Scaling properties of saddle-node bifurcations on fractal basin boundaries

Abstract

We analyze situations where a saddle-node bifurcation occurs on a fractal basin boundary. Specifically, we are interested in what happens when a system parameter is slowly swept in time through the bifurcation. Such situations are known to be indeterminate in the sense that it is difficult to predict the eventual fate of an orbit that tracks the pre-bifurcation node attractor as the system parameter is swept through the bifurcation. In this paper we investigate the scaling of (1) the fractal basin boundary of the static (i.e., unswept) system near the saddle-node bifurcation, (2) the dependence of the orbit's final destination on the sweeping rate, (3) the dependence of the time it takes for an attractor to capture a swept orbit on the sweeping rate, and (4) the dependence of the final attractor capture probability on the noise level. With respect to noise, our main result is that the effect of noise scales with the 5/6 power of the parameter drift rate. Our approach is to first investigate all these issues using one-dimensional map models. The simplification of treatment inherent in one dimension greatly facilitates analysis and numerical experiment, aiding us in obtaining the new results listed above. Following our one-dimensional investigations, we explain that these results can be applied to two-dimensional systems. We show, through numerical experiments on a periodically forced second order differential equation example, that the scalings we have found also apply to systems that result in two dimensional maps.

The Nature of Attractor Basins in Multistable Systems

“…3. The jump in the slope of the uncertainty exponent is related to the fact that the new attractor is born in the subcritical pitchfork bifurcation [Breban et al, 2003] with a fractal basin of attraction. The uncertainty exponent shows values close to zero when one of the nonsynchronous nonchaotic attractor becomes chaotic around α ∼ 3.505.…”

Controlling systems that drift through a tipping point

“…This probability was shown in Ref. 13 to be a function of the scaled variable A/δµ 5/6 for a fixed x ′ nc . We see that in order for p(x ′ nc , n c ) > p th , one needs to have x ′ nc "deep enough" in a basin interval, which is impossible if the interval is too narrow compared to the noise amplitude.…”

“…The structure of basin boundaries slowly varies with time because of the drifting parameter. If the boundary is fractal, then the bifurcation can be indeterminate, in the sense that the fate of the system after the bifurcation (the final attractor) can depend on small noise or be extremely sensitive to the specific rate of parameter variation [8][9][10][11][12][13][14] . Thus, prediction of the final attractor can be difficult.…”

Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

“…A number of researchers have explored the adverse coupling of non-stationary and non-deterministic influences on the activation of the saddle-node bifurcations by introducing scaling laws [50][51][52] and by analytically and numerically integrating the associated Fokker-Plank equation of the stochastic normal form [53]. Recently, the distribution of the escape events induced by saddle-node bifurcations has been approximately derived and numerically validated by Miller and Shaw [54].…”