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We investigate a model for the dynamics of ellipsoidal microswimmers in an externally imposed, laminar Kolmogorov flow. Through a phase-space analysis of the dynamics without noise, we find that swimmers favor either cross-stream or rotational drift, depending on their swimming speed and aspect ratio. When including noise, i.e., rotational diffusion, we find that swimmers are driven into certain parts of phase space, leading to a nonuniform steady-state distribution. This distribution exhibits a transition from swimmer aggregation in low-shear regions of the flow to aggregation in high-shear regions as the swimmer’s speed, aspect ratio, and rotational diffusivity are varied. To explain the nonuniform phase-space distribution of swimmers, we apply a weak-noise averaging principle that produces a reduced description of the stochastic swimmer dynamics. Using this technique, we find that certain swimmer trajectories are more favorable than others in the presence of weak rotational diffusion. By combining this information with the phase-space speed of swimmers along each trajectory, we predict the regions of phase space where swimmers tend to accumulate. The results of the averaging technique are in good agreement with direct calculations of the steady-state distributions of swimmers. In particular, our analysis explains the transition from low-shear to high-shear aggregation.
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Variations in stability revealed by temporal asymmetries in contraction of phase space flow
Abstract Empirical diagnosis of stability has received considerable attention, often focused on variance metrics for early warning signals of abrupt system change or delicate techniques measuring Lyapunov spectra. The theoretical foundation for the popular early warning signal approach has been limited to relatively simple system changes such as bifurcating fixed points where variability is extrinsic to the steady state. We offer a novel measurement of stability that applies in wide ranging systems that contain variability in both internal steady state dynamics and in response to external perturbations. Utilizing connections between stability, dissipation, and phase space flow, we show that stability correlates with temporal asymmetry in a measure of phase space flow contraction. Our method is general as it reveals stability variation independent of assumptions about the nature of system variability or attractor shape. After showing efficacy in a variety of model systems, we apply our technique for measuring stability to monthly returns of the S&P 500 index in the time periods surrounding the global stock market crash of October 1987. Market stability is shown to be higher in the several years preceding and subsequent to the 1987 market crash. We anticipate our technique will have wide applicability in climate, ecological, financial, and social systems where stability is a pressing concern.
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- Award ID(s):
- 1715638
- PAR ID:
- 10217043
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 11
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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