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Global stability properties of the complex Lorenz model. (English) Zbl 0887.34048
Summary: By means of a quadratic Lyapunov function it is shown that the complex Lorenz model is globally stable in the sense that the trajectories asymptotically settle within a finite region around the zero fixed point. Thus, similar to the original Lorenz model, the five-dimensional complex model does not have runaway solutions. For conditions appropriate to model single mode lasers an analytical expression is given for the upper bound on the magnitude of the time-dependent electric field as a function of the model parameters (pumping rate, detuning and relaxation constants). When compared with the time-dependent solutions found by direct integration, the upper bounds exceed the maximum values reached by the asymptotic solutions by factors between 3 and 6, or between 2 and 3, for conditions below and above the second threshold, respectively. In extreme cases of transient evolution, the solutions approach within 20% of the predicted upper bounds.
MSC:
34D20Stability of ODE
81V80Applications of quantum theory to quantum optics
37N99Applications of dynamical systems