We describe a framework for bounding extreme values of quantities on global attractors of differential dynamical systems. A global attractor is the minimal set that attracts all bounded sets; it contains all forward-time limit points. Our approach uses (generalized) Lyapunov functions to find attracting sets, which must contain the global attractor, and the choice of Lyapunov function is optimized based on the quantity whose extreme value one aims to bound. We also present a non-global framework for bounding extrema over the minimal set that is attracting in a specified region of state space. If the dynamics are governed by ordinary differential equations, and the equations and quantities of interest are polynomial, then our methods can be implemented computationally using polynomial optimization. In particular, we enforce nonnegativity of certain polynomial expressions by requiring them 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 lets one construct complicated polynomial Lyapunov functions. Computations are illustrated using three examples. The first is the chaotic Lorenz system, where we bound extreme values of various monomials of the coordinates over the global attractor. In the second example we bound extreme values in a nine-mode truncation of fluid dynamics which displays long-lived chaotic transients. The third example has two locally stable limit cycles, each with its own basin of attraction. We apply our non-global framework to construct bounds for one basin that do not apply to the other. For each example we compute Lyapunov functions of polynomial degrees up to at least eight. In cases where we can judge the sharpness of our bounds, they are sharp to at least three digits when the polynomial degree is at least four or six.

中文翻译：

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使用多项式优化在全局吸引子上界定极值

我们描述了一个框架，用于在微分动力系统的全局吸引子上限制数量的极值。全局吸引子是吸引所有有界集的最小集；它包含所有前进时间限制点。我们的方法使用（广义）Lyapunov 函数来寻找必须包含全局吸引子的吸引集，并且 Lyapunov 函数的选择基于其极值要限制的数量进行优化。我们还提出了一个非全局框架，用于在状态空间的指定区域中吸引的最小集合上界定极值。如果动力学由常微分方程控制，并且感兴趣的方程和数量是多项式的，那么我们的方法可以使用多项式优化在计算上实现。特别是，我们通过要求某些多项式表达式可以表示为平方和来强制它们的非负性，从而导致凸优化问题可以重铸为半定程序并通过计算解决。这种计算机辅助可以构造复杂的多项式李雅普诺夫函数。使用三个示例来说明计算。第一个是混沌洛伦兹系统，我们在全局吸引子上绑定坐标的各种单项式的极值。在第二个示例中，我们在流体动力学的九模式截断中绑定了极值，该截断显示了长期存在的混沌瞬态。第三个例子有两个局部稳定的极限环，每个都有自己的吸引力盆。我们应用我们的非全局框架来构建一个盆地的界限，而这些界限不适用于另一个盆地。对于每个示例，我们计算多项式次数至少为 8 的 Lyapunov 函数。在我们可以判断边界锐度的情况下，当多项式次数至少为四或六时，它们至少锐利到三位数。