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 flow 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.

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为离散时间动力系统计算完整的Lyapunov函数

完整的Lyapunov函数可表征一般离散时间动力系统的行为。特别地，它将状态空间划分为链递归集合，其中完整的Lyapunov函数沿轨迹恒定，而部分流体呈梯度状且完整的Lyapunov函数沿解严格减小。此外，一个完整的李雅普诺夫函数的水平集提供了有关吸引子，驱蚊器和吸引盆的信息。我们提出了两类新颖的方法来为迭代给出的一般离散时间动力系统计算完整的李雅普诺夫函数。第一类方法使用无网格搭配，通过近似一个不适定方程的离散轨道导数的解来计算一个完整的Lyapunov函数。第二类方法计算一个完整的Lyapunov函数，作为再现内核Hilbert空间中最小化问题的解决方案。我们将这两种方法都应用于几个示例。