The second presentation in the A2IR2 webinar series was successfully delivered by Dr. A. Jubril on Wednesday, 5th May, 2021.

Titled “Modal Description of Nonlinear Dynamical Systems with Koopman  Operator Theory”, the talk was designed as an introduction to Koopman operator theoretic methods, with an emphasis on potential application areas for researchers.

Most real and natural processes, like those in climate science, population dynamics in ecology, motion dynamics, complex social network, can be considered as complex systems. The governing dynamics of such complex systems are generally nonlinear  and unknown; and such that are known are intractable for effective computational usefulness. Also, such complex system only allow access to only few of their output variables. Local identification of such systems for prediction, estimation, control and uncertainty quantification over a limited operation range is very common.

Koopman operator approach to the analysis of complex nonlinear allows the transformation by nonlinear lifting of the finite dimensional nonlinear system into an equivalent linear system embedded in an infinite dimensional function space. This transformation allows the matured linear analysis tools to be applicable for the understanding of the global dynamical of the system. Since the infinite dimensionality still makes the analysis of the system intractable, a finite dimensional approximation of the Koopman operator is usually sought which is invariant over a finite dimensional subspace which well approximates the system. While there are different approximation of the Koopman operator, the data driven method of the dynamic mode decomposition (DMD) provides a good road to computing finite dimensional approximation of the Koopman operator.

This talk therefore provided an introductory access to the door of opportunity and  power of Koopman mode decomposition for the analysis of nonlinear problems in  time series and complex dynamical systems.