Center for Applied Mathematics Colloquium
Excitability is a very common phenomenon in the dynamics of many natural and engineered systems; examples are neurons, certain chemical reactions and laser systems. Being at equilibrium, an excitable system reacts to a sufficiently large perturbation by suddenly releasing a pulse of stored energy. Then the system needs some time to recover its level of stored energy. When excitable systems are coupling to themselves or to each other, they receive feedback with a delay time that is considerably larger than the pulse length. This may lead to very interesting pulsing dynamics. We demonstrate this here with an excitable micropillar laser with a feedback loop, or external cavity, generated by a regular mirror, which has been shown experimentally to be able to sustain trains of optical pulses. These can be triggered largely independently by optical perturbations injected into the laser, and they are then sustained simultaneously via feedback from the external cavity. A bifurcation analysis of a rate-equation model shows that the system has a number of periodic solutions with different numbers of equally spaced pulses as its only attractors. Hence, although coexisting pulse trains can seem independent on the timescale of the experiment, they correspond to very long transient dynamics. We determine the switching dynamics by studying the associated basins of attraction, which demonstrates that timing is everything when it comes to triggering or erasing pulse trains.
Bernd Krauskopf received his PhD in Mathematics from the University of Groningen in 1995. Subsequently, he spent one year at Cornell University and CAM and two years as a Postdoctoral Research Fellow at the Department of Physics and Astronomy of Vrije Universiteit Amsterdam before taking up a permanent position at the Department of Engineering Mathematics at the University of Bristol. In 2011 he moved to the University of Auckland, where he is a Professor of Applied Mathematics and leads a highly visible research group. Bernd’s research interests are in Applied Dynamical Systems, all the way from underlying theory, via numerical methods to real-world applications, including the nonlinear dynamics of laser systems, aircraft ground maneuvering and the role of feedback mechanisms in control and climate modeling. He has published over 170 academic journal papers and has been supervisor of 37 PhD students and 13 postdoctoral fellows.