Related papers: Unfolding codimension-two subsumed homoclinic conn…
Bifurcation with symmetry is considered in the case of an isotropy subgroup with a two-dimensional fixed point subspace and non-zero quadratic terms. In general, there are one or three branches of solutions, and five qualitatively different…
Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for $k$-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of $2$-parameter…
The mode-locking regions of a dynamical system are the subsets of the parameter space of the system within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a curious chain…
The saddle-node bifurcation is the simplest example of a generic bifurcation in smooth ordinary differential equations, and is associated with the creation or destruction of a pair of equilibria. In this paper we examine the unfolding of…
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the…
In this paper we study a beyond all order phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated normal form of…
The limiting slow dynamics of slow-fast, piecewise-linear, continuous systems of ODEs occurs on critical manifolds that are piecewise-linear. At points of non-differentiability, such manifolds are not normally hyperbolic and so the…
For dynamical systems that switch between different modes of operation, parameter variation can cause periodic solutions to lose or acquire new switching events. When this causes the eigenvalues (stability multipliers) associated with the…
Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark-Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a…
A six-dimensional reversible normal form system occurs in B{\'e}nard-Rayleigh convection between parallel planes, when we look for domain walls intersecting orthogonally (see Buffoni et al [1]). On the truncated system, we prove…
The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on $\mathbb{R}^2$ can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove that throughout an open…
Unfolding homoclinic tangencies is the main source of bifurcations in 2-dimensional (real or complex) dynamics. When studying this phenomenon, it is common to assume that tangencies are quadratic and unfold with positive speed. Adapting to…
In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs…
This paper is devoted to the study of codimension two holomorphic foliations and distributions. We prove the stability of complete intersection of codimension two distributions and foliations in the local case. Converserly we show the…
We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable…
In a Vlasov equation, the destabilization of a homogeneous stationary state is typically described by a continuous bifurcation characterized by strong resonances between the unstable mode and the continuous spectrum. However, when the…
Homoclinic bifurcation of 4-dimensional symplectic mappings is asymptotically studied. We construct the 2-dimensional stable and unstable manifolds near the sub-manifolds which experience exponentially small splitting, and successfully…
We give a local characterization of codimension two submanifolds which are marginally trapped in Robertson-Walker spaces, in terms of an algebraic equation to be satisfied by the height function. We prove the existence of a large number of…
We study bifurcations of area-preserving maps, both orientable (symplectic) and non-orientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of…
There are few examples of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. We analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere.…