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 (US)
Article number022906
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume91
Issue number2
DOIs
StatePublished - Feb 9 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

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Statistics and Probability

Cite this

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

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 91, No. 2, 022906, 09.02.2015.

Research output: Contribution to journalArticle

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