Local solutions of the fast-slow model of an optoelectronic oscillator with delay (Q2047666)

In this paper, the author studies the local dynamics of a difference-differential model of an optoelectronic oscillator (a modification of the Ikeda equation with delay) in a neighborhood of the zero equilibrium state. This model after the change of the variables and differentiation can be written in the form \[ \varepsilon \nu^{-1}\ddot x+\dot x+\nu x=b_1\dot x(t-1)+2b_2x(t-1)\dot x(t-1)+3b_3x^2(t-1)\dot x(t-1)+\cdots, \] \[ b_1=-\beta\sin(2\phi),\ b_2=-\beta\cos(2\phi),\ b_3=\frac23\beta\sin(2\phi), \] where the function \(x\) describes the voltage at the Mach-Zehnder modulator input, \(\beta\) is the feedback coefficient, \(\phi\) is the phase determining the operating point of the modulator, \(\varepsilon\) is the frequency cutoff coefficient and \(\nu\) is the delay time in the feedback circuit, which are both assumed to take small positive values. The dot denotes the derivative with respect to \(t.\) The author explicitly formulated results on the correspondence between the solutions of the original system and those of special equations for slow amplitudes, which play the role of normal forms.