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

Author(s): Park, Youngyong; Do, Younghae; Altmeyer, Sebastian A; Lai, Yingcheng; Lee, Gyuwon
Article Title: Early effect in time-dependent, high-dimensional nonlinear dynamical systems with multiple resonances
Affiliation IST Austria
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.
Keywords: Kinetics; vorticity; Bifurcation (mathematics); Flow of fluids; Navier Stokes equations; Kinetic energy; Resonance; Nonlinear dynamical systems; Adiabatic parameters; Driving frequencies; Heuristic analysis; Instantaneous frequency; Multiple resonances; Numerical characterization; Physical quantities; Time independent system
Journal Title: Physical Review E Statistical Nonlinear and Soft Matter Physics
Volume: 91
Issue 2
ISSN: 1539-3755
Publisher: American Institute of Physics  
Date Published: 2015-02-09
Start Page: Article number: 022906
DOI: 10.1103/PhysRevE.91.022906
Notes: 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).
Open access: no