HOMOCLINIC AND HETEROCLINIC BIFURCATIONS CLOSE TO A TWISTED HETEROCLINIC CYCLE

We study the interaction of a transcritical (or saddle-node) bifurcati on with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by num erical observations on the traveling wave ODE of a reaction-diffusion equation. The manifold organization is such that two branches of homoc linic orbits to each fixed point are created when varying the two para meters controlling the codimension-2 loop. It is shown that the homocl inic orbits may become degenerate in an orbit-flip bifurcation. We est ablish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurc ate in an inclination-flip bifurcation, where a Smale's horseshoe is f ound.