MS15.3: Time-periodic systems

One of the manifestations of parametric excitation in dynamical systems is the normal mode splitting. This phenomenon has been known for long in other disciplines as quantum optics and cavity optomechanics. It appears as a splitting in the resonance peak of an oscillating mode when coupled to another and excited parametrically at the difference frequency. The coupling strength between both modes dictates the width of the splitting, thus, the phenomenon is helpful for its estimation. In this contribution, the phenomenon is investigated analytically, where effect could be traced in the parameter space of the parametric and forced excitation frequencies. In addition, an experiment is conducted on a two-mode dielectrically actuated nanostring and the obtained model could be validated.

We are interested in a first-principle time-delayed model for the study of active mode-locking. It allows us to access the typical regimes encountered in semiconductor lasers and to perform an extended bifurcation analysis. In particular, close to the harmonic resonances and to the lasing threshold, we recover the Hermite-Gauss solutions. However, the presence of the linewidth enhancement factor induces complex regimes in which even the fundamental solution may become unstable. In addition, we discover a global bifurcation scenario in which a single pulse can jump, over a slow time scale, between the different minima of the modulation potential. Finally, we derive a Haus master equation close to the lasing threshold which shows a good agreement with the original time-delayed model.

Mechanical models for non-linear dynamic systems are defined by differential equations of second order where frequently also first order differential equations are present. Differential equations of first order occur if some degrees of freedom have no corresponding mass, if the stiffness parameters vanish in some of the equations or if a controller is implemented in the system. For linear systems of first and second order various numerical procedures for solving the differential equations are available. A semi- analytical method is presented which is exact for the linear dynamic and decoupled systems of first and second order. A modal transformation of the partitioned system equations is necessary for each part. After a discretization in the time-domain the relevant equations for a suitable and effective time-integration algorithm are defined taking non-linearity and the variable mass into account. The resulting procedure is derived and it turns out that the formulation is analogous to a BEM-formulation in time as Green functions are used. The method is extended to coupled non-linear differential equations of first and second order and is applied to a system with two degrees of freedom having time-periodic mass.