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Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval$$[0,1]$$in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure$$\nu $$. We consider a class of non-square-integrable observables$$\phi $$, mostly of form$$\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$$, where$$x_0$$is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index$$\alpha \in (0,2)$$. The two types of maps we concatenate are a class of piecewise$$C^2$$expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and$$\alpha $$, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by$$\nu $$. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.
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Hydrodynamics and scaling laws for intermittent S-start swimming
The hydrodynamics of a self-propelling swimmer undergoing intermittent S-start swimming are investigated extensively with varying duty cycle$$DC$$, swimming period$$T$$, and tailbeat amplitude$$A$$. We find that the steady time-averaged swimming speed$$\bar {U}_x$$increases directly with$$A$$, but varies inversely with$$DC$$and$$T$$, where there is a maximal improvement of$$541.29\,\%$$over continuous cruising swimming. Our results reveal two scaling laws, in the form of input versus output relations, that relate the swimmer's kinematics to its hydrodynamic performance: swimming speed and efficiency. A smaller$$DC$$causes increased fluctuations in the swimmer's velocity generation. A larger$$A$$, on the other hand, allows the swimmer to reach steady swimming more quickly. Although we set out to determine scaling laws for intermittent S-start swimming, these scaling laws extend naturally to burst-and-coast and continuous modes of swimming. Additionally, we have identified, categorized and linked the wake structures produced by intermittent S-start swimmers with their velocity generation.
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- Award ID(s):
- 1931368
- PAR ID:
- 10573856
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 984
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2024.103
