Efficient computation of Lyapunov functions for Morse decompositions | Zendy

Arnaud Goullet | Zendy; Shaun Harker | Zendy; Konstantin Mischaikow | Zendy; William D. Kalies | Zendy; Dinesh Kasti | Zendy

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Efficient computation of Lyapunov functions for Morse decompositions

Author(s) -

Arnaud Goullet,

Shaun Harker,

Konstantin Mischaikow,

William D. Kalies,

Dinesh Kasti

Publication year - 2015

Publication title -

discrete and continuous dynamical systems - b

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.864

H-Index - 53

eISSN - 1553-524X

pISSN - 1531-3492

DOI - 10.3934/dcdsb.2015.20.2419

Subject(s) - lyapunov function , piecewise , mathematics , sequence (biology) , lyapunov exponent , constant (computer programming) , nonlinear system , lyapunov equation , metric (unit) , lyapunov redesign , control lyapunov function , metric space , constant function , computation , algorithm , computer science , mathematical analysis , physics , quantum mechanics , operations management , biology , economics , genetics , programming language

We present an efficient algorithm for constructing piecewise constant Lyapunov functions for dynamics generated by a continuous nonlinear map defined on a compact metric space. We provide a memory efficient data structure for storing nonuniform grids on which the Lyapunov function is defined and give bounds on the complexity of the algorithm for both time and memory. We prove that if the diameters of the grid elements go to zero, then the sequence of piecewise constant Lyapunov functions generated by our algorithm converge to a continuous Lyapunov function for the dynamics generated the nonlinear map. We conclude by applying these techniques to two problems from population biology.

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