Low-dimensional Linearized Models for Systems with Periodic Orbits, with Application to the Ginzburg-Landau Equation

We introduce a general procedure for obtaining a low-dimensional linear time-periodic model from a very high-dimensional nonlinear system that has an asymptotically stable periodic orbit. Our goal is to develop models that are suitable for designing feedback controllers for fluids systems with periodic orbits, such as periodically shedding wakes, or flow control problems where periodic actuation is introduced. In our method, we first linearize the nonlinear system about its asymptotically stable periodic orbit. We then compute a projection to project out the one-dimensional neutrally stable eigenspace appearing in the linear model corresponding to perturbations along the direction of the periodic orbit. Finally, we apply the method of snapshot-based balanced truncation for the high-dimensional linear periodic system to obtain a reduced-order model. We illustrate the method by developing reduced-order models for the complex Ginzburg-Landau equation.