Nonlinear dynamics in a laser with a negative delayed feedback
The paper deals with the theoretical investigation of laser dynamics with a nonlinear element whose losses depend on the intensity of radiation passing through the feedback loop. From the mathematical point of view, the subject of the authors' consideration is the system of equations (1) \(\dot u=v[k(t)-1-\alpha u(t-\tau)]u(t)+vu_ 0\), \(\dot k=k_ 0-k(t)- u(t)k(t)\), i.e. a differential equation with a delayed argument. Here the running time \(t\) and the delay time \(\tau\) are normalized to some relaxation time; \(u\) is the ratio of the radiation intensity to the saturation intensity, \(k(k_ 0)\) is the ratio of the gain factor (unsaturated) to the loss factor independent of the radiation intensity; \(v=T_ a/T_ c\) is the ratio of the relaxation time of the gain \(T_ a\) to the damping time of radiation in the cavity \(T_ c\), \(\alpha\) is the parameter characterizing the feedback depth. The phase space of the system (1) is the direct product of the Banach space \(C_{[-\tau,0]}\) of continuous functions on \([-\tau,0]\) into \(\mathbb{R}^ 1\). The authors introduce the subset \(S_ 0\in C_{[-\tau,0]}\) of functions having the properties (i) \(\psi(0)=1\); (ii) \(0\leq\psi(s)\leq 1\), \(s\in[-\tau,0]\); (iii) \(\int^ 0_{-\tau}\psi(s)ds\leq[k_ 0v(1-e^{-\tau})]^{-1}\), and define the set of initial conditions as \(S(h)=S_ 0\times k_ 0h\), \(h\in(0,1)\), i.e. they consider solutions of (1) with the initial conditions \(u(s)=\psi(s)\in S_ 0\), \(s\in[-\tau,0]\) and \(k(0)=k_ 0h\). Then the system (1) is solved by the method of sequential integration steps, and different classes of solutions are considered under additional assumptions on the parameters \(k_ 0\) and \(u_ 0\). Parameter domains with different order of instability of the system steady state are determined. Also the transition to dynamic chaos by a sequence of period- dubling bifurcations is numerically investigated.
differential equations with a delayed argument
scientific article; zbMATH DE number 86307