Bifurcations and Attractor-Repeller Splittings of Non-Saddle Sets

Cech cohomology, homoclinic trajectories and robustness of non-saddle sets

“…We call dissonant points to those points in ∂H(K) which are not in K. The previous discussion illustrates that all the interesting dynamical features in I(K) \ K occur in the components containing dissonant points. In fact, in absence of dissonant points, it has been seen in [6,Theorem 19] the dynamics in I(K) \ K is qualitatively the same as the dynamics in the basin attraction of an isolated attractor without external explosions studied in [47].…”

“…More specifically, the complexity of a flow having an isolated non-saddle set relies in the structure of the region of influence of the non-saddle set which is especially rich in presence of the so-called dissonant points, a phenomenon that does not appear in the basin of attraction of an attractor neither stable nor unstable. The region of influence of an isolated non-saddle set has been deeply studied in [5,6].…”

Čech cohomology, homoclinic trajectories and robustness of non-saddle sets

“…We call dissonant those points in ∂H(K) which are not in K. The previous discussion illustrates that all the interesting dynamical features in I(K) \ K occur in the components containing dissonant points. In fact, in the absence of dissonant points it has been seen in [5,Theorem 19] the dynamics in I(K) \ K is qualitatively the same as the dynamics in the basin attraction of an isolated attractor without external explosions studied in [23].…”

“…This is especially rich in presence of the so-called dissonant points, a phenomenon that does not appear in the basin of attraction of an attractor neither stable nor unstable. The region of influence of an isolated non-saddle set has been deeply studied in [5,6].…”