Lifting Nonlinear Systems with Multiple Equilibria
This presentation explores a fundamental challenge in dynamical systems theory: how to apply Koopman operator methods to systems with multiple invariant sets. The authors demonstrate that discontinuous observables and symmetry constraints enable effective linear reconstruction of complex nonlinear dynamics, overturning conventional assumptions about the limitations of Koopman-based approaches for systems with disjoint basins of attraction.Script
Can you make a nonlinear system behave like a linear one, even when it has completely separate equilibria pulling trajectories in different directions? The Koopman operator promises exactly this transformation, but a persistent misconception claims it fails when faced with multiple invariant sets.
The Koopman operator elegantly transforms chaos into linearity by tracking how observable functions evolve. Yet theorists argued that systems with multiple equilibria, like a Duffing oscillator with two stable fixed points, cannot be reconstructed using continuous functions alone. Meanwhile, practitioners kept building models that worked.
The resolution lies in abandoning an unnecessary constraint.
The authors show that discontinuous observables, specifically indicator functions that mark which basin a trajectory occupies, solve the reconstruction problem. These functions act like switches, cleanly separating dynamics in different regions while preserving the Koopman framework's linearity.
Here we see the lifting mechanism in action for a Duffing oscillator with two stable fixed points. The observable functions Phi embed the original nonlinear state space into a higher-dimensional space where evolution becomes linear. The reconstruction map Psi then recovers the physical state. Notice how this framework accommodates the separated equilibria that supposedly made Koopman methods unworkable.
But the authors go further. When invariant sets exhibit symmetry, like rotational symmetry in the Lorenz attractor, this structure can be explicitly encoded into the Koopman eigenfunctions. Symmetry-constrained learning dramatically reduces the dimension of the lifted system and improves how well models generalize to unseen initial conditions.
The experimental validation is striking. Across radial basis functions, trigonometric observables, and polynomials, symmetry-constrained extended dynamic mode decomposition consistently outperforms the vanilla approach. The mean squared error on unseen trajectories drops significantly, and the improvement is robust to the choice of observable function.
This visualization reveals how symmetry augmentation works in practice for the chaotic Lorenz system. By rotating training data 180 degrees around the vertical axis, the researchers effectively double their dataset. Even more impressively, they can halve the original data, augment it through symmetry, and still achieve comparable coverage of the attractor's structure.
This work reshapes how we think about linearizing complex dynamics. The framework scales to systems that were previously considered intractable, opening pathways for improved prediction and control in engineering and science. The main limitation is practical: discovering the right symmetries in a new system still requires insight or systematic exploration.
Discontinuous functions and symmetry transform a theoretical impossibility into a computational advantage. Visit EmergentMind.com to explore more cutting-edge research and create your own video presentations.
