Role of saddle tori in the mutual synchronization of periodic oscillations

We show that in the mutual synchronization of periodic oscillators, besides an attracting torus, there is also a saddle torus that plays an equally important role. We demonstrate that the saddle and stable tori form an elegant structure, allowing for a variety of phenomena, both known and new, related to the origin and evolution of coexisting synchronous regimes (phase multistability).

Figure 1

(Color online) Central figure: bifurcation diagram showing the synchronization region for linear coupling, on the $p\text{−}{C}_1$ (detuning-coupling strength) plane. In the shaded areas there are no 1:1 synchronous solutions. The full lines are fold bifurcations, and the heavy dashed lines are Neimark-Sacker bifurcations leading to torus birth. The dotted lines show the two routes A and B across the diagram that are illustrated in the insets. The filled black circle 1 marks the parameter values for which the structure of phase space is shown in Fig. 5b. Inset A shows how the positions of the maxima $x_{\max }$ of the stable $S_{1,2}$ and single-saddle $S_{1,2}^{*}$ cycles depend on $p$ for $C_1=0.15$. Inset B shows how $x_{\max }$ of all six cycles depend on $C_1$ for $p=1.002$. In insets full lines represent $S_{1,2}$, dashed lines $S_{1,2}^{*}$, and grey lines the double-saddle cycles $U_{1,2}$.

Figure 2

(Color online) Bifurcation diagrams in the $p\text{−}{C}_2$ parameter plane for four different values of $C_1$. In the shaded areas there are no 1:1 synchronous solutions. Full black lines are fold bifurcations; dashed lines show Neimark-Sacker bifurcation; full grey line for $C_1=0.2$ (see also Fig. 3) shows fold bifurcations for stable and saddle tori.

Figure 3

(Color online) Bifurcation diagram on the $p\text{−}{C}_2$ plane for $C_1=0.2$. In the shaded areas there are no 1:1 synchronous solutions. Full black lines are fold bifurcations for the pairs of cycles $S_{1,2}$ and $S_{1,2}^{*}$, and $S_{1,2}^{*}$ and $U_{1,2}$. Dashed lines are Neimark-Sacker bifurcations for the cycles $S_{1,2}$. A full grey (green online) line is the fold bifurcation for a saddle torus $T^{*}$ and a stable torus $T_2$ born from $S_2$. In the region bounded by the dashed line from below and by the full grey line from above (see the inset), a stable cycle $S_1$, and a stable torus $T_2$ coexist.

Figure 4

(Color online) Sketches of Poincaré sections of (a) the attracting resonant torus $T$, (b) the saddle resonant torus $T^{*}$ shown as a closed grey (green online) curve, and its manifolds shown as surfaces. The circles indicate cycles: filled ones represent the stable cycles $S_{1,2}$; open ones represent the single-saddle cycles $S_{1,2}^{*}$; and grey (green online) ones represent the double-saddle cycles $U_{1,2}$.

Figure 5

(Color online) Poincaré sections of objects in the phase space. (a) Hypothesized structure of phase space inside the region where phase locking of the mutually coupled periodic oscillators occurs. A dotted line shows the “sphere,” which in fact corresponds to the cylinder in Fig. 4b that is closed from below and above. (b) A numerically revealed structure of the phase space at parameter values corresponding to the point labeled 1 in Fig. 1. The circles indicate cycles: filled ones are the stable cycles $S_{1,2}$; open ones are the single-saddle cycles $S_{1,2}^{*}$; grey (green online) are the double-saddle cycles $U_{1,2}$. The full closed curves show tori: the black ones are stable $T$, and grey (green online) ones are saddle $T^{*}$. For details, see the text.

Figure 6

(Color online) Poincaré sections of objects in the phase space. (a) A hypothesized structure of phase space inside the suppression regime for the mutually coupled periodic oscillators, in particular, inside the region containing points 2 and 3 in the inset of Fig. 3, where a stable cycle coexists with a stable torus. (b) and (c) A numerically revealed structure of the phase space at parameter values corresponding to points 2 and 3 in Fig. 3, respectively. Designations are as in Fig. 5. The full black line here shows the newly born stable torus $T_2$.

Figure 7

(Color online) Poincaré sections of objects in the phase space of two coupled FitzHugh-Nagumo systems, Eqs. (2). Designations are as in Fig. 5.