Finding characteristically rich nonlinear solution space: a statistical mechanics approach

In this work, the modally equivalent perturbed system (MEPS) which was originally developed for finding the parametrically rich solution space of linear time-invariant systems is modified for time-varying cases and applied to find the characteristically rich nonlinear solution space given arbitrary initial or boundary conditions, or system inputs. An integral form of the non-Hamiltonian Liouville equation is derived such that a rich ensemble average of its solutions covers a broad range of the modal space when a maximum uncertainty is present in the solutions. The MEPS degenerates the integrated Liouville equation into a linear differential equation with the Gauge Modal Invariance, a newly found field property that allows extending the application beyond the initial conditions or impulse inputs, making it possible to calculate the rich set of basis modes by taking snapshots of the linear responses at a considerably low computational cost. The proposed theory and algorithm are demonstrated using a computational model of a two-dimensional incompressible, viscous flow at low Reynolds numbers. It is shown that the basis modes obtained herein, when used in conjunction with a low dimensional modeling, reproduce time simulation results very accurately for a wide range of Reynolds numbers and boundary conditions.