Showcase of Blue Sky Catastrophes

Let a system of differential equations possess a saddle periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases of the breakdown of such a homoclinic connection that causes the blue sky catastrophe, as well as the onset of complex dynamics. The birth of an invariant torus and a Klein bottle is also described.

EXISTENCE OF HORSESHOES AND HOMOCLINIC CONNECTIONS IN DC/DC CONVERTERS

In this letter, chaos in a current-mode controlled boost converter is studied. Firstly, the existence of chaos is proven theoretically in this system. The proof consists of showing that the dynamics of the system is semiconjugate to that of a one-sided shift map, which implies positive entropy of the system and hence chaotic behavior. The essential tool is the horseshoe hypotheses proposed by Kennedy and Yorke, which will be reviewed prior to the discussion of the main finding. Then, the existence of chaos is illustrated in the light of homoclinic connection. Furthermore, global chaos resulting from homoclinic intersection of stable and unstable manifolds are illustrated numerically.

HOMOCLINIC CONNECTIONS NEAR A BELYAKOV POINT IN CHUA'S EQUATION

In this work, the distribution and organization of different homoclinic orbits (double- and triple-pulse) in a ℤ2-symmetric three-dimensional system are studied in the vicinity of a Belyakov point, that is, a point where the involved equilibrium in the homoclinic connection changes from saddle-node to saddle-focus. The analytical results are successfully applied in the study of such degeneration in Chua's equation.