Title: Uniformly Bounded Sets in Quasiperiodically Forced Dynamical Systems,
Abstract: This paper addresses structures of state space in quasiperiodically forced
dynamical systems. We develop a theory of ergodic partition of state space in a
class of measure-preserving and dissipative flows, which is a natural extension
of the existing theory for measure-preserving maps. The ergodic partition
result is based on eigenspace at eigenvalue 0 of the associated Koopman
operator, which is realized via time-averages of observables, and provides a
constructive way to visualize a low-dimensional slice through a
high-dimensional invariant set. We apply the result to the systems with a
finite number of attractors and show that the time-average of a continuous
observable is well-defined and reveals the invariant sets, namely, a finite
number of basins of attraction. We provide a characterization of invariant sets
in the quasiperiodically forced systems. A theorem on uniform boundedness of
the invariant sets is proved. The series of analytical results enables
numerical analysis of invariant sets in the quasiperiodically forced systems
based on the ergodic partition and time-averages. Using this, we analyze a
nonlinear model of complex power grids that represents the short-term swing
instability, named the coherent swing instability. We show that our analytical
results can be used to understand stability regions in such complex systems.