758 research outputs found
Optimal Piecewise-Linear Approximation of the Quadratic Chaotic Dynamics
This paper shows the influence of piecewise-linear approximation on the global dynamics associated with autonomous third-order dynamical systems with the quadratic vector fields. The novel method for optimal nonlinear function approximation preserving the system behavior is proposed and experimentally verified. This approach is based on the calculation of the state attractor metric dimension inside a stochastic optimization routine. The approximated systems are compared to the original by means of the numerical integration. Real electronic circuits representing individual dynamical systems are derived using classical as well as integrator-based synthesis and verified by time-domain analysis in Orcad Pspice simulator. The universality of the proposed method is briefly discussed, especially from the viewpoint of the higher-order dynamical systems. Future topics and perspectives are also provide
Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization
We present a flexible data-driven method for dynamical system analysis that
does not require explicit model discovery. The method is rooted in
well-established techniques for approximating the Koopman operator from data
and is implemented as a semidefinite program that can be solved numerically.
Furthermore, the method is agnostic of whether data is generated through a
deterministic or stochastic process, so its implementation requires no prior
adjustments by the user to accommodate these different scenarios. Rigorous
convergence results justify the applicability of the method, while also
extending and uniting similar results from across the literature. Examples on
discovering Lyapunov functions, performing ergodic optimization, and bounding
extrema over attractors for both deterministic and stochastic dynamics
exemplify these convergence results and demonstrate the performance of the
method.Comment: We have significantly expanded the presentation. This has improved
the presentation and made the paper more readable. Comments welcom
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