  The dynamics of molecular-scale enzymes and molecular motors are activated by
thermal noise, and driven out-of-equilibrium by local energy dissipation.
Because the energies dissipated in these systems are comparable to the thermal
energy, one would generally expect their dynamics to be highly stochastic.
Here, by studying a thermodynamically-consistent model of two coupled
noise-activated oscillators, we show that this is not always the case. Thanks
to a novel phenomenon that we term topological phase locking (TPL), the coupled
dynamics become quasi-deterministic, resulting in a greatly enhanced average
speed of the oscillators. TPL is characterized by the emergence of a band of
periodic orbits that form a torus knot in phase space, along which the two
oscillators advance in rational multiples of each other. The effectively
conservative dynamics along this band coexists with the basin of attraction of
the dissipative fixed point. We further show that TPL arises as a result of a
complex, infinite hierarchy of global bifurcations. Our results have
implications for understanding the dynamics of a wide range of systems, from
biological enzymes and molecular motors to engineered nanoscale electronic,
optical, or mechanical oscillators.
