Nonlinear model reduction to temporally aperiodic spectral submanifolds

Area of Science:

• Dynamical Systems Theory
• Mathematical Physics
• Applied Mathematics

Background:

• Spectral submanifolds (SSMs) offer a powerful model-reduction technique for autonomous systems.
• Extending SSMs to non-autonomous systems with time-varying parameters presents significant theoretical challenges.
• Applications in structural dynamics, fluid-structure interactions, and control problems necessitate methods for analyzing time-dependent systems.

Purpose of the Study:

• To generalize the theory of spectral submanifolds (SSMs) to non-autonomous dynamical systems.
• To develop a mathematically justified model-reduction technique for time-dependent physical systems.
• To investigate the persistence and approximation of SSMs under various forcing conditions.

Main Methods:

• Extension of SSM theory to weakly forced or slowly varying non-autonomous systems.
• Construction of time-dependent SSMs that are normally hyperbolic.
• Derivation of formal asymptotic expansions for approximating SSMs under stronger forcing.

Main Results:

• Demonstrated the existence and persistence of time-dependent SSMs under specific assumptions.
• Developed methods to approximate SSMs and their trajectories for more general forcing scenarios.
• Illustrated the application of these techniques to mechanical systems with chaotic forcing.

Conclusions:

• The developed theory provides a robust model-reduction framework for a broad class of non-autonomous dynamical systems.
• Time-dependent SSMs offer persistent and accurate reduced models, even beyond the precise existence theory.
• The methods are applicable to complex physical systems with moderate to strong time dependence.