Breathing torus near double Hopf bifurcation

Bifurcation and Continuation

By Allan R. Willms
Breathing torus near double Hopf bifurcation
The normal form near a double supercritical Hopf bifurcation for a system with two symmetrically coupled identical oscillators can be written as
$$\dot{r}_1 = r_1\left[\mu_1 - r_1^2+ (b_{12} + B_1\cos\theta)r_2^2\right],$$
$$\dot{r}_2 = r_2 \left[\mu_2 + (b_{21} + B_2\cos(\theta + \phi))r_1^2 - r_2^2\right],$$
$$\dot{\theta}_1 = \omega +\gamma_0 + c_{11}r_1^2 + \left[c_{12} - B_1\sin\theta\right] r_2^2,$$
$$\dot{\theta}_2 = \omega + \eta_0 + \left[c_{21} +B_2\sin(\theta+\phi)\right]r_1^2 + c_{22} r_2^2,$$
This system admits 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, and quasiperiodic solutions in an invariant 3-torus. These latter solutions may be visualized in three dimensions as trajectories on a breathing 2-torus, where the two principal radii of the 2-torus slowly vary in time. The three-dimensional projection of solutions of the above system is given by
\(x = (r_1 + r_2 \cos\theta_2 ) \cos\theta_1 ,   y = (r_1 + r_2 \cos\theta_2 ) \sin\theta_1 ,   z = r_2 \sin\theta_2.\)
At the beginning of the video, the relationship between the polar coordinates \((r_1,r_2,\theta_1,\theta_2)\) of the normal form and the cartesian coordinates \((x,y,z)\) of the plot are illustrated. The 2-torus at \(t=0\) is drawn and its cross-sections at \(x=0\), \(y=0\) and \(z=0\) are projected onto the background planes. The distances \(r_1\) and \(r_2\) are drawn as black rods on the \(x=0\) and \(y=0\) projections. Time then progresses from \(t=0\) and the leading edge of the trajectory is drawn in blue. The trailing edge of the trajectory disappears in order to keep the central part of the figure uncluttered. As time increases the 2-torus (and its cross-sections) are slowly changing since both \(r_1\) and \(r_2\) vary slowly with time. The leading edge of the trajectory remains on the surface of the breathing 2-torus. At some points in the video it can be seen that before the trailing edge of the trajectory disappears, it is off the instantaneous 2-torus. The projections of the entire trajectory are shown in red overlaid on the \(x=0\) and \(z=0\) cross sections.
Author Institutional AffiliationUniversity of Guelph
Author Email
Author Postal MailDept. of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada
For more information see:
P.M. Kitanov, W.F. Langford, and A.R. Willms, "Double Hopf Bifurcation with Huygens Symmetry", SIAM J. App. Dyn. Syst. 12 (1) (2013) 126-174.
Keywordstoriodal breather, double Hopf bifurcation, coupled identical oscillators

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