Non-Markovian processes on heteroclinic networks

Sets of saddle equilibria connected by trajectories are known as heteroclinic networks. Trajectories near a heteroclinic network typically spend a long period of time near one of the saddles before rapidly transitioning to the neighborhood of a different saddle. The sequence of saddles visited by a trajectory can be considered a stochastic sequence of states. In the presence of small-amplitude noise, this sequence may be either Markovian or non-Markovian, depending on the appearance of a phenomenon called lift-off at one or more saddles of the network. In this paper, we investigate how lift-off occurring at one saddle affects the dynamics near the next saddle visited, how we might determine the order of the associated Markov chain of states, and how we might calculate the transition probabilities of that Markov chain. We first review methods developed by Bakhtin to determine the map describing the dynamics near a linear saddle in the presence of noise and extend the results to include three different initial probability distributions. Using Bakhtin's map, we determine conditions under which the effect of lift-off persists as the trajectory moves past a subsequent saddle. We then propose a method for finding a lower bound for the order of this Markov chain. Many of the theoretical results in this paper are only valid in the limit of small noise, and we numerically investigate how close simulated results get to the theoretical predictions over a range of noise amplitudes and parameter values.