Nonautonomous driving induces stability in network of identical oscillators

Nonautonomous driving of an oscillator has been shown to enlarge the Arnold tongue in parameter space, but little is known about the analogous effect for a network of oscillators. To test the hypothesis that deterministic nonautonomous perturbation is a good candidate for stabilizing complex dynamics, we consider a network of identical phase oscillators driven by an oscillator with a slowly time-varying frequency. We investigate both the short- and long-term stability of the synchronous solutions of this nonautonomous system. For attractive couplings we show that the region of stability grows as the amplitude of the frequency modulation is increased, through the birth of an intermittent synchronization regime. For repulsive couplings, we propose a control strategy to stabilize the dynamics by altering very slightly the network topology. We also show how, without changing the topology, time-variability in the driving frequency can itself stabilize the dynamics. As a byproduct of the analysis, we observe chimeralike states. We conclude that time-variability-induced stability phenomena are also present in networks, reinforcing the idea that this is a quite realistic scenario for living systems to use in maintaining their functioning in the face of ongoing perturbations.

Figure 1

Regions of existence of synchronous solutions. Region I: no synchronization (SNS). Region II: synchronization (SS). Region III: intermittent synchronization (SIS). Parameter values used ${\omega }_0=2$ and $k=0.4$.

Figure 2

Finite-time dynamics and stability, in the attractive case $D<0$. (a) Typical dynamics of the phase difference ${\psi }_i$ in the three regimes: synchronized (solid, $\gamma =3)$, intermittently synchronized (dashed, $\gamma =2)$, and not synchronized (dotted, $\gamma =1)$. Panels (b)–(d) only consider the intermittently synchronized trajectory. (b) Instantaneous stability, measured by the instantaneous LE (gray) given by Eq. (6) and the adiabatic approach (dashed black) of Eq. (11). The average LE, as defined in Eq. (12), is negative (dotted red). (c) Time-frequency representation computed applying continuous Morlet wavelet transform with central frequency 3 to the signal $y\left(t\right)=sin{\theta }_1\left(t\right)$. Epochs of boundedness (unbounded drift) ${\psi }_i$ correspond to negative (zero) $\lambda \left(t\right)$ and (no) frequency entrainment in (c) (respectively). (d) Convergence of two different initial conditions, in red and black, respectively, in the intermittent synchronization regime. Each initial condition is a quasisynchronous $N$-dimensional state. Each state first quickly becomes synchronous, and then converges to the synchronized solution. Parameters are $N=20,D=-0.5,k=0.4,\mathrm{\Delta }\omega =2,{\omega }_0=1$. (e) Network used for the numerics: random with connection probability $p=0.5$ and ${\mathrm{\Lambda }}^N=-15.9$. Note that the results do not depend on the network considered.

Figure 3

Stability region increases with amplitude of nonautonomicity $k$, in the attractive case $D<0$. (a–c) Quantitative stability measured by the LE of Eq. (12): the region of stability (shades of blue) increases with $k$. Gray values represent zero values of the LE. (d) Same LE values as in (a–c), shown only for $\gamma =2.5$, for the different $k$ values: 0.0 (solid), 0.4 (dashed), and 0.8 (dotted). The range of $\mathrm{\Delta }\omega$ for which ${\lambda }_{\text{max}}<0$ increases with $k$. Parameters are $D=-0.5,w_0=2$. This picture holds true for any network topology.

Figure 4

Stability region shrinks as ${\mathrm{\Lambda }}^N$ becomes more negative, i.e., as the largest degree of connectivity is increased, in the repulsive, $D>0$, autonomous, $k=0$, case. (a) Theoretical qualitative regions. Region I: no synchronization (light gray). Region IIa: synchronization unstable (dark gray vertically hatched). Region IIb: synchronization stable (dark gray horizontally hatched). (b–d) Quantitative stability measured by the LE: the region of stability IIb (shades of blue) decreases with ${\mathrm{\Lambda }}^N$, while the region of instability IIa (shades of red) increases. White values represent zero values of the LE. Above each panel, an example network that has the corresponding value of ${\mathrm{\Lambda }}^N$ is shown. Parameters are $D=0.5,w_0=2$.

Figure 5

Control of the stability by cutting a few chosen links, in the repulsive case $D>0$. (a, c) Trajectory of a synchronous synchronized initial condition of the phase difference ${\psi }_i$, in (b, d) the corresponding network, respectively. In the initial network (b), the synchronous state is unstable (a). By cutting a few chosen links (red), in the new network (d), the synchronous state is made stable (c). The initial network is a Barabási-Albert with $N=30$, and five links are cut. Other parameters are $D=0.5,\gamma =6.5,\mathrm{\Delta }\omega =2,{\omega }_0=1,k=0$.

Figure 6

Stability regions, in the repulsive, $D>0$, nonautonomous, $k=3$, case. (a) Theoretical qualitative regions. Birth of region IIc (square hatch), where the SS is alternates between stability and instability. Other regions are I (light gray) and III (medium gray) [see Fig. 3], and IIa (vertical hatch) and IIb (horizontal hatch) [see Fig. 4]. (b) Quantitative stability measured by the LE. White values represent zero values of the LE. (c) LE for a fixed $\gamma =2.5$ and different values of $k$. Other parameters are $D=0.5,w_0=1$.

Figure 7

Control strategy: fully synchronous state enforced intermittently by nonautonomous driving. Trajectory of a synchronous initial condition for $N=20$ oscillators for (a) constant and (b) modulated driving frequency. (a) Synchronous solution is unstable and yields patterns. (b) Time-variability of the driving forces state to be synchronous intermittently. Other parameters are $D=0.5,\gamma =4.6,\mathrm{\Delta }\omega =2,{\omega }_0=2$. Network used is random with $p=0.2$ and ${\mathrm{\Lambda }}^N=-8.68$.