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We derive and study a model describing passive mode locking-a set of differential equations with time delay. Unlike classical mode locking models based on the Haus master equation, this model does not assume small gain and loss per cavity round trip. Therefore, it is valid in a parameter range typical of semiconductor lasers. The limit of a slow saturable absorber is analyzed analytically. Bifurcations responsible for the appearance and breakup of the mode locking regime are studied numerically.

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Recent results describing non-trivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather non-trivial features which are in a sharp distinction with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors. (c) 1996 American Institute of Physics.

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ 1 , is positive. Concerning the next Lyapunov exponent, Λ 2 , there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ 2 | < ρ, where ρ is some tolerance ranging between 10 −5 and 10 −6 ). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.