological Attractors and historic behavior
• Existence and non-existence of length averages for foliations
• Historic Behaviour for Random Expanding Maps on the Circle
• Strange attractors and non wandering domains near a homoclinic cycle to a bifocus
• On the Almost Everywhere Stability of Discrete-Time Dynamical Systems
• Takens' Last Problem and strong pluripotency