Early effect in time-dependent, high-dimensional nonlinear dynamical systems with multiple resonances

Youngyong Park, Younghae Do, Sebastien Altmeyer, Ying-Cheng Lai, GyuWon Lee

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Abstract

We investigate high-dimensional nonlinear dynamical systems exhibiting multiple resonances under adiabatic parameter variations. Our motivations come from experimental considerations where time-dependent sweeping of parameters is a practical approach to probing and characterizing the bifurcations of the system. The question is whether bifurcations so detected are faithful representations of the bifurcations intrinsic to the original stationary system. Utilizing a harmonically forced, closed fluid flow system that possesses multiple resonances and solving the Navier-Stokes equation under proper boundary conditions, we uncover the phenomenon of the early effect. Specifically, as a control parameter, e.g., the driving frequency, is adiabatically increased from an initial value, resonances emerge at frequency values that are lower than those in the corresponding stationary system. The phenomenon is established by numerical characterization of physical quantities through the resonances, which include the kinetic energy and the vorticity field, and a heuristic analysis based on the concept of instantaneous frequency. A simple formula is obtained which relates the resonance points in the time-dependent and time-independent systems. Our findings suggest that, in general, any true bifurcation of a nonlinear dynamical system can be unequivocally uncovered through adiabatic parameter sweeping, in spite of a shift in the bifurcation point, which is of value to experimental studies of nonlinear dynamical systems.
Original languageEnglish
Article number022906
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume91
DOIs
Publication statusPublished - 9 Feb 2015

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Nonlinear Dynamical Systems
dynamical systems
High-dimensional
Bifurcation
Sweeping
Instantaneous Frequency
Bifurcation Point
Faithful
Kinetic energy
Vorticity
Control Parameter
Navier-Stokes equation
vorticity
fluid flow
Fluid Flow
Experimental Study
Navier-Stokes Equations
kinetic energy
Heuristics
boundary conditions

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Early effect in time-dependent, high-dimensional nonlinear dynamical systems with multiple resonances. / Park, Youngyong; Do, Younghae; Altmeyer, Sebastien; Lai, Ying-Cheng; Lee, GyuWon.

In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, Vol. 91, 022906, 09.02.2015.

Research output: Contribution to journalArticle

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abstract = "We investigate high-dimensional nonlinear dynamical systems exhibiting multiple resonances under adiabatic parameter variations. Our motivations come from experimental considerations where time-dependent sweeping of parameters is a practical approach to probing and characterizing the bifurcations of the system. The question is whether bifurcations so detected are faithful representations of the bifurcations intrinsic to the original stationary system. Utilizing a harmonically forced, closed fluid flow system that possesses multiple resonances and solving the Navier-Stokes equation under proper boundary conditions, we uncover the phenomenon of the early effect. Specifically, as a control parameter, e.g., the driving frequency, is adiabatically increased from an initial value, resonances emerge at frequency values that are lower than those in the corresponding stationary system. The phenomenon is established by numerical characterization of physical quantities through the resonances, which include the kinetic energy and the vorticity field, and a heuristic analysis based on the concept of instantaneous frequency. A simple formula is obtained which relates the resonance points in the time-dependent and time-independent systems. Our findings suggest that, in general, any true bifurcation of a nonlinear dynamical system can be unequivocally uncovered through adiabatic parameter sweeping, in spite of a shift in the bifurcation point, which is of value to experimental studies of nonlinear dynamical systems.",
author = "Youngyong Park and Younghae Do and Sebastien Altmeyer and Ying-Cheng Lai and GyuWon Lee",
note = "ACKNOWLEDGMENT This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. NRF-2013R1A1A2010067). Y.-C.L. was supported by AFOSR under Grant No. FA9550-12- 1-0095. G.W.L. was supported by the Korea Institute of Construction Technology (project name: Development of a Micro Raingauge Using Electromagnetic Wave).",
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AU - Lee, GyuWon

N1 - ACKNOWLEDGMENT This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. NRF-2013R1A1A2010067). Y.-C.L. was supported by AFOSR under Grant No. FA9550-12- 1-0095. G.W.L. was supported by the Korea Institute of Construction Technology (project name: Development of a Micro Raingauge Using Electromagnetic Wave).

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N2 - We investigate high-dimensional nonlinear dynamical systems exhibiting multiple resonances under adiabatic parameter variations. Our motivations come from experimental considerations where time-dependent sweeping of parameters is a practical approach to probing and characterizing the bifurcations of the system. The question is whether bifurcations so detected are faithful representations of the bifurcations intrinsic to the original stationary system. Utilizing a harmonically forced, closed fluid flow system that possesses multiple resonances and solving the Navier-Stokes equation under proper boundary conditions, we uncover the phenomenon of the early effect. Specifically, as a control parameter, e.g., the driving frequency, is adiabatically increased from an initial value, resonances emerge at frequency values that are lower than those in the corresponding stationary system. The phenomenon is established by numerical characterization of physical quantities through the resonances, which include the kinetic energy and the vorticity field, and a heuristic analysis based on the concept of instantaneous frequency. A simple formula is obtained which relates the resonance points in the time-dependent and time-independent systems. Our findings suggest that, in general, any true bifurcation of a nonlinear dynamical system can be unequivocally uncovered through adiabatic parameter sweeping, in spite of a shift in the bifurcation point, which is of value to experimental studies of nonlinear dynamical systems.

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