We study one parameter families of vector fields that are defined on three dimensional
manifolds and whose nonwandering sets are structurally stable.
As families, these families may not be structurally stable; heteroclinic
bifurcations that give rise to moduli can occur.
Some but not all moduli are related to the geometry of stable
and unstable manifolds. We study a notion of stability, weaker
then structural stability, in which geometry and dynamics on stable
and unstable manifolds are reflected.
We classify the families from the above mentioned class of families
that are stable in this
We consider a class of step skew product systems of interval diffeomorphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present in a simplified setting intriguing phenomena of intermingled basins, master-slave synchronization and on-off intermittency. We provide a self-contained discussion of these phenomena.
We discuss one parameter families of unimodal maps,
with negative Schwarzian derivative, unfolding a
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
an atomic measure supported on the orbit of the saddle-node point.
Using these measures we analyze the intermittent time series that result
from the destruction of the periodic attractor
in the saddle-node bifurcation and prove
asymptotic formulae for the frequency with which
orbits visit the region previously occupied by the periodic attractor.
We discuss iterates of random circle diffeomorphisms with identically distributed noise, where the noise is bounded and absolutely continuous. Using arguments of B. Deroin, V.A. Kleptsyn and A. Navas, we provide precise conditions under which random attracting fixed points or random attracting periodic orbits exist. Bifurcations leading to an explosion of the support of a stationary measure from a union of intervals to the circle are treated. We show that this typically involves a transition from a unique random attracting periodic orbit to a unique random attracting fixed point.
We study Hopf-Andronov bifurcations in a class of random differential
equations (RDEs) with bounded noise. We observe that when an ordinary differential
equation that undergoes a Hopf bifurcation is subjected to bounded noise
then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant