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

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

Moduli of stability for heteroclinic cycles of periodic solutions

We consider $C^2$ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.Comment: 23 pages, 6 figure

Transitions of bifurcation diagrams of a forced heteroclinic cycle

A family of periodic perturbations of an attracting robust heteroclinic cycle defined on the two-sphere is studied by reducing the analysis to that of a one-parameter family of maps on a circle. The set of zeros of the family forms a bifurcation diagram on the cylinder. The different bifurcation diagrams and the transitions between them are obtained as the strength of attraction of the cycle and the amplitude of the periodic perturbation vary. When the cycle is weakly attracting further transitions are found giving rise to a frequency locked invariant torus and to a frequency locked suspended horseshoe, arising from heteroclinic tangencies in the family of maps. We determine a threshold in the cycle's attraction strength above which there are no other transitions in the bifurcation diagrams. Above this threshold and as the period of the perturbation decreases, frequency locked periodic solutions with arbitrarily long periods bifurcate from the cycle

Acerca do Último Problema de Takens

No contexto de equações diferenciais autónomas, uma solução limitada com comportamento histórico é aquela para a qual as médias de Birkhoff não convergem. O Último Problema de Takens descrito em (F. Takens (2008), Nonlinearity 21(3), T33–T36) questiona a existência de sistemas dinâmicos suaves onde o comportamento histórico ocorre persistentemente num conjunto de condições iniciais com medida de Lebesgue positiva. Tendo este desafio como mote, nesta nota pretende-se fazer uma pequena digressão sobre o problema, assim como referir alguns dos desenvolvimentos recentes do tema em fluxos com ciclos heteroclínicos associados a soluções periódicas com multiplicadores de Floquet reais