On Takens' last problem: tangencies and time averages near heteroclinic networks

• Type: Article in International Scientific Journal
• Date: 2017

Authors

Isabel S Labouriau (Author) FCUP; Alexandre A P Rodrigues (Author) FCUP

Journal

Title: Nonlinearity

Vol. 30

Pages: 1876-1910

ISSN: 0951-7715

Publisher: Institute of Physics Publishing

Indexing

ISI Web of Knowledge - 21 Citations

Scopus - 19 Citations

Other information

Authenticus ID: P-00M-P00

DOI: 10.1088/1361-6544/aa64e9

Abstract (EN): 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 Takens' last problem (Takens 2008 Nonlinearity 21 T33-6). 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 SO(2)-equivariant vector field.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 35