# American Institute of Mathematical Sciences

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Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares
October  2015, 20(8): 2419-2451. doi: 10.3934/dcdsb.2015.20.2419

## Efficient computation of Lyapunov functions for Morse decompositions

 1 Rutgers University, 110 Frelinghusen Road, Piscataway, NJ 08854, United States, United States 2 Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, 33431 Boca Raton 3 Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, United States

Received  June 2014 Revised  February 2015 Published  August 2015

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.
Citation: Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, William D. Kalies, Dinesh Kasti. Efficient computation of Lyapunov functions for Morse decompositions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2419-2451. doi: 10.3934/dcdsb.2015.20.2419
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