TCC Transitions over a saddle
This TCC module will run in Autumn 2025.
Transition over a saddle is a common scenario in a wide variety of dynamical contexts, from chemical reactions to ship capsize.
Mathematically, there are two main strands: deterministic dynamics and stochastic dynamics. We will study both and achieve a synthesis.
On the deterministic side, the historical starting point is reversible 2DoF autonomous Hamiltonian systems with an index-1 saddle, but the picture is easily extended to non-reversible and N-DoF Hamiltonian systems. If the accessible regions on both sides of the saddle are bounded, one can have integrable behaviour or near-perfect chaos. Furthermore, it is relatively easy to extend the analysis to non-Hamiltonian (think crazy golf) and non-autonomous dynamical systems (think ship capsize) and with neutral directions (ship capsize again).
On the stochastic side, the main focus is on forcing by Gaussian white noise, for which one can deduce large deviation results in the limit of small noise, giving the asymptotic distribution of the most likely paths over the saddle. A simple but important extension we will make is to filtered noise.
A synthesis is obtained by considering the non-autonomous dynamics under samples of a stochastic forcing.
I’ll possibly also address the hybrid quantum-classical case, connecting with Aubry’s theory of electron-transfer reactions.
Prerequisites: Some basic dynamical systems theory, including Hamiltonian systems. Elements of stochastic dynamics in continuous-time and associated theory of optimal control.
References:
RS MacKay, Flux over a saddle, Phys Lett A 145 (1990) 425–7
D Pinheiro, RS MacKay, Interaction of two charges in a uniform magnetic field: II spatial case, J Nonlin Sci 18 (2008) 615–666.
RS MacKay, DC Strub, Bifurcations of transition states: Morse bifurcations, Nonlinearity 27 (2014) 859–95
RS MacKay, DC Strub, Morse bifurcations of transition states in bimolecular reactions, Non- linearity 28 (2015) 4303–29
LM Bujorianu, RS MacKay, T Grafke, S Naik, E Boulougouris, A new stochastic framework for ship capsizing (2021), STAB&S21 conf, online proceedings; arxiv:2105.05965
JW Burby, RS MacKay, S Naik, Isodrastic magnetic fields for suppressing transitions in guiding-centre motion, Nonlinearity 36 (2023) 5884–5954
McSweeney-Davis A, MacKay RS, Naik S, Escape from a potential well under forcing and dissipation with applications to ship capsize; arxiv:2311.00065
Burri M, MacKay RS, Motion in Aubry’s galaxy, arxiv
Shao JY, Grafke T, MacKay RS, Escape over a saddle by coloured noise: theory and numerics, arxiv:2509.03538