标题： 洛伦兹系统作为梯度类似系统 显示英文标题

标题： The Lorenz system as a gradient-like system

摘要： 我们为连续时间动力系统制定一个充分条件，使其成为梯度类似系统，即所有有界轨迹趋近于平衡点，因此周期轨道、混沌吸引子等不存在。 该条件基于在系统状态空间中定义的一个辅助函数，其方式类似于平衡点稳定性的李雅普诺夫函数。 对于多项式系统，可以通过使用平方和优化计算找到李雅普诺夫函数。 我们通过为洛伦兹系统找到这样的辅助函数来演示此方法。 我们能够证明当$0\leq\rho\leq12$时，$\sigma=10$且$\beta=8/3$，显著扩展了以前的结果。 这些结果通过一种新颖的程序进行了严格验证：首先，使用有限精度浮点数平方和优化找到近似数值解。 然后，我们使用区间算术证明存在一个接近该解的精确解。 显示英文摘要

摘要： We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc. do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for $0\leq\rho\leq12$ when $\sigma=10$ and $\beta=8/3$, significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.