Double Hopf Bifurcation with Huygens Symmetry

Abstract

Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens' clocks, is studied using the codimension-three Elphick-Huygens equivariant normal form presented here. The focus is on the effect that the Huygens symmetry assumption has on the dynamic behavior of the system. Periodic solutions include the classical in-phase and anti-phase normal modes that are forced by the symmetry, as well as pairs of mixed mode phase-locked periodic solutions. The escapement paradox is explained. A theorem based on topological degree theory establishes the existence of quasi-periodic solutions in an invariant 3-torus that resembles a 2-torus slightly thickened to a solid toroidal shell, with the two principal radii of the 2-torus slowly modulated in time-that is, a toroidal breather. Secondary bifurcations from the in-phase and anti-phase normal modes are explored, of codimension one and two, and it is shown that an Arnold tongue plays a fundamental role in the determination of whether secondary bifurcation gives birth to phase-locked periodic solutions or to quasi-periodic solutions. Detailed numerical analysis, using MATLAB, extends the local bifurcation analysis to a more global picture that includes coexistence of multiple stable solutions and a "swallowtail" bifurcation of periodic solutions.

Huygens’ clocks revisited

“…A detailed description of all of the solution types, a bifurcation analysis, and a stability analysis are presented in § 5 . The results in this section extend and adapt the results in [ 31 ] to the case of Huygens' clocks, augment the stability analysis and clarify the behaviour of the toroidal breather solutions. § 6 applies the results to Huygens' experimental situation.…”

“…This minimum number of parameters is called the codimension of the bifurcation. For most bifurcations of practical interest the codimension is a finite number; for the present case (a system with Huygens symmetry) the codimension is three [ 31 ]. However, the model ( 2.1 )–( 2.2 ) of Huygens' clocks is not a versal unfolding, as the following thought experiment shows.…”

“…In fact, the Huygens symmetry of the system has forced J to be semisimple (i.e. diagonalizable over ) as verified in [ 31 ]. Thus, the mathematical model of § 2 is an example of an equivariant Hopf bifurcation, where the equivariance is given by Huygens permutation symmetry.…”

Bifurcation Controller Designs for the Generalized Cusp Plants of Bogdanov--Takens Singularity with an Application to Ship Control

“…Each change in the qualitative dynamics of a singular system in the vicinity of an equilibrium is called a local bifurcation. The most common local bifurcations in nonlinear planar systems are the changes in the number of equilibria, limit cycles, homoclinic and heteroclinic cycles, and changes in their stabilities and/or the existence of bi-stabilities; e.g., see [12,13,20,23,28,36]. An important challenging problem in engineering control is to predict and locate possible trajectories of these types in an engineering singular plant.…”

Turing-Hopf instabilities through a combination of diffusion, advection, and finite size effects

“…In what follows we will show that the drift term in the reaction diffusion system is exactly such a parameter and at a critical value of it there is the possibility of a double Hopf instability. Double Hopf bifurcations have usually been associated with oscillators with time delays [25][26][27] and some swirling flows in hydrodynamics. 28,29 Our point is that the drift induced pattern formation in reaction diffusion systems provides a natural setting for the double Hopf bifurcation.…”