Bounding extreme values on attractors using sum-of-squares optimization, with application to the Lorenz attractor.
Bounding extreme values on attractors using sum-of-squares optimization, with application to the Lorenz attractor.
复制标题
使用平方和优化限制吸引子上的极值，并应用于洛伦兹吸引子。
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
D. Goluskin
中科院分区：
文献类型：
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作者：
D. Goluskin
我们描述的方法来界定的数量的极值的整体吸引子的微分动力系统。特别地，这样的界限在足够晚的时间沿着沿着每个轨迹应用。该方法利用李雅普诺夫函数寻找包含全局吸引子的吸收集，并根据目标函数极值的大小优化李雅普诺夫函数的选择。当控制方程和感兴趣的量是多项式时，优化约束要求两个多项式表达式是非负的。我们通过要求这些多项式可表示为平方和来强制非负性，从而导致凸优化问题，该问题可以被改写为半定规划并通过计算解决。这种计算机辅助使得构造复杂的多项式李雅普诺夫函数成为可能。我们将这些方法应用于混沌洛伦兹吸引子，边界极值的各种时刻的坐标（x，y，z）使用多项式次数的李雅普诺夫函数高达8。在所有情况下，我们得到的界限是尖锐的三个或更多的有效数字，其中大多数是尖锐得多比以前的结果。某些构造的吸收集也给出了整个吸引子的精确局部化。
We describe methods for bounding extreme values of quantities on global attractors of differential dynamical systems. Such bounds apply, in particular, along every trajectory at sufficiently late times. The methods use Lyapunov functions to find absorbing sets that contain the global attractor, and the choice of Lyapunov function is optimized based on the quantity whose extreme value one aims to bound. When the governing equations and quantities of interest are polynomials, the optimization constraints require two polynomial expressions to be nonnegative. We enforce nonnegativity by requiring these polynomials to be representable as sums of squares, leading to a convex optimization problem that can be recast as a semidefinite program and solved computationally. This computer assistance makes it possible to construct complicated polynomial Lyapunov functions. We apply these methods to the chaotic Lorenz attractor, bounding extreme values of various moments of the coordinates (x,y,z) using Lyapunov functions of polynomial degrees up to 8. In all cases we obtain bounds that are sharp to three or more significant figures, most of which are much sharper than prior results. Some of the absorbing sets constructed also give precise localizations of the attractor as a whole.
DOI：
10.1137/15m1053347
发表时间：
2015-12
期刊：
SIAM J. Appl. Dyn. Syst.
影响因子：
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作者：
Giovanni Fantuzzi;D. Goluskin;Deqing Huang;Sergei I. Chernyshenko
通讯作者：
Giovanni Fantuzzi;D. Goluskin;Deqing Huang;Sergei I. Chernyshenko