Amplitude Death, Bifurcations, and Basins of Attraction of a Planar Self-Sustained Oscillator with Delayed Feedback

Abstract

We investigate the nonlinear dynamics of a two-dimensional self-sustained oscillator subject to a delayed feedback by performing a phase reduction analysis and considering two cases of amplitude variations, which represent weakly and strongly nonlinear cases. We investigate the amplitude death phenomenon and show that the feedback phase takes a relevant role to the suppression of the oscillations when amplitude variations are taking into account. In particular, we show that amplitude death can only occur in certain ranges of the feedback phase, in which destructive interferences are more pronounced. We analytically compute codimension-one and codimension-two bifurcations of the steady states and show parameter space maps with the number of steady-state solutions. We pay an special attention to the feedback phase in antiphase configuration, in which a richer bifurcation scenario is observed. We also numerically compute basins of attraction for the investigated delayed-feedback models, which is a challenge task for time-delay systems, shedding some light to the domains of initial conditions leading to each stable phase-locked state. Finally, we briefly discuss the effects of the shear parameter, which describes the amplitude-phase coupling, in the models in which amplitude variations are taking into account.

Appendix

A class-A laser [68] with delayed feedback is described by

$$\begin{aligned} \dot{\tilde{A}} = \tilde{A}\frac{(P - |\tilde{A}|^2)}{1+|\tilde{A}|^2}(1+i\alpha ) + \eta e^{-i\theta } \tilde{A}', \end{aligned}$$

(45)

where $\tilde{A}\equiv \tilde{A}(\tilde{t})$ and $\tilde{A}'\equiv \tilde{A}(\tilde{t}-\tilde{\tau })$ represent the value of the electric field at the actual time and the delayed time, respectively. $\tau$ is the delay time, $\eta$ is the feedback strength, P is the pump parameter, $\theta$ is the feedback phase, and $\alpha$ is the atomic detuning parameter. In the situation in which the electric field is not so large, we can approximate the first term of Eq. (45)

$$\begin{aligned} \frac{(P - |\tilde{A}|^2)}{1+|\tilde{A}|^2} = \frac{P - |\tilde{A}|^2 + 1 - 1}{1+|\tilde{A}|^2} =\frac{P +1}{1+|\tilde{A}|^2} - 1. \end{aligned}$$

(46)

By expanding the term below in a series Taylor, we have at first order

$$\begin{aligned} \frac{1}{1+|\tilde{A}|^2} \approx (1-|\tilde{A}|^2). \end{aligned}$$

(47)

By applying the earlier approximation in Eq. (46), we have

$$\begin{aligned} (P+1)(1-|\tilde{A}|^2) - 1=P\bigg [1-\frac{(P+1)}{P}|\tilde{A}|^2\bigg ]. \end{aligned}$$

(48)

Therefore,

$$\begin{aligned} \frac{P - |\tilde{A}|^2}{1+|\tilde{A}|^2}\approx P\bigg [1-\frac{(P+1)}{P}|\tilde{A}|^2\bigg ]. \end{aligned}$$

(49)

By using the earlier approximation in Eq. 45, we obtain

$$\begin{aligned} \dot{\tilde{A}}&= \tilde{A} P \bigg [1-\frac{(P+1)}{P}|\tilde{A}|^2 \bigg ](1+i\alpha ) + \eta e^{-i\theta } \tilde{A}'. \end{aligned}$$

(50)

By redefining the electric field as

$$\begin{aligned} \tilde{A}= \sqrt{\frac{P}{P+1}}{A}, \;\;\; \tilde{A}'= \sqrt{\frac{P}{P+1}}{A'}, \end{aligned}$$

(51)

we obtain a new system of equations known as the cubic laser model, which is known in literature as the simplest laser model [68]

$$\begin{aligned} \dot{A} = AP(1-|A|^2)(1+i\alpha ) + \eta e^{-i\theta } A'. \end{aligned}$$

(52)

Here, in the cubic laser model given by Eq. (52), we have included the atomic detuning parameter and the delayed feedback. Rewriting Eq. (52) in terms of amplitude and phase, we have

$$\begin{aligned} \dot{a}&= aP(1 - a^2) + \eta a'\cos (\theta + \phi - \phi '), \nonumber \\ \dot{\phi }&= \alpha (1 - a^2) - \eta \frac{a'}{a}\sin (\theta + \phi - \phi '), \end{aligned}$$

(53)

where the variables $a=a(t)$ and $a'\equiv a(t-\tau )$ represent the amplitude of the electric field at the actual time and the delayed time, and $\phi \equiv \phi (t)$ and $\phi '\equiv \phi (t-\tau )$ represent the phase of the electric field at the actual time and the delayed time, respectively. If the feedback strength is small, the amplitude of the electric field almost does not change. Therefore, by considering $a=1$ and neglecting amplitude variations, the laser dynamics is described by a single phase equation

$$\begin{aligned} \dot{\phi }&= -\eta \sin ( \theta + \phi - \phi '). \end{aligned}$$

(54)