Dimensionality reduction of dynamical systems with parameters: a geometric approach. (English) Zbl 1348.37034
Summary: A method for obtaining a low-dimensional description of a family of attractors produced by continuous-time nonlinear dynamical systems with static parameters is developed. A geometric approach is taken, which allows for the reduction of general state space manifolds, not restricted to \(\mathbb{R}^n\). An existing secant-based projection method, utilizing optimization over Grassmann manifolds, is extended for use with multiple parameter values and data sets. A family of reduced vector fields is obtained, with parameterization by the original parameter space, that reproduces the projected attractors and their dynamics in a low-dimensional space. We illustrate the method with several examples. The Rössler system demonstrates the accurate reproduction of period-doubling bifurcations; a forced damped double pendulum demonstrates a nonautonomous system with angular variables; and the Brusselator demonstrates application to a high-dimensional system.
MSC:
| 37C10 | Dynamics induced by flows and semiflows |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 34D45 | Attractors of solutions to ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
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