Delay Embedding of Periodic Orbits Using a Fixed Observation Function

2017/09/30 by Raymundo Navarrete, Navarrete, Raymundo, Divakar Viswanath +1 · 1 citation
Computer Science · Physics and Astronomy · #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #Advanced Thermodynamics and Statistical Mechanics

paper · pdf · doi:10.48550/arxiv.1710.00128

Abstract

Delay coordinates are a widely used technique to pass from observations of a dynamical system to a representation of the dynamical system as an embedding in Euclidean space. Current proofs show that delay coordinates of a given dynamical system result in embeddings generically over a space of observations (Sauer, Yorke, Casdagli, J. Stat. Phys., vol. 65 (1991), p. 579-616). Motivated by applications of the embedding theory, we consider the situation where the observation function is fixed. For example, the observation function may simply be some fixed coordinate of the state vector. For a fixed observation function (any nonzero linear combination of coordinates) and for the special case of periodic solutions, we prove that delay coordinates result in an embedding generically over the space of flows in the C^r topology with r≥2.

Cited by

• Koopman Operator Based Time-Delay Embeddings and State History Augmented LQR for Periodic Hybrid Systems: Bouncing Pendulum and Bipedal Walking

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