Dynamical characterization of stochastic bifurcations in a random logistic map

The interaction of noise and non-linearity can cause sharp dynamical transitions, as appear in phenomena such as stochastic resonance or noise-induced synchronization or chaos. The theory of such phenomena is still not well-developed, but stands to advance through studies of random dynamical systems, where recent work has characterised bifurcations as abrupt topological changes in system attractors. In a recent pre-print, LML External Fellows Yuzuru Sato, Jeroen Lamb and colleagues build on these insights to characterise noise-induced transitions in one archetypal system – the logistic map with additive noise. As a function of increasing noise, they find three distinct phases of random dynamics.
For sufficiently weak noise, they observe a random point attractor with fully periodic motion. In this regime, the support of the invariant density for the attractor consists of three mutually disjoint intervals. The random attractor consists of three random points within the respective intervals. Increasing noise brings about a bifurcation into a second dynamical phase, where the support of the invariant measure abruptly explodes to the entire interval [0,1]. In contrast to the well-defined periodic behaviour in phase I, where two orbits having initial conditions in different connected intervals never converge towards one another, in phase II the distance between any two orbits with different initial conditions eventually tends to zero. This phase corresponds to noise-induced synchronization. Finally, a second bifurcation at a higher noise level leads to phase III, with a random strange attractor, where the density of the stationary measure becomes more uniformly spread.
As the authors note, the existence of the lower noise threshold may have implications for the management of real world systems. As noise increases just beyond this critical noise threshold, chaotic bursts in the dynamics become possible. Scrutiny of the finite-time Lyapunov exponents may offer a way to detect early warning signals of such bursty behaviour, and may be useful in safeguarding systems where bursts must be avoided, as in electrical power or traffic systems, or in computer networks.
A pre-print of the paper is available at https://arxiv.org/pdf/1811.03994.pdf