This report presents a modern approach to the theoretical and experimental study of complex nonlinear behavior of a semiconductor laser with optical injection—an example of a widely applied and ...technologically relevant forced nonlinear oscillator. We show that the careful bifurcation analysis of a rate equation model yields (i) a deeper understanding of already studied physical phenomena, and (ii) the discovery of new dynamical effects, such as multipulse excitability.
Different instabilities, cascades of bifurcations, multistability, and sudden chaotic transitions, which are often viewed as independent, are in fact logically connected into a consistent web of bifurcations via special points called organizing centers. This theoretical bifurcation analysis has predictive power, which manifests itself in good agreement with experimental measurements over a wide range of parameters and diversity of dynamics.
While it is dealing with the specific system of an optically injected laser, our work constitutes the state-of-the-art in the understanding and modeling of a nonlinear physical system in general.
We study a generic model for the interaction of negative delayed feedback and periodic forcing that was first introduced by Ghil et al (2008 Nonlinear Process. Geophys. 15 417-33) in the context of ...the El Niño Southern Oscillation climate system. This model takes the form of a delay differential equation and has been shown in previous work to be capable of producing complicated dynamics, which is organised by resonances between the external forcing and dynamics induced by feedback. For certain parameter values, we observe in simulations the sudden disappearance of (two-frequency dynamics on) tori. This can be explained by the folding of invariant tori and their associated resonance tongues. It is known that two smooth tori cannot simply meet and merge; they must actually break up in complicated bifurcation scenarios that are organised within so-called resonance bubbles first studied by Chenciner. We identify and analyse such a Chenciner bubble in order to understand the dynamics at folds of tori. We conduct a bifurcation analysis of the Chenciner bubble by means of continuation software and dedicated simulations, whereby some bifurcations involve tori and are detected in appropriate two-dimensional projections associated with Poincaré sections. We find close agreement between the observed bifurcation structure in the Chenciner bubble and a previously suggested theoretical picture. As far as we are aware, this is the first time the bifurcation structure associated with a Chenciner bubble has been analysed in a delay differential equation and, in fact, for a flow rather than an explicit map. Following our analysis, we briefly discuss the possible role of folding tori and Chenciner bubbles in the context of tipping.
Modeling and parameter estimation to capture the dynamics of physical systems are often challenging because many parameters can range over orders of magnitude and are difficult to measure ...experimentally. Moreover, selecting a suitable model complexity requires a sufficient understanding of the model's potential use, such as highlighting essential mechanisms underlying qualitative behavior or precisely quantifying realistic dynamics. We present an approach that can guide model development and tuning to achieve desired qualitative and quantitative solution properties. It relies on the presence of disparate time scales and employs techniques of separating the dynamics of fast and slow variables, which are well known in the analysis of qualitative solution features. We build on these methods to show how it is also possible to obtain quantitative solution features by imposing designed dynamics for the slow variables in the form of specified two-dimensional paths in a bifurcation-parameter landscape.
We report on the first experimental observation of spontaneous mirror symmetry breaking (SSB) in coherently driven-dissipative coupled optical cavities. SSB is observed as the breaking of the spatial ...or mirror Z_{2} symmetry between two symmetrically pumped and evanescently coupled photonic crystal nanocavities, and manifests itself as random intensity localization in one of the two cavities. We show that, in a system featuring repulsive boson interactions (U>0), the observation of a pure pitchfork bifurcation requires negative photon hopping energies (J<0), which we have realized in our photonic crystal molecule. SSB is observed over a wide range of the two-dimensional parameter space of driving intensity and detuning, where we also find a region that exhibits bistable symmetric behavior. Our results pave the way for the experimental study of limit cycles and deterministic chaos arising from SSB, as well as the study of nonclassical photon correlations close to SSB transitions.
A full-scale experimental test for large and complex structures is not always achievable. This can be due to many reasons, the most prominent one being the size limitations of the test. Real-time ...dynamic substructuring is a hybrid testing method where part of the system is modelled numerically and the rest of the system is kept as the physical test specimen. The numerical–physical parts are connected via actuators and sensors and the interface is controlled by advanced algorithms to ensure that the tested structure replicates the emulated system with sufficient accuracy. The main challenge in such a test is to overcome the dynamic effects of the actuator and associated controller, that inevitably introduce delay into the substructured system which, in turn, can destabilize the experiment. To date, most research concentrates on developing control strategies for stable recreation of the full system when the interface location is given a priori. Therefore, substructurability is mostly studied in terms of control. Here, we consider the interface location as a parameter and study its effect on the stability of the system in the presence of delay due to actuator dynamics and define substructurability as the system’s tolerance to delay in terms of the different interface locations. It is shown that the interface location has a major effect on the tolerable delays in an experiment and, therefore, careful selection of it is necessary.
We present a continuation method that enables one to track or continue branches of periodic orbits directly in an experiment when a parameter is changed. A control-based setup in combination with ...Newton iterations ensures that the periodic orbit can be continued even when it is unstable. This is demonstrated with the continuation of initially stable rotations of a vertically forced pendulum experiment through a fold bifurcation to find the unstable part of the branch.
A piecewise-linear model with a single degree of freedom is derived from first principles for a driven vertical cantilever beam with a localized mass and symmetric stops. The aim is to show that this ...model constitutes a considerable step toward developing a vibro-impact model that is able to make qualitative and quantitative predictions of the observed dynamics. The resulting piecewise-linear dynamical system is smoothed by a switching function (nonlinear homotopy). For the chosen smoothing function, it is shown that the smoothing can induce bifurcations in certain parameter regimes. These induced bifurcations disappear when the transition of the switching is sufficiently and increasingly localized as the impact becomes harder. The bifurcation structure of the impact oscillator response is investigated via the one- and two-parameter continuation of periodic orbits in the driving frequency and/or forcing amplitude. The results are in good agreement with experimental measurements.
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