In these talks we consider dynamics near heteroclinic cycles between two hyperbolic equilibria with different saddle indices.
For that purpose, in the first talk we establish Lin's method as a unified approach to study nonwandering dynamics near those networks.
Afterwards we consider codimension-two T-points and their unfoldings in Rn. Here the constituting heteroclinic connections Gamma_1 and Gamma_2 are assumed to be such that one of them, say Gamma_1, breaks up under perturbations while the second one is transverse and isolated. The robustness of Gamma_2 is due to the transversal intersection of the corresponding stable and unstable manifolds of the equilibria. Such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions, and called T-points in this context.

In our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria. We show that in the real eigenvalue case the dynamics is rather tame while in the presence of complex eigenvalues shift dynamics does occur. To a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane. In the second talk we consider non-elementary T-points in reversible systems in R2n+1. Here the corresponding orbit Gamma_2 is related to stable and unstable manifolds of the equilibria which have a quadratic tangency. So Gamma_2 is no longer robust but unfolds to two robust heteroclinic connections. We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.

Concluding we present a model system that allows to discuss the dissolution of shift dynamics in more detail.