Living cells, ecosystems, stock markets, the weather and society are all examples of complex systems – large aggregations of many smaller interacting parts. These parts may be e.g. species, stocks and investors, air particles or individuals. There is one particular property that secludes a complex system from one that is merely complicated: emergence. Emergence is the appearance of behavior that cannot be anticipated from the behavior of one of the constituents of the system alone. In complex systems this behavior appears through self-organization, i.e. there is no external entity engineering the appearance of emergent features. Instead, patterns appear spontaneously due to the interaction of each part with its surroundings. Such patterns will not arise if the various parts are simply coexisting; the presence and nature of the interaction of these parts is central.
Although complex systems clearly include living biological systems, we focus on emergent structures in non-living systems in this second branch of research in the lab. This is of course a bit an arbitrary classification, and our research entitled “Dynamics in Biology” could be described here similarly. In any case, it is interesting to see that very similar patterns are seen in wildly different contexts, such as e.g. the stripes of a zebra and the ripple patterns in sand dunes. It turns out to be common for a given pattern or dynamical behavior to show up in several different systems, and many aspects of these dynamics are not dependent on the finer details of the system and can be understood from the underlying symmetry and interaction patterns. Therefore, we combine methods of nonlinear dynamics and bifurcation theory to find universal descriptions of pattern formation and their dynamics.
More specifically, apart from studying generic model systems, we are currently also focusing our attention to the field of optics, where localized light pulses can be observed in nonlinear optical (micro-) cavities. Such light pulses can circulate in optical cavities without changing their shape thanks to the interaction between dispersion and nonlinearity, and this in the absence of any mechanism that continues to excite these pulses locally. Recently, it has been found that these traveling light pulses correspond to so-called frequency combs. A frequency comb consists in a broad optical spectrum of sharp comb lines with an equidistant frequency spacing that can be used to perform ultra-precise measurements of optical frequencies, and has numerous other applications in spectroscopy, optical clocks and waveform synthesis. Therefore, we study the origin and stability of such light pulses to gain new information about the dynamics of optical frequency combs.
||| Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion |
Physical Review E, volume 97, pp. 042204, 2018.
||| Origin and stability of dark pulse Kerr combs in normal dispersion resonators |
Optics Letters, volume 41, pp. 2402–2405, 2016.
||| Dynamics of localized and patterned structures in the Lugiato-Lefever equation determine the stability and shape of optical frequency combs |
Physical Review A, volume 89, pp. 043813, 2014.
||| Dynamics of one-dimensional Kerr cavity solitons |
Optics Express, volume 21, pp. 9180–9191, 2013.
||| Exploring multistability in semiconductor ring lasers: Theory and experiment |
Physical Review Letters, volume 102, pp. 193904, 2009.