Related papers: Computing connecting orbits to infinity associated…
We study homoclinic orbits of the Swift-Hohenberg equation near a Hamiltonian-Hopf bifurcation. It is well known that in this case the normal form of the equation is integrable at all orders. Therefore the difference between the stable and…
We numerically study bifurcations of attractors of the H\'enon map with additive bounded noise with spherical reach. The bifurcations are analysed using a finite-dimensional boundary map. We distinguish between two types of bifurcations:…
Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle…
We use the `moving puncture' approach to perform fully non-linear evolutions of spinning quasi-circular black-hole binaries with individual spins not aligned with the orbital angular momentum. We evolve configurations with the individual…
In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be…
A systematic study of closed classical orbits of the hydrogen atom in crossed electric and magnetic fields is presented. We develop a local bifurcation theory for closed orbits which is analogous to the well-known bifurcation theory for…
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.…
We show that any combinatorial triangulation on n vertices can be transformed into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give an example of an infinite family of triangulations that requires this many flips…
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…
Homoclinic and heteroclinic orbits provide a skeleton of the full dynamics of a chaotic dynamical system and are the foundation of semiclassical sums for quantum wave packet, coherent state, and transport quantities. Here, the homoclinic…
We report on the experimental investigation of gluing bifurcations in the analog electronic circuit which models a dynamical system of the third order: Lorenz equations with an additional quadratic nonlinearity. Variation of one of the…
A boundary equilibrium bifurcation (BEB) in a hybrid dynamical system occurs when a regular equilibrium collides with a switching surface in phase space. This causes a transition to a pseudo-equilibrium embedded within the switching…
Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every $n$-vertex triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in O(n)…
Presented in this work are some results relative to sequences found in the logistic equation bifurcation diagram, which is the unimodal quadratic map prototype. All of the different saddle-node bifurcation cascades, associated to every last…
The aim of this paper is to study global bifurcations of non-constant solutions of some nonlinear elliptic systems, namely the system on a sphere and the Neumann problem on a ball. We study the bifurcation phenomenon from families of…
We give criteria for the existence of bifurcations of symmetric periodic orbits in reversible Hamiltonian systems in terms of local equivariant Lagrangian Rabinowitz Floer homology. As an example, we consider the family of the direct…
We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure.…
Symbolic dynamics for homoclinic orbits in the two-dimensional symmetric map, $x_{n+1}+cx_{n}+x_{n-1}=3x_{n}^3$, is discussed. Above a critical $c^{\ast}$, the system exhibits a fully-developed horse-shoe so that its global behavior is…
Studying 2 degree-of-freedom (DOF) Hamiltonian dynamical systems often involves the computation of stable & unstable manifolds of periodic orbits, due to the homoclinic & heteroclinic connections they can generate. Such study is generally…
We prove that for a $C^\infty$-generic contact form defining a given co-oriented contact structure on a closed $3$-manifold, every hyperbolic periodic Reeb orbit admits a transverse homoclinic connection in each of the branches of its…