Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions

Abstract

Small noise can induce rare transitions between metastable states, which can be characterized by Maximum Likelihood Paths (MLPs). Nongradient systems contrast gradient systems in that MLP does not have to cross the separatrix at a saddle point, but instead possibly at a point on a hyperbolic periodic orbit. A numerical approach for identifying such unstable periodic orbits is proposed based on String method. In a special class of nongradient systems ('orthogonal-type'), there are provably local MLPs that cross such saddle point or hyperbolic periodic orbit, and the separatrix crossing location determines the associated local maximum of transition rate. In general cases, however, the separatrix crossing may not determine a unique local maximum of the rate, as we numerically observed a counter-example in a sheared 2D-space Allen-Cahn SPDE. It is a reasonable conjecture that there are always local MLPs associated with each attractor on the separatrix, such as saddle point or hyperbolic periodic orbit; our numerical experiments did not disprove so.

Computing the quasipotential for nongradient SDEs in 3D

“…We have applied olim3D to two of Tao's examples [36] where the value of the quasipotential is known analytically at hyperbolic periodic orbits serving as transition states between two attractors. Finally, we have applied olim3D to a genetic switch model (Lv et al [21]).…”

“…We have conducted an extensive numerical study of the proposed solver. Our set of examples includes a genetic switch model [21], systems with hyperbolic periodic orbits [36], and a series of linear and nonlinear SDEs with different ratios of magnitudes of their rotational and potential components. The practical results of this study are: (i) a guideline for choosing the update parameter K, (ii) the conclusion that a local factoring may or may not be helpful, depending on the problem, and (iii) a way to estimate the numerical error using the integration along the MAP where the quasipotential is not given analytically.…”

“…Path evolution can stall when the true MAP spirals near an attractor or a transition state, or if the MAP has a kink. Improved path-based methods (e. g. [36,35]) have been proposed to tackle these issues in some common special cases.…”

“…We have tested olim3D on a collection of examples that includes: • Two of Tao's examples [36] (Examples 6-7) with two point attractors and hyperbolic periodic orbits serving as transition states between them. In these examples, the quasipotential can be found at the transition states using certain orthogonal decompositions (Section 4.3).…”

“…The function v(x) is not necessarily the quasipotential, however, it is a solution of the Hamilton-Jacobi (6) which is easy to check. Let x * be one of the asymptotically stable equilibria ofẋ = b(x), It is shown in [36] that the quasipotential at y with respect to x * coincides with v(y) − v(x * ). This fact can be useful in the case where the quasipotential is hard or impossible to find analytically, but an orthogonal decomposition of the form (58) is easy to spot.…”