Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

Abstract

We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions. Nonnegativity is enforced by requiring the polynomials to be sums of squares, a condition which is then formulated as a semidefinite program (SDP) that can be solved computationally. Although such computations are subject to numerical error, we demonstrate two ways to obtain rigorous results: using interval arithmetic to control the error of an approximate SDP solution, and finding exact analytical solutions to relatively small SDPs. Previous formulations are extended to allow for bounds depending analytically on parametric variables. These methods are illustrated using the Lorenz equations, a system with three state variables (x, y, z) and three parameters $(\beta ,\sigma ,r)$. Bounds are reported for infinite-time averages of all eighteen moments $x^ly^mz^n$ up to quartic degree that are symmetric under $(x,y)\mapsto (-x,-y)$. These bounds apply to all solutions regardless of stability, including chaotic trajectories, periodic orbits, and equilibrium points. The analytical approach yields two novel bounds that are sharp: the mean of $z^3$ can be no larger than its value of $(r-1)^3$ at the nonzero equilibria, and the mean of $xy^3$ must be nonnegative. The interval arithmetic approach is applied at the standard chaotic parameters to bound eleven average moments that all appear to be maximized on the shortest periodic orbit. Our best upper bound on each such average exceeds its value on the maximizing orbit by less than 1%. Many bounds reported here are much tighter than would be possible without computer assistance.

Appendices

Appendix A: Details of Verified Computations Using VSDP

When computing the bounds reported in Tables 2 and 3 with the software VSDP, we most often rescaled the Lorenz system by $\mathbf {x}\mapsto 20\mathbf {x}$ and then converted bounds back to the original scaling. Various researchers using SOS methods to study dynamics (although not to bound time averages) have found that rescaling the governing equations improves the convergence of SDP solvers. A heuristic that is often used (e.g., Henrion and Korda 2014) is to rescale each coordinate of $\mathbf {x}\in \mathbb R^n$ so that the relevant dynamics occur approximately in the cube $[-1,1]^n$. Our rescaling is similar, putting most of the Lorenz attractor in the domain $[-1,1]^2\times [0,2]$. We find that rescaling by 10 or 40 instead of 20 usually gives more conservative upper enclosures $U^+$. Results become more sensitive to the rescaling when the auxiliary function V reaches degree 8 or 10, at which point slightly different scalings can significantly affect the conservativeness of the enclosures. The degree-10 bounds in Table 3 were produced after rescaling by either 25 (for $x^4$, $x^2y^2$, $xy^3$) or 30, and the degree-8 lower enclosure in Table 2 was produced after rescaling by 10. The time required to compute each tabulated bound on a single processor ranged from seconds to minutes.

Results reported here for V of degree 6, 8, or 10 are not quite rigorous to the standard of a computer-assisted proof. This is because we have used the software YALMIP (Löfberg 2004, 2009) to automatically parse the SOS conditions, formulate corresponding SDPs, and pass them to the VSDP software. This incurs roundoff errors that are not accounted for since YALMIP does not use interval arithmetic. A parser that uses interval arithmetic is under development. Until it is available, rigorous results can be found by formulating the relevant SDPs manually. We have done this only for V of degree 4, in which cases roundoff errors introduced by YALMIP have all been orders of magnitude smaller than the margins of the enclosures generated by VSDP.

Appendix B: Relations Between Mean Moments

On any trajectory of the Lorenz system, all mean moments up to degree 2, 3, or 4 that are symmetric under $(x,y)\mapsto (-x,-y)$ can be expressed as linear combinations of $\big \{ \overline{z}, \overline{z^2} \big \}$, $\big \{ \overline{z}, \overline{z^2}, \overline{y^2z}, \overline{z^3} \big \}$, or $\big \{ \overline{z}, \overline{z^2}, \overline{y^2z}, \overline{z^3}, \overline{y^2z^2}, \overline{z^4} \big \}$, respectively. Many of these relations are not useful for constructing bounds, although they are derived using the same basic identity $\overline{\mathbf {f}\cdot \nabla V}=0$ that is central to our bounding methods. When proving bounds we have sought V such that $\varphi +\mathbf {f}\cdot \nabla V$ obeys the desired bound at all points in phase space. For the different objective of expressing various moments in terms of a smaller subset of moments, V must be chosen so that $\varphi +\mathbf {f}\cdot \nabla V$ contains only moments from this subset. Table 5 gives relations for symmetric moments of the Lorenz system, as well as the choices of V that yield these relations. About half of these relations have appeared in the literature for decades (Lücke 1976). Every tabulated moment is expressed as a linear combination of moments that either are in the minimal set $\big \{ \overline{z}, \overline{z^2}, \overline{y^2z}, \overline{z^3}, \overline{y^2z^2}, \overline{z^4} \big \}$ or are expressed in this same way higher in the table. Each moment can be expressed using only the minimal set after some further substitution: the $\overline{x^2}$ relation and the one above it give $\overline{x^2}=\beta \overline{z}$, the $\overline{y^2}$ relation and the ones above it give $\overline{y^2}=\beta \big (r\overline{z}-\overline{z^2}\big )$, the $\overline{x^2z}$ relation and the ones above it give $\overline{x^2z}=\beta (1+\sigma )(r-1)\overline{z}-\beta \sigma \overline{z^2}$, and so on.

Table 5 Symmetric mean moments of the Lorenz system up to quartic degree, expressed as linear combinations of moments that either are in the minimal set $\big \{ \overline{z}, \overline{z^2}, \overline{y^2z}, \overline{z^3}, \overline{y^2z^2}, \overline{z^4} \big \}$ or are expressed in this same way higher in the table. Each equality is an instance of the identity $\overline{\varphi }=\overline{\varphi +\mathbf {f}\cdot \nabla V}$ with $\varphi$ and V defined as shown