Computation of attractor dimension and maximal sums of Lyapunov exponents using polynomial optimization

TL;DR

The paper tackles the challenge of obtaining global upper bounds on sums of Lyapunov exponents and the Lyapunov dimension for autonomous ODEs, independent of any particular trajectory. It develops two computational schemes—sphere projection (time-averaged bounds on projected tangent dynamics) and shifted spectrum (stability-based bounds via spectral shifting)—and renders them tractable via SOS relaxations for polynomial right-hand sides. The main contributions show that these bounds are sharp as polynomial degree grows and that they can closely match exact values in nontrivial systems, demonstrated on the Duffing oscillator and a six-dimensional Hamiltonian model, where periodic orbits saturate the bounds. The work provides a systematic framework to derive global dynamical bounds, with potential for rigorous computer-assisted proofs and extensions to broader dynamical settings. These results connect global attractor geometry, via Lyapunov dimension, to computable polynomial optimizations, enabling practical estimation of fractal dimensions of attractors from first principles.

Abstract

Two approaches are presented for computing upper bounds on Lyapunov exponents and their sums, and on Lyapunov dimension, among all trajectories of a dynamical system governed by ordinary differential equations. The first approach expresses a sum of Lyapunov exponents as a time average in an augmented dynamical system and then applies methods for bounding time averages. This generalizes the work of Oeri & Goluskin (Nonlinearity 36:5378-5400, 2023), who bounded the single leading Lyapunov exponent. The second approach considers a different augmented dynamical system, where bounds on sums of Lyapunov exponents are implied by stability of certain sets, and such stability is verified using Lyapunov function methods. Both of our approaches also can be adapted to directly compute bounds on Lyapunov dimension, which in turn implies bounds on the fractal dimension of a global attractor. For systems of ordinary differential equations with polynomial right-hand sides, all of our bounding formulations lead to polynomial optimization problems with sum-of-squares constraints. These sum-of-squares problems can be solved computationally for any particular system to yield numerical bounds, provided the number of variables and degree of polynomials is not prohibitive. Most of our upper bounds are proven to be sharp under relatively weak assumptions. In the case of the polynomial optimization problems, sharpness means that upper bounds converge to the exact values as polynomial degrees are raised. Computational examples demonstrate upper bounds that are sharp to several digits, including for a six-dimensional dynamical system where sums of Lyapunov exponents are maximized on periodic orbits.

Figures (2)

• Figure 1: A chaotic trajectory (grey) in the system \ref{['eq:duffingsys']}, and the unstable periodic orbit of period $2\pi$ discussed in the text (blue).
• Figure 2: Plots of the position variables $(x_1,x_2,x_3)$ for the Hamiltonian system \ref{['eq:triaxial']} at the energy level $H=1$. Left: The family of periodic orbits $S_1$. Right: The family of periodic orbits $S_2$. Trajectories are confined to stay within the region $\frac{1}{2}\left(x_1^2+x_2^2+x_3^2\right)+\left(x_1^2x_2^2+x_2^2x_3^2+x_3^2x_1^2\right)\leq 1$, which is bounded by the shaded surface. On $S_1$, the LE sums are $M_1=M_2=M_3=M_4=M_5=0.42913$ and on $S_2$, $M_1=M_5=0.25919$ and $M_2=M_3=M_4=0.51837$.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Remark 2.5
• Remark 2.5
• Proposition 2.6
• Proposition 2.7
• proof