8 results found
Webster K, 2021, Low-Rank kernel approximation of Lyapunov functions using neural networks, Journal of Computational Dynamics, ISSN: 2158-2505
Giesl P, Hamzi B, Rasmussen M, et al., 2020, Approximation of Lyapunov functions from noisy data, Journal of Computational Dynamics, Vol: 7, Pages: 57-81, ISSN: 2158-2491
Methods have previously been developed for the approximation of Lyapunovfunctions using radial basis functions. However these methods assume that theevolution equations are known. We consider the problem of approximating a givenLyapunov function using radial basis functions where the evolution equationsare not known, but we instead have sampled data which is contaminated withnoise. We propose an algorithm in which we first approximate the underlyingvector field, and use this approximation to then approximate the Lyapunovfunction. Our approach combines elements of machine learning/statisticallearning theory with the existing theory of Lyapunov function approximation.Error estimates are provided for our algorithm.
Rasmussen M, Rieger J, Webster KN, 2018, A reinterpretation of set differential equations as differential equations in a Banach space, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, Vol: 148, Pages: 429-446, ISSN: 1473-7124
Set differential equations are usually formulated in terms of theHukuhara differential. As a consequence, the theory of set differentialequations is perceived as an independent subject, in which all resultsare proved within the framework of the Hukuhara calculus.We propose to reformulate set differential equations as ordinarydifferential equations in a Banach space by identifying the convex andcompact subsets ofRdwith their support functions. Using this rep-resentation, standard existence and uniqueness theorems for ordinarydifferential equations can be applied to set differential equations. Weprovide a geometric interpretation of the main result, and we demon-strate that our approach overcomes the heavy restrictions the use ofthe Hukuhara differential implies for the nature of a solution.
knobloch J, Lamb JSW, webster KN, 2017, Shift dynamics near non-elementary T-points with real eigenvalues, Journal of Difference Equations and Applications, Vol: 24, Pages: 609-654, ISSN: 1563-5120
We consider non-elementary T-points in reversible systems in R2n+1. We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.
Rasmussen M, Rieger J, Webster KN, 2016, Approximation of reachable sets using optimal control and support vector machines, Journal of Computational and Applied Mathematics, Vol: 311, Pages: 68-83, ISSN: 0377-0427
We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples.
Knobloch J, Lamb JSW, Webster KN, 2014, Using Lin's method to solve Bykov's problems, JOURNAL OF DIFFERENTIAL EQUATIONS, Vol: 257, Pages: 2984-3047, ISSN: 0022-0396
Lamb JSW, Teixeira MA, Webster KN, 2005, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R-3, JOURNAL OF DIFFERENTIAL EQUATIONS, Vol: 219, Pages: 78-115, ISSN: 0022-0396
Webster KN, Elgin JN, 2003, Asymptotic analysis of the Michelson system, NONLINEARITY, Vol: 16, Pages: 2149-2162, ISSN: 0951-7715
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