Resonance bifurcation from homoclinic cycles (2009)
Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R-4 there...
Essentially asymptotically stable homoclinic networks (2009)
Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835-844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which...
Homburg, A.J., Jukes, A.C., Knobloch, J., Lamb, J.S.W.
In unfoldings of resonant homoclinic bellows interesting bifurcation phenomena occur: two suspensed Smale horseshoes can collide and disappear in saddle-node bifurcations (all periodic orbits...
Intermittency and Jakobson's theorem near saddle-node bifurcations (2007)
Abstract: 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...
Bifurcations of stationary measures of random diffeomorphisms (2007)
Abstract: Random diffeomorphisms with bounded absolutely continuous noise are known to possess a finite number of stationary measures. We discuss the dependence of stationary measures on an auxiliary...
Bifurcations of stationary measures of random diffeomorphisms (2007)
Abstract: Random diffeomorphisms with bounded absolutely continuous noise are known to possess a finite number of stationary measures. We discuss the dependence of stationary measures on an auxiliary...
Intermittency and Jakobson's theorem near saddle-node bifurcations (2007)
Abstract. 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...
Multiple homoclinic orbits in conservative and reversible systems. (2006)
Abstract: We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the...
Symmetric homoclinic tangles in reversible systems (2006)
We study the dynamics near transverse intersections of stable and unstable manifolds of sheets of symmetric periodic orbits in reversible systems. We prove that the dynamics near such homoclinic and...
Invariant manifolds near hyperbolic fixed points (2006)
Abstract: In these notes, we discuss obstructions to the existence of local invariant manifolds of some smoothness class, near hyperbolic fixed points of diffeomorphisms. We present an elementary...
Bellows bifurcating from degenerate homoclinic orbits in conservative systems. (2005)
We study the dynamics near degenerate homoclinic orbits in conservative systems. Typically bellows bifurcate from those orbits. We show the existence of one parameter families of suspended...
Abstract We present a global bifurcation study of a four-dimensional system of differential equations, proposed by F.H. Busse and coworkers, modeling instabilities of convection rolls in the...
We consider a family of differential equations that describes traveling waves in a reaction-diffusion equation modeling oxidation of carbon oxide on a platinum surface, near the onset of...
We consider a system of differential equations proposed by Busse et al (1992 Physica D 61 94–105) to describe the development of spatio-temporal structures in Rayleigh–Bénard convection, near...
Multiple Homoclinic Orbits in Conservative and Reversible Systems (2002)
We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the...