Lyapunov exponents play a central role in dynamical systems theory, and offer a measure of the local instability that lies behind deterministic chaos and the sensitive dependence of trajectories on initial conditions. Most dynamical systems theory applies to systems operating in the absence of noise, but mathematical models often arise for dynamical flows which are also strongly influenced by noise. These can also be referred to as “random dynamical systems.” Unfortunately, one consequence of unbounded noise is a joining together of all the attractors and other invariant sets of the deterministic flow, which implies that the generalisation of the Lyapunov exponent in the presence of noise has only global, not local, properties. This has hampered efforts to extend the bifurcation theory for deterministic dynamical systems to the setting of stochastic flows, as the key local building blocks of bifurcation theory cease to exist.
In a new paper, LML Fellow Jeroen Lamb and colleagues report a strategy to generalise Lyapunov exponents in a different way. They study so-called killed Markov processes – those for which the phase space contains certain traps from which the process can never escape (i.e. once entering a trap, the process is killed). In this context, their analysis focuses on the theory of so-called conditioned processes, first pioneered in 1947, and receiving renewed attention in recent years. The idea of this approach is to restrict attention within the full flow only to that subset of trajectories which never get killed. As Lamb and colleagues show, due to the loss of mass by absorption – the loss of killed trajectories – there does not exist any stationary distribution for the flow. But the concept of stationarity is replaced with a still useful quasi-stationarity, defined in a particular sense. Using this perspective, they derive an expression for a conditioned Lyapunov exponent, which measures the asymptotic instability of typical surviving trajectories. In effect, these exponents give a measure of “chaos” of the “non-escaping” dynamics. The authors hope this new definition may provide a useful pathway in the further pursuit of a bifurcation theory for random dynamical systems.
A pre-print of the paper is available at https://arxiv.org/abs/1805.07177
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