Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation

Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation

The saddle-node Hopf bifurcation (SNH) is a generic codimensiontwo bifurcation of equilibria of vector fields in dimension at least three. It has been identified as an organizing centre in numerous vector field models arising in applications. We consider here the case that there is a global reinjection mechanism, because the centre manifold of the zero eigenvalue returns to a neighbourhood of t...

Jakobson’s Theorem near saddle-node bifurcations

We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. It was previously shown that for a parameter set of positive Lebesgue density at the bifurcation, the maps possess attracting periodic orbits of high period. We show that there is also a parameter set of positive density at the bifurcation, for which the maps exhibit abs...

Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

Homoclinic bifurcations are important phenomena that cause global re-arrangements of the dynamics in phase space, including changes to basins of attractions and the generation of chaotic dynamics. We consider here a homoclinic (or connecting) orbit that converges in both forward and backward time to a saddle equilibrium of a three-dimensional vector field. We assume that the saddle is such that...

Intermittency and Jakobson's theorem near saddle-node bifurcations

Intermittency and Jakobson’s theorem near saddle-node bifurcations

We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to a...