Identifying nonlinear dynamics with high confidence from sparse data. (English) Zbl 1536.37022
Summary: We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (assumed to be a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.
MSC:
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
| 37M10 | Time series analysis of dynamical systems |
| 62R40 | Topological data analysis |
| 68T09 | Computational aspects of data analysis and big data |
Software:
GitHubReferences:
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