[Paper Review] Bounding extreme values on attractors using sum-of-squares optimization, with application to the Lorenz attractor

David Goluskin|arXiv (Cornell University)|Jul 25, 2018

Quantum chaos and dynamical systems4 citations

TL;DR

This paper presents a sum-of-squares optimization framework to bound extreme values of dynamical quantities on global attractors of polynomial differential systems, such as the Lorenz attractor. By constructing optimized polynomial Lyapunov functions that define absorbing sets, the method computes sharp bounds—accurate to three or more significant figures—on moments of state variables, significantly improving prior results.

ABSTRACT

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.

Motivation & Objective

• To develop a computational method for bounding extreme values of quantities on global attractors of differential systems.
• To improve upon existing bounds for chaotic systems like the Lorenz attractor by leveraging polynomial Lyapunov functions.
• To optimize the choice of Lyapunov function specifically for the quantity of interest, enhancing bound sharpness.
• To localize the global attractor precisely by constructing tight absorbing sets via convex optimization.
• To enable the use of high-degree polynomial Lyapunov functions (up to degree 8) through semidefinite programming.

Proposed method

• The method uses Lyapunov functions to define absorbing sets that contain the global attractor, ensuring all trajectories eventually enter and remain within these sets.
• The choice of Lyapunov function is optimized to minimize upper or lower bounds on specific quantities of interest, such as moments of (x, y, z).
• Nonnegativity of polynomial expressions in the optimization constraints is enforced via sum-of-squares representations, transforming the problem into a convex semidefinite program.
• The resulting convex optimization problem is solved computationally, enabling the construction of complex, high-degree polynomial Lyapunov functions.
• The method applies to systems with polynomial vector fields and polynomial quantities of interest, allowing rigorous bounding of extreme values.
• The approach localizes the attractor by constructing tight absorbing sets that constrain the region where trajectories can reside.

Experimental results

Research questions

• RQ1 Can sum-of-squares optimization be used to compute sharp bounds on extreme values of dynamical quantities on global attractors?
• RQ2 How does optimizing the Lyapunov function for a specific quantity improve bound sharpness compared to generic constructions?
• RQ3 To what extent can high-degree polynomial Lyapunov functions enhance localization of the attractor in chaotic systems?
• RQ4 How do the computed bounds compare quantitatively to previously known results for the Lorenz system?
• RQ5 Can the method precisely localize the global attractor by constructing tight absorbing sets?

Key findings

• The method computes bounds on moments of the Lorenz system's coordinates that are sharp to three or more significant figures.
• For all tested moments, the bounds are significantly sharper than those reported in prior studies.
• Absorbing sets constructed via the method localize the global attractor with high precision, providing tight geometric constraints.
• Polynomial Lyapunov functions of degree up to 8 were successfully constructed and used, demonstrating the method's scalability to complex functions.
• The use of sum-of-squares representations enabled the solution of a non-convex problem class via convex semidefinite programming, ensuring computational tractability.
• The approach achieves rigorous bounds through computer-assisted optimization, with all results verified via convex programming.