Large-shear chaos conjecture for stochastically perturbed Hopf systems

Consider the random dynamical system induced by the stochastic differential equation~, and fix $a>0$ and $\beta\in\mathbb{R}$. Let $\Lambda_1$ denote its largest Lyapunov exponent. Large-shear chaos conjecture. There exists a function

$$C:\mathbb{R}\times\mathbb{R}^+\to\mathbb{R}^+$$

such that, whenever

$$b\geq C(\alpha,\sigma),$$

the largest Lyapunov exponent $\Lambda_1$ is positive. Numerical evidence suggests that sufficiently large shear produces chaotic behavior, but an analytical proof has generally remained out of reach except for a strongly simplified model; the conjecture is therefore unresolved in the stated generality.