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 inphase and antiphase normal modes that are forced by the symmetry, as well as pairs of mixed mode phaselocked periodic solutions, and quasiperiodic solutions in an invariant 3torus.
These latter solutions may be visualized in three dimensions as trajectories on a
breathing 2torus, where the two principal radii of the 2torus slowly vary in time. The threedimensional 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 2torus at \(t=0\) is drawn and its crosssections 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 2torus (and its crosssections) 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 2torus. At some points in the video it can be seen that before the trailing edge of the trajectory disappears, it is off the instantaneous 2torus. The projections of the entire trajectory are shown in red overlaid on the \(x=0\) and \(z=0\) cross sections.
Author Institutional Affiliation  University of Guelph 
Author Email  
Author Postal Mail  Dept. of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada 
Notes  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) 126174. 
Keywords  toriodal breather, double Hopf bifurcation, coupled identical oscillators 