Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

We investigate two coupled oscillators, each of which shows an attracting heteroclinic cycle in the absence of coupling. The two heteroclinic cycles are nonidentical. Weak coupling can lead to the elimination of the slowing-down state that asymptotically approaches the heteroclinic cycle for a single cycle, giving rise to either quasiperiodic motion with separate frequencies from the two cycles or periodic motion in which the two cycles are synchronized. The synchronization transition, which occurs via a Hopf bifurcation, is not induced by the commensurability of the two cycle frequencies but rather by the disappearance of the weaker frequency oscillation. For even larger coupling the motion changes via a resonant heteroclinic bifurcation to a slowing-down state corresponding to a single attracting heteroclinic orbit. Coexistence of multiple attractors can be found for some parameter regions. These results are of interest in ecological, sociological, neuronal, and other dynamical systems, which have the structure of coupled heteroclinic cycles.

Figure 1

Dynamics of two coupled heterocycles [Eq. (2)] for different coupling strengths: $\xi =5\times {10}^{-5}$ for the first row (a)–(d); $\xi =0.09072$, which is slightly smaller than the critical value for the bifurcation to a periodic orbit, for the second row (e)–(h); and $\xi =0.09537$, which is slightly larger than the critical value, for the third (i)–(l). The first column shows the time series. The second column shows the projections of the phase space onto the $X_1-Y_1$ plane, with the inset in panel (b) expanding the region near a saddle to show the quasiperiodic nature of the orbit. The third column exhibits the Poincaré section: $A_1=A_2,A_3>A_2$. The fourth column shows the Fourier transform of the two oscillators. Other parameter values are ${\tau }_a={\tau }_b=1$, ${\varepsilon }_a=1$, ${\varepsilon }_b=4$, $g_{-}=0.5$, and $g_{+}=2$.

Figure 2

Dependence of the periods $2\pi /\omega$ on the logarithm of the coupling strength ${\log }_{10}\xi$. Parameter values for the single cycle are the same as in Fig. 1. The inset shows the ratio of the two frequencies in the quasiperiodic state.

Figure 3

Average period of a single heteroclinic cycle with added noise, Eqs. (1) and (4, 5, 6). The period exhibits a linear dependence on the logarithmic noise intensity. The parameters are $\varepsilon =1$, with other values the same as in Fig. 2. Fitted values for $K_0$ and $K_1$ of Eq. (7) are $2.242$ and $6.916$, respectively.

Figure 4

Dynamics on the $X_2-X_1$ plane for the same three values of $\xi$ and the same values of the other parameters as in Fig. 1.

Figure 5

Ratio of $B_1$ to $A_1$ for the periodic orbit at (a) $\xi =0.33$ and (b) $\xi =0.35$. Other parameters are the same as Fig. 3. The horizontal lines are theoretical predictions of the ratio near the saddles. Noticing that there are two constant values of $B_1/A_1$ given by the eigenvectors in each subspace $\{A_i,B_i\}$ near the saddle $S_i$, the dashed and dotted lines are, respectively, the values given by the eigenvectors corresponding to the larger eigenvalue near $S_2$ and $S_3$. The solid line is the value $c_1$ calculated from Eq. (9) near the saddle $S_1$.

Figure 6

Values of the amplitude $\widetilde{B_1}$ at the saddle $S_1$ and the minimum and maximum amplitudes of $B_1$ in the trajectory near the saddle $S_1$ as a function of the coupling strength $\xi$. The values ${\xi }_c$ and ${\xi }_c^{\prime }$ give the bifurcation from quasiperiodic to periodic, and from periodic to the slowing-down state, respectively.

Figure 7

Phase diagram varying $\xi$ and ${\varepsilon }_b$ while fixing ${\tau }_a={\tau }_b=1$, ${\varepsilon }_a=1$, $g_{-}=0.5$, $g_{+}=2$. Filled circles are the Hopf bifurcation points from quasiperiodic to periodic oscillation, and unfilled circles are the critical points to the slowing-down state, both of which are numerically calculated with the initial conditions near the point $\{A_1=A_2=A_3=\sqrt{{\varepsilon }_a}/3,B_1=B_2=B_3=\sqrt{{\varepsilon }_b}/3\}$. The triangles are the critical points to the slowing-down state when we use states near a saddle as the initial conditions for simulation. The dashed line is calculated from Eq. (14) (${\lambda }_{21}+{\lambda }_{31}<0$ to the left), and the solid line is ${\lambda }_{22}=0$ (${\lambda }_{22}<0$ above the line). Below the intersection of these two lines, the unfilled circles diverge from the dashed line. In the region between the unfilled circles and triangles multiple attractors coexist.

Figure 8

An example of multiple attractors when ${\tau }_a={\tau }_b=1$, ${\varepsilon }_a=1$, ${\varepsilon }_b=1.9$, $g_{-}=0.5$, $g_{+}=2$, $\xi =0.016$: (a) slowing-down state with the initial condition near a saddle; (b) stable quasiperiodic orbit with the initial condition near $A_1=A_2=A_3=\sqrt{{\varepsilon }_a}/3,B_1=B_2=B_3=\sqrt{{\varepsilon }_b}/3$. If the state shown in panel (b) is used as the initial condition and $\xi$ is increased to $0.01815$, the dynamics is driven to the attracting heteroclinic cycle, but the transient looks quite similar to quasiperiodic oscillation. This is shown in panel (c).

Figure 9

Fourier transforms of one of the amplitudes in the noisy system for parameter values ${\tau }_a={\tau }_b=1$, ${\varepsilon }_a=1$, $g_{-}=0.5$, $g_{+}=2$, and Gaussian noise intensity ${10}^{-10}$. Note the two base frequencies when ${\varepsilon }_b=1.9$, $\xi =0.02$ (a), but only a single base frequency when ${\varepsilon }_b=4$, $\xi =0.34$ (b).