title: Early effect in time-dependent, high-dimensional nonlinear dynamical systems with multiple resonances.
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.
authors.affiliation: Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 702-701, South Korea.
authors.forename: Youngyong
authors.initials: Y
authors.lastname: Park
authors.affiliation: Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 702-701, South Korea.
authors.forename: Younghae
authors.initials: Y
authors.lastname: Do
authors.affiliation: Institute of Science and Technology Austria (IST Austria), 3400 Klosterneuburg, Austria.
authors.forename: Sebastian
authors.initials: S
authors.lastname: Altmeyer
authors.affiliation: School of Electrical, Computer, and Energy Engineering, Department of Physics, Arizona State University, Tempe, Arizona 85287, USA.
authors.forename: Ying-Cheng
authors.initials: YC
authors.lastname: Lai
authors.affiliation: Department of Astronomy and Atmospheric Sciences, Center for Atmospheric Remote Sensing (CARE), Kyungpook National University, Daegu 702-701, South Korea.
authors.forename: GyuWon
authors.initials: G
authors.lastname: Lee