## Abstract

A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the ow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction. We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples.

Original language | English |
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Pages (from-to) | 299-336 |

Number of pages | 38 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2021 |

### Bibliographical note

Publisher Copyright:© 2021 American Institute of Mathematical Sciences. All rights reserved.

## Other keywords

- Complete Lyapunov function
- Discrete-time dynamical system
- Meshfree collocation
- Quadratic programming