Asymptotic analysis of the Michelson system

Abstract

We consider the dynamical system x ttt = c 2 − 1 2 x 2 − x t for the parameter c close to zero. We perform a multiple time scale analysis to provide analytic forms for all bounded solutions of the formal normal form in the phase space, in a neighbourhood of the origin (x,c)=(0,0). These results are contrasted with a numerical simulation of the system.

On the Hopf-zero bifurcation of the Michelson system

“…For c > 0 small numerical experiments (see for instance Kent and Elgin [3]) and asymptotic expansions in sinus series (see Michelson [8] in 1986 and Webster and Elgin [11] in 2003) revealed the existence of a Hopf-zero bifurcation at the origin for c = 0. But their results do not provide an analytic proof on the existence of such Hopf-zero bifurcation.…”

Central Configurations, Periodic Orbits, and Hamiltonian Systems

“…Introduction. The classical theory Webster and Elgin [120] in 2003) revealed the existence of a zero-Hopf bifurcation at the origin for c = 0. But their results do not provide an analytic proof on the existence of such zero-Hopf bifurcation.…”

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

“…An interesting phenomenon related to periodic orbits is the so-called noose bifurcation [10], that appears in the well-known Michelson system [1, 12,18,7,22],         ẋ = y,…”

“…The existence of the noose bifurcation is based on numerical computations and no theoretical proofs are given. In fact, although there are many interesting works about periodic behavior and global connections in the Michelson system (see, for instance, [1, 6,11,14,17,21,22,23]), as far as we know, the existence of the noose bifurcation has not been proved yet.…”