Publications

Using Covariant Lyapunov Vectors to Quantify High Dimensional Chaos with a Conservation Law

Abstract: We explore the high-dimensional chaos of a one-dimensional lattice of diffusively coupled tent maps using the covariant Lyapunov vectors (CLVs). We investigate the connection between the dynamics of the maps in the physical space and the dynamics of the covariant Lyapunov vectors and covariant Lyapunov exponents that describe the direction and growth (or decay) of small perturbations in the tangent space. We explore the tangent space splitting into physical and transient modes and find that the splitting persists for all of the conditions we explore. In general, the leading CLVs are highly localized in space and the CLVs become less localized with increasing Lyapunov index. We consider the dynamics with a conservation law whose strength is controlled by a parameter that can be continuously varied. Our results indicate that a conservation law delocalizes the spatial variation of the CLVs. We find that when a conservation law is present, the leading CLVs are entangled with fewer of their neighboring CLVs than in the absence of a conservation law.

Citation: J. Barbish, and M. R. Paul, “Using Covariant Lyapunov Vectors to Quantify High Dimensional Chaos with a Conservation Law," Physical Review E, vol. 108, p. 054202 Nov. 2023.

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Multi-Mode Brownian Dynamics of a Nanomechanical Resonator in a Viscous Fluid

Abstract: Brownian motion imposes a hard limit on the overall precision of a nanomechanical measurement. Here, we present a combined experimental and theoretical study of the Brownian dynamics of a quintessential nanomechanical system, a doubly clamped nanomechanical beam resonator, in a viscous fluid. Our theoretical approach is based on the fluctuation-dissipation theorem of statistical mechanics: we determine the dissipation from fluid dynamics; we incorporate this dissipation into the proper elastic equation to obtain the equation of motion; and the fluctuation-dissipation theorem then directly provides an analytical expression for the position-dependent power spectral density (PSD) of the displacement fluctuations of the beam. We compare our theory to experiments on nanomechanical beams immersed in air and water and obtain excellent agreement. Within our experimental parameter range, the Brownian-force noise driving the nanomechanical beam has a colored PSD due to the “memory” of the fluid; the force noise remains mode independent and uncorrelated in space. These conclusions are not only of interest for nanomechanical sensing but also provide insight into the fluctuations of elastic systems at any length scale.

Citation: H. Gress, J. Barbish, C. Yanik, I. I. Kaya, R. T. Erdogan, M. S. Hanay, M. Gonzalez, O. Svitelskiy, M. R. Paul, and K. L. Ekinci, “Multi-mode Brownian Dynamics of a Nanomechanical Resonator in a Viscous Fluid," Physical Review Applied, vol. 20, no. 4, p. 044061, Oct. 2023.

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The Dynamics of an Externally Driven Nanoscale Beam that is Under High Tension and Immersed in a Viscous Fluid

Abstract: We explore the dynamics of a nanoscale doubly clamped beam that is under high tension, immersed in a viscous fluid, and driven externally by a spatially varying drive force. We develop a theoretical description that is valid for all possible values of tension, includes the motion of the higher modes of the beam, and accounts for a harmonic force that is applied over a limited spatial region of the beam near its ends. We compare our theoretical predictions with experimental measurements for a nanoscale beam that is driven electrothermally and immersed in air and water. The theoretical predictions show good agreement with experiments, and the validity of a simplified string approximation is demonstrated.

Citation: J. Barbish, C. Ti, K. L. Ekinci, and M. R. Paul, "The Dynamics of an Externally Driven Nanoscale Beam that is Under High Tension and Immersed in a Viscous Fluid" Journal of Applied Physics, vol. 132, no. 3, p. 034501, Jul. 2022.

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