Dynamics on invariant tori emerging through forced symmetry breaking in phase oscillator networks

Figure 1.  Dynamics of system (41) for $ \alpha = \beta = \frac{\pi}{2} $, $ r = 0.2 $ and $ \delta = 0.01 $. Panel (a) shows solutions to the reduced system (51) on the torus $ \mathbb{T}^2 $ described by coordinates $ (\phi_2, \phi_3) $ representing the '$ \mathrm{D} $' populations; these periodic orbits wind around $ \mathbb{T}^2 $. Shown are two stable periodic orbits (blue) at $ \psi\in\{\frac{\pi}{2}, \frac{3\pi}{2}\} $ and two unstable periodic orbits (red) at $ \psi\in\{0, \pi\} $. Panel (b) shows a periodic orbit in the full system that lies on $ \mathrm{SDD} $ for $ \phi_3-\phi_2 = 0 $. Note that the two '$ \mathrm{D} $' populations have approximately the same phase

Figure 2.  Bifurcations can stabilize periodic orbits of the full system (16) stemming from $ \mathrm{SDD} $ for a perturbation determined by (19) and (26) that is sufficiently large $ \delta>0 $. The rest of the parameters are as in (52). Panel (a) shows the four branches of hyperbolic periodic orbits $ \gamma_0, \gamma_{\frac{\pi}{2}}, \gamma_{\pi} $ and $ \gamma_{\frac{3\pi}{2}} $ emanating from $ \psi^* \in \big\{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\big\} $ at $ \delta = 0 $, bifurcations that change their stability in the full system, and secondary solution branches. The vertical green line at $ \delta = 0 $ represents the unperturbed case, in which the torus $ \mathrm{SDD} $ is foliated by neutrally stable periodic orbits. Panel (b) highlights bifurcations at $ \delta = \delta^* $ that stabilize the branches $ \gamma_0 $ and $ \gamma_{\pi} $ in the full system and corresponding secondary branches of periodic orbits forming isolas

Figure 3.  Stable synchrony patterns bifurcating from primary branches of periodic orbits. Panels (a)–(d) show the phase evolution of oscillators in '$ \mathrm{D} $' populations along one period on coexisting stable periodic orbits lying on the secondary branches emanating from $ \psi = 0 $ (bottom) and $ \psi = \pi $ (top) for $ \delta\approx 0.40372 $; see the magnified parameter range in Figure 2(b). Shading indicates the deviation from $ \pi $ of each phase oscillator, where black indicates that $ \theta_{\sigma, k} = \pi $ and white $ \theta_{\sigma, k} = 0 $ or $ 2\pi $

Figure 4.  Some codimension-two bifurcations of system (16) with the perturbation driven by (19) and (26). Panel (a) shows curves of saddle-node bifurcation arising on the primary (red) and secondary (magenta) branches that organize the parameter plane $ (r, \delta) $. See the main text for an elaboration on the codimension-two points shown. Panel (b) shows the origin of the primary saddle-node bifurcation as one continues a primary branch of periodic orbits in parameter $ r $, with $ r $ decreasing. This bifurcation gives rise to the curve $ \mathsf{SN}_p $ in Panel (a)

Figure 5.  Evolution of periodic orbits along the overlapping principal saddle-node curves $ \mathsf{SN}_p $ in Figure 4. Panels (a)–(d) correspond to the points $ r_a, r_b, r_c, r_d $, respectively. Each panel shows (1) a time series of the coordinate $ \theta_{2, 1} $ with respect to the rescaled {\sc Auto} integration time for orbits on each principal saddle-node curve in green and red, (2) a relative position between the two '$ \mathrm{D} $' populations on each principal saddle-node curve in green and red, and (3) how each '$ \mathrm{D} $' populations deviate from $ \pi $ as $ (r, \delta) $ moves away from the origin

Figure 6.  Bifurcations of periodic orbits of the full system (16) with the perturbation driven by (19) and (26) for $ \delta>0 $ for the perturbed torus $ \mathrm{SSD} $. The rest of the parameters are as in Figure 2. Panel (a) shows four branches of hyperbolic periodic orbits $ \gamma_0, \gamma_{\frac{\pi}{2}}, \gamma_{\pi}, \gamma_{\frac{3\pi}{2}} $ emanating from $ \psi^*\in \big\{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\big\} $ at $ \delta = 0 $. The vertical green line located at $ \delta = 0 $ represents the unperturbed case, in which the torus $ \mathrm{SSD} $ is foliated by neutrally stable periodic orbits. Panel (b) highlights the isola of secondary branches of periodic orbits emanating from the branch $ \gamma_{\pi} $, and Panel (c) shows a zoom of the origin of the branch of secondary periodic orbits that emanates from $ \gamma_{0} $