TY - JOUR

T1 - Slim fractals

T2 - The geometry of doubly transient chaos

AU - Chen, Xiaowen

AU - Nishikawa, Takashi

AU - Motter, Adilson E.

PY - 2017/6/8

Y1 - 2017/6/8

N2 - Traditional studies of chaos in conservative and driven dissipative systems have established a correspondence between sensitive dependence on initial conditions and fractal basin boundaries, but much less is known about the relation between geometry and dynamics in undriven dissipative systems. These systems can exhibit a prevalent form of complex dynamics, dubbed doubly transient chaos because not only typical trajectories but also the (otherwise invariant) chaotic saddles are transient. This property, along with a manifest lack of scale invariance, has hindered the study of the geometric properties of basin boundaries in these systems-most remarkably, the very question of whether they are fractal across all scales has yet to be answered. Here, we derive a general dynamical condition that answers this question, which we use to demonstrate that the basin boundaries can indeed form a true fractal; in fact, they do so generically in a broad class of transiently chaotic undriven dissipative systems. Using physical examples, we demonstrate that the boundaries typically form a slim fractal, which we define as a set whose dimension at a given resolution decreases when the resolution is increased. To properly characterize such sets, we introduce the notion of equivalent dimension for quantifying their relation with sensitive dependence on initial conditions at all scales. We show that slim fractal boundaries can exhibit complex geometry even when they do not form a true fractal and fractal scaling is observed only above a certain length scale at each boundary point. Thus, our results reveal slim fractals as a geometrical hallmark of transient chaos in undriven dissipative systems.

AB - Traditional studies of chaos in conservative and driven dissipative systems have established a correspondence between sensitive dependence on initial conditions and fractal basin boundaries, but much less is known about the relation between geometry and dynamics in undriven dissipative systems. These systems can exhibit a prevalent form of complex dynamics, dubbed doubly transient chaos because not only typical trajectories but also the (otherwise invariant) chaotic saddles are transient. This property, along with a manifest lack of scale invariance, has hindered the study of the geometric properties of basin boundaries in these systems-most remarkably, the very question of whether they are fractal across all scales has yet to be answered. Here, we derive a general dynamical condition that answers this question, which we use to demonstrate that the basin boundaries can indeed form a true fractal; in fact, they do so generically in a broad class of transiently chaotic undriven dissipative systems. Using physical examples, we demonstrate that the boundaries typically form a slim fractal, which we define as a set whose dimension at a given resolution decreases when the resolution is increased. To properly characterize such sets, we introduce the notion of equivalent dimension for quantifying their relation with sensitive dependence on initial conditions at all scales. We show that slim fractal boundaries can exhibit complex geometry even when they do not form a true fractal and fractal scaling is observed only above a certain length scale at each boundary point. Thus, our results reveal slim fractals as a geometrical hallmark of transient chaos in undriven dissipative systems.

KW - Nonlinear dynamics

UR - http://www.scopus.com/inward/record.url?scp=85021312731&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021312731&partnerID=8YFLogxK

U2 - 10.1103/PhysRevX.7.021040

DO - 10.1103/PhysRevX.7.021040

M3 - Review article

AN - SCOPUS:85021312731

VL - 7

JO - Physical Review X

JF - Physical Review X

SN - 2160-3308

IS - 2

M1 - 021040

ER -