Heteroclinic bifurcations of $\Omega$-stable vector fields on 3-manifolds
Ale Jan Homburg
Discrete & Continuous Dynamical Systems - A 1998, 4(3): 559-580 doi: 10.3934/dcds.1998.4.559
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 sense.
keywords: Heteroclinic bifurcations $\Omega$-stable vector fields.
Random interval diffeomorphisms
Masoumeh Gharaei Ale Jan Homburg
Discrete & Continuous Dynamical Systems - S 2017, 10(2): 241-272 doi: 10.3934/dcdss.2017012

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.

keywords: Iterated function systems interval diffeomorphisms Lyapunov exponents synchronization intermittency
Intermittency and Jakobson's theorem near saddle-node bifurcations
Ale Jan Homburg Todd Young
Discrete & Continuous Dynamical Systems - A 2007, 17(1): 21-58 doi: 10.3934/dcds.2007.17.21
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 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.
keywords: absolutely continuous invariant measure saddle node bifurcation nonuniform hyperbolicity.
Dynamics and bifurcations of random circle diffeomorphism
Hicham Zmarrou Ale Jan Homburg
Discrete & Continuous Dynamical Systems - B 2008, 10(2&3, September): 719-731 doi: 10.3934/dcdsb.2008.10.719
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.
keywords: Random dynamical systems bifurcations. circle diffeomorphisms
The Hopf bifurcation with bounded noise
Ryan T. Botts Ale Jan Homburg Todd R. Young
Discrete & Continuous Dynamical Systems - A 2012, 32(8): 2997-3007 doi: 10.3934/dcds.2012.32.2997
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 set.
keywords: stationary measure Random dynamical system minimal forward invariant set. random differential equation

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