Rigorous verification of cocoon bifurcations in the Michelson system

Abstract

We prove the existence of cocoon bifurcations for the Michelson system ẋ = y,, where (x, y, z) ∈ R 3 and c ∈ R + is a parameter, based on the theory given in (Dumortier et al 2006 Nonlinearity 19 305-28). The main difficulty lies in the verification of the (topological) transversality of some invariant manifolds in the system. The proof is computer-assisted and combines topological tools including covering relations and the smooth ones using the cone conditions. These new techniques developed in this paper will have broader applicability to similar global bifurcation problems.

On the Hopf-zero bifurcation of the Michelson system

“…The first study on system (1) goes back to Michelson [8] who proved that if c > 0 is sufficiently large, system (1) has a unique bounded solution, which is the transversal heteroclinic orbit connecting the two finite singularities. When c decreases there will appears the cocoon bifurcation, which was verified in [8,7,4] using computer assistance. As for the appearance of the cocoon bifurcation, Remark 1.6 of [2] explained that if there exists a saddle-node periodic orbit γ at some value c 0 > 0 of the parameter which is symmetric with respect to the involution R, then for c on the one side of c 0 the saddle-node periodic orbits split into two limit cycles, while for c on the other side of c 0 no periodic orbits will be present near γ, but a cocooning cascade appears.…”

Recent development in rigorous computational methods in dynamical systems

“…However, the question of whether the Michelson system indeed has the cusp-transverse heteroclinic chain remained open until very recently [23]. In this section, we show how such a question can be answered by the use of rigorous numerical computation.…”

“…Nevertheless, we can prove the existence of a CTHC and obtain the following theorem: Theorem 2.4. (See [23].) The Michelson system (1) exhibits a cocooning cascade of heteroclinic tangencies centered at a c ∞ ∈ [1.2662323370670545, 1.2662323370713253].…”

“…Since the saddle-node periodic orbit is not hyperbolic, it is not straightforward how to carry out this strategy. For this purpose, we use two additional ideas in [23]: h-sets with cones and a Lyapunov function near the saddle-node periodic orbit. The former carries the information on the derivative of the dynamics in the form of cones, and thus controls how a piece of stable or unstable manifolds can be mapped nicely to the next h-set.…”

“…Combining this information, we can prove that certain part of the stable manifold of the hyperbolic equilibrium point x − intersects topologically transversely, in an h-set, with the unstable set of the saddle-node periodic point, as it is shown in Figure 3. See [23] for the detailed proof and related information. Finally, we remark that the above methods are general and can be applied to other systems which exhibit a similar bifurcation.…”