Nonlinear phase-amplitude reduction of delay-induced oscillations

Spontaneous oscillations induced by time delays are observed in many real-world systems. Phase reduction theory for limit-cycle oscillators described by delay-differential equations (DDEs) has been developed to analyze their synchronization properties, but it is applicable only when the perturbation applied to the oscillator is sufficiently weak. In this study, we formulate a nonlinear phase-amplitude reduction theory for limit-cycle oscillators described by DDEs on the basis of the Floquet theorem, which is applicable when the oscillator is subjected to perturbations of moderate intensity. We propose a numerical method to evaluate the leading Floquet eigenvalues, eigenfunctions, and adjoint eigenfunctions necessary for the reduction and derive a set of low-dimensional nonlinear phase-amplitude equations approximately describing the oscillator dynamics. By analyzing an analytically tractable oscillator model with a cubic nonlinearity, we show that the asymptotic phase of the oscillator state in an infinite-dimensional state space can be approximately evaluated and nontrivial bistability of the oscillation amplitude caused by moderately strong periodic perturbations can be predicted on the basis of the derived phase-amplitude equations. We further analyze a model of gene-regulatory oscillator and illustrate that the reduced equations can elucidate the mechanism of its complex dynamics under nonweak perturbations, which may be relevant to real physiological phenomena such as circadian rhythm sleep disorders.

Figure 1

(a) Time course of a scalar DDE with a cubic nonlinearity, Eq. (47), showing a slow convergence of the system state to the limit cycle. (b) Time course of the oscillator state projected onto the $(x,dx/dt)$ plane. [(c), (d)] Extended adjoint method for calculating $q_1^{\left(t\right)}$. (c) Peak heights of the time course of $Y^{(t=nT)}\left(0\right)$ measured at each period vs $t$. The red squares are the data from which the Floquet exponent ${\lambda }_1$ is evaluated. (d) Time evolution of $exp\left(-{\lambda }_{1}t\right)Y^{\left(t\right)}\left(0\right)$ after compensating the exponential decay. (e) Eigenfunctions and adjoint eigenfunctions associated with ${\lambda }_0=0$ plotted as functions of $\varphi$. The functions $q_0^{\left(\varphi \right)}\left(0\right)$ and $q_0^{\left(\varphi \right)*}\left(0\right)$ are analytically derived, while ${\overline{q}}_0^{\left(\varphi \right)}\left(0\right)$ and ${\overline{q}}_0^{\left(\varphi \right)*}\left(0\right)$ are numerically obtained by the extended adjoint method. (f) Eigenfunctions $q_1^{\left(\varphi \right)}\left(0\right)$ and adjoint eigenfunctions $q_1^{{\left(\varphi \right)}^{*}}\left(0\right)$ associated with ${\lambda }_1$.

Figure 2

Evaluation of the asymptotic phase $\mathrm{\Phi }$ from $\varphi$ and $R$. (a) Difference $\varphi -\mathrm{\Phi }$ between $\varphi$ and $\mathrm{\Phi }$ plotted on the $(\varphi ,R)$ plane. The data are obtained by direct numerical integration from initial system states given by $x^{(t=0)}\left(\sigma \right)=x_0^{\left(\varphi \right)}\left(\sigma \right)+R{q}_1^{\left(\varphi \right)}\left(\sigma \right)$. (b) Difference $\varphi -\widehat{\mathrm{\Phi }}$ between $\varphi$ and $\widehat{\mathrm{\Phi }}$ estimated by using Eq. (61). (c) Absolute difference $|\mathrm{\Phi }-\widehat{\mathrm{\Phi }}|$ between the asymptotic phase $\mathrm{\Phi }$ measured directly by numerical integration and $\widehat{\mathrm{\Phi }}$ estimated by using Eq. (61). (d) Asymptotic phase of the initial system states given by $x^{(t=0)}\left(\sigma \right)=sin\sigma +psin(\sigma /2)$, which is not on the plane spanned by the first two Floquet eigenfunctions. The black points indicate the numerical results, the blue line indicates the phase $\varphi$ evaluated using the linearized isochrons, and the red line indicates the analytical estimation of asymptotic phase $\mathrm{\Phi }$.

Figure 3

Maximum amplitude of the DDE (47) subjected to a periodic input. (a) Dependence of maximum amplitude on $G_0$ at $r=1$. (b) Dependence of maximum amplitude on $r$ at $G_0=0.1$. Blue points show numerical results obtained by direct numerical integration of the original DDE (47). The red lines are analytical predictions by the linear approximation, while black lines are numerical solutions of the phase-amplitude equations (38) and (39).

Figure 4

(a) Stable and unstable fixed points of the amplitude $R$ plotted against the frequency detuning $r$ at $G_0=0.02$. (b) Stable and unstable fixed points at $G_0=0.1$. A bistable region exists near $r=1.052$. (c) Nullclines and stable fixed points on the ($\psi$, $R$) plane at $r=1.052$ and $G_0=0.1$. The gray broken lines show the nullclines satisfying $\dot{\psi }=0$ and the gray solid line shows $\dot{R}=0$. The orange and green dots show the stable fixed points (sp1 and sp2) and black lines show the trajectories started from $(-2,0)$ and $(-2.5,0)$. Sp1 is located at $(-0.722,0.992)$ and sp2 is at $(-2.647,-0.064)$. (d) Time course of Eq. (47) with $G(x,t)=0.1sin\left(1.052t\right)$. The orange line corresponds to the sp1, while the green line corresponds to the sp2 in panel (c).

Figure 5

(a) Time course of the gene-regulatory oscillator Eq. (63) without perturbation ($G=0$), showing a slow convergence to the limit cycle. (b) Trajectory of the system state projected onto the $(x,dx/dt)$ plane. (c) Floquet and adjoint eigenfunctions $q_0^{\left(\varphi \right)}\left(0\right)$ and $q_0^{{\left(\varphi \right)}^{*}}\left(0\right)$ associated with ${\lambda }_0=0$. (d) Floquet and adjoint eigenfunctions $q_1^{\left(\varphi \right)}\left(0\right)$ and $q_1^{{\left(\varphi \right)}^{*}}\left(0\right)$ associated with ${\lambda }_1$.

Figure 6

(a) Asymptotic phase values of initial functions $x^{(t=0)}\left(\sigma \right)\equiv p$ far from the limit cycle. The black points indicate the asymptotic phase obtained by direct numerical integration, the blue line indicates analytical estimation of the phase $\widehat{\varphi }$, and the red line indicates analytical estimation of the asymptotic phase $\widehat{\mathrm{\Phi }}$. (b) Stable and unstable fixed points plotted with respect to $r$ at $G_0=0.05$. (c) Fixed points at $G_0=0.4$. A bistable region exists around $r=0.9911$. (d) Time course of the DDE (63) with $G_0=0.4$ and $r=0.9911$. The red line shows the result for the initial condition $x^{(t=0)}\left(\sigma \right)=X_0^{({\varphi }_{\mathrm{ini}}=0.6T)}\left(\sigma \right)$, while the blue shows the result for $x^{(t=0)}\left(\sigma \right)=X_0^{({\varphi }_{\mathrm{ini}}=0.2T)}\left(\sigma \right)$ ($-\tau \le \sigma \le 0$). The green line represents the external force.