MS03.5: Computational Methods

Many important statements about dynamical systems can be proved by finding scalar-valued auxiliary functions whose time evolution along trajectories obeys certain pointwise inequalities that imply the desired result. The most familiar of these auxiliary functions is a Lyapunov function to prove steady-state stability, but such functions can also be used to bound averages of ergodic systems, define trapping boundaries, and so much more. In this talk I will highlight a method of identifying auxiliary functions from data using polynomial optimization. The method leverages recent advances in approximating the Koopman operator from data, so-called extended dynamic mode decomposition, to provide system-level information without system identification. The result is a flexible, data-driven, model-agnostic computational method that does not require explicit model discovery. Furthermore, it can be applied to data generated through a deterministic or stochastic process with no prior adjustments to the implementation. It can be used to bound quantities of interest, develop optimal state-dependent feedback controllers, and discover invariant measures.

In the domain of vibrations, the responses of a mechanical system are often depicted as the amplitude versus the excitation frequency, namely the Frequency Response Function (FRF). For a given value of the excitation frequency, the uniqueness of the solution to the structural dynamics problem is not ensured in a nonlinear framework. Given this property, continuation techniques should be employed as a way of computing the entire response of the system. Moreover, several branches of solutions may exist due to the appearance of bifurcation points. Strategies have been proposed to obtain such solutions. Besides, it was shown in the literature that some branches of solutions-named isolas-can even be isolated from the primary resonance curve of the system. The systematic computation of isolated branches of solutions remains an active field of research. Among the main techniques, one can find for instance, the Melnikov analysis and Groebner basis. In this work, a novel approach is proposed. Contrary to traditional FRFs, the excitation frequency is kept here as a constant and the force level becomes an unknown of the problem. This new set of equations now takes into account for an imposed displacement on a control degree of freedom. Varying the value of the displacement parameter, through a continuation algorithm for example, allows for obtaining curves of solutions. If one is interested in traditional FRF the new methodology can provide several initialization points to a continuation algorithm which lead to possible multiple branches of solutions and isolated curves.