Phase Reduction Method for Strongly Perturbed Limit Cycle Oscillators

The phase reduction method for limit cycle oscillators subjected to weak perturbations has significantly contributed to theoretical investigations of rhythmic phenomena. We here propose a generalized phase reduction method that is also applicable to strongly perturbed limit cycle oscillators. The fundamental assumption of our method is that the perturbations can be decomposed into a slowly varying component as compared to the amplitude relaxation time and remaining weak fluctuations. Under this assumption, we introduce a generalized phase parameterized by the slowly varying component and derive a closed equation for the generalized phase describing the oscillator dynamics. The proposed method enables us to explore a broader class of rhythmic phenomena, in which the shape and frequency of the oscillation may vary largely because of the perturbations. We illustrate our method by analyzing the synchronization dynamics of limit cycle oscillators driven by strong periodic signals. It is shown that the proposed method accurately predicts the synchronization properties of the oscillators, while the conventional method does not.

Figure 1

Phase dynamics of a modified Stuart-Landau oscillator. (a) A schematic diagram of the extended phase space ${\mathbb{R}}^n\times A$ with $n=2$ and $m=1$. (b) Frequency $\omega (I)$. (c) $I$-dependent stable limit cycle solutions ${\mathit{X}}_0(\theta ,I)$. (d),(e) Sensitivity functions $\zeta (\theta ,I)$ and $\xi (\theta ,I)$. (f),(g) Time series of the phase $\theta (t)$ of the oscillator driven by (f) a periodically varying parameter $I^{(1)}(t)$ or (g) a chaotically varying parameter $I^{(2)}(t)$. For each of these cases, results of the conventional (top panel) and proposed (middle panel) methods are shown. Evolution of the conventional phase $\tilde{\theta }(t)=\Theta (\mathit{X}(t),{\mathit{q}}_c)$ and the generalized phase $\theta (t)=\Theta \mathbf{(}\mathit{X}(t),\mathit{q}(ϵt)\mathbf{)}$ measured from the original system (lines) is compared with that of the conventional and generalized phase equations (circles). Time series of the state variable $x(t)$ (red solid line) and time-varying parameter $I(t)$ (blue dashed line) are also depicted (bottom panel). The periodically varying parameter is given by $I^{(1)}(t)=q^{(1)}(ϵt)+\sigma p^{(1)}(t)$ with $q^{(1)}(ϵt)=0.05\sin (0.5t)+0.02\sin (t)$ and $\sigma p^{(1)}(t)=0.02\sin (3t)$, and the chaotically varying parameter is given by $I^{(2)}(t)=q^{(2)}(ϵt)+\sigma p^{(2)}(t)$ with $q^{(2)}(ϵt)=0.007L_1(0.3t)$ and $\sigma p^{(2)}(t)=0.001L_2(t)$, where $L_1(t)$ and $L_2(t)$ are independently generated time series of the variable $x$ of the chaotic Lorenz equation [3], $\dot{x}=10(y-x)$, $\dot{y}=x(28-z)-y$, and $\dot{z}=xy-8z/3$.

Figure 2

Phase locking of the modified Stuart-Landau oscillator. Four types of periodically varying parameters $I^{(j)}$ ($j=3$, 4, 5, 6) are applied, which lead to $1:1$ phase locking to $I^{(3)}(t)$ [(a), (e), and (i)], $1:1$ phase locking to $I^{(4)}(t)$ [(b), (f), and (j)], $1:2$ phase locking to $I^{(5)}(t)$ [(c), (g), and (k)], and failure of phase locking to $I^{(6)}(t)$ [(d), (h), and (l)]. (a)–(d) Time series of the state variable $x(t)$ of a periodically driven oscillator (red solid line) and periodic external forcing (blue dashed line). (e)–(h) Dynamics of the phase difference ${\psi }_{1,2}$ with an arrow representing a stable fixed point (top panel) and time series of ${\psi }_{1,2}$ with 20 different initial states (bottom panel). (i)–(l) Orbits of a periodically driven oscillator (blue line) on the cylinder of the limit cycles (light blue line) plotted in the extended phase space. The parameter $I^{(j)}(t)$ is given by $I^{(j)}(t)=q^{(j)}(ϵt)+\sigma p^{(j)}(t)$, $q^{(j)}(ϵt)={\alpha }^{(j)}\sin ({\omega }_I^{(j)}t)$, and $\sigma p^{(j)}(t)=0.02\sin (5{\omega }_I^{(j)}t)$ with ${\alpha }^{(3,4,5,6)}=0.1$, 0.3, 0.4, 0.4 and ${\omega }_I^{(3,4,5,6)}=1.05$, 1.10, 0.57, 0.51.