Accumulating regions of winding periodic orbits in optically driven lasers

We investigate the route to locking in class B lasers subject to optically injected light for injection strengths and detunings near a codimension-two saddle-node Hopf point. This is the parameter region where the Adler approximation is not valid and where Yeung and Strogatz recently reported a self-similar cascade of periodic orbits in the case of a solid-state laser. We explain this cascade as an accumulation of large regions bounded by saddle-node bifurcations of periodic orbits, but also containing further bifurcations, such as period-doubling, torus bifurcations and small pockets of chaos. In the vicinity of the simultaneous saddle-node and Hopf bifurcations, successive periodic orbits wind more and more near the point in phase space where the saddle-node bifurcation is about to occur. This leads to a self-similar period-adding cascade. By varying the linewidth enhancement parameter α from zero, the case of a solid-state or C

Unnested islands of period-doublings in an injected semiconductor laser

We present a theoretical study of unnested period-doubling islands in three-dimensional rate equations modeling a semiconductor laser subject to external optical injection. In this phenomenon successive curves of period doublings are not arranged in nicely nested islands, but intersect each other. This overall structure is globally organized by several codimension-2 bifurcations. As a consequence, the chaotic region existing inside an unnested island of period doublings can be entered not only via a period-doubling cascade but also via the breakup of a torus, and even via the sudden appearance of a chaotic attractor. In order to fully understand these different chaotic transitions we reveal underlying global bifurcations and we show how they are connected to codimension-2 bifurcation points. Unnested islands of period doublings appear to be generic and hence must be expected in a large class of dynamical systems

Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping

We discuss the nonlinear phenomena of irreversible tipping for non-autonomous systems where time-varying inputs correspond to a smooth "parameter shift" from one asymptotic value to another. We express tipping in terms of pullback attraction and present some results on how nontrivial dynamics for non-autonomous systems can be deduced from analysis of the bifurcation diagram for an associated autonomous system where parameters are fixed. In particular, we show that there is a unique local pullback point attractor associated with each linearly stable equilibrium for the past limit. If there is a smooth stable branch of equilibria over the range of values of the parameter shift, the pullback attractor will remain close to (track) this branch for small enough rates, though larger rates may lead to rate-induced tipping. More generally, we show that one can track certain stable paths that go along several stable branches by pseudo-orbits of the system, for small enough rates. For these local pullback point attractors, we define notions of bifurcation-induced and irreversible rate-induced tipping of the non-autonomous system. In one-dimension, we give a number of sufficient conditions for the presence or absence of rate-induced tipping, and we discuss some applications of our results to give criteria for irreversible rate-induced tipping in a conceptual climate model example