Bifurcation analysis of a semiconductor laser subject to phase conjugate feedback
In this thesis we present a detailed bifurcation analysis of a semiconductor laser
subject to phase-conjugate feedback (PCF). Mathematically, lasers with feedback are
modelled by delay differential equations (DDEs) with an infinite-dimensional phase
space. This is why, in the past, systems described by DDEs were only studied by
numerical simulation. We employ new numerical bifurcation tools for DOEs that go
much beyond mere simulation. More precisely, we continue steady states and periodic
orbits, irrespective of their stability with the package DDE-BIFTOOL, and present here
the first algorithm for computing unstable manifolds of saddle-periodic orbits with one
unstable Floquet multiplier in systems of DDEs. Together, these tools make it possible,
for the first time, to numerically study global bifurcations in ODEs.
Specifically, we first show how periodic solutions of the PCF laser are all connected
to one another via a locked steady state solution. A one-parameter study of these steady
states reveals heteroclinic bifurcations, which tum out to be responsible for bistability
and excitability at the locking boundaries. We then perform a full two-parameter
investigation of the locking range, where we continue bifurcations of steady states and
heteroclinic bifurcations. This leads to the identification of a number of codirnensiontwo
bifurcation points. Here, we also make a first attempt at providing a two-parameter
study of bifurcations of periodic orbits in a system of DDEs. Finally, our new method
for the computation of unstable manifolds of saddle periodic orbits is used to show how
a torus breaks up with a sudden transition to chaos in a crisis bifurcation.
In more general terms, we believe that the results presented in this thesis showcase
the usefulness of continuation and manifold computations and will contribute to the
theory of global bifurcations in DDEs.