Global Attractor for a Low Order ODE Model Problem for Transition to Turbulence


Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy-conserving, non-quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples.


Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research


This work was supported in part by the National Science Foundation [grant DMS-1217122].

Keywords and Phrases

Energy conservation; Incompressible flow; Ordinary differential equations; Turbulence; Complex fluid flow; Energy estimates; Energy-conserving; Global attractor; Numerical results; Quadratic nonlinearities; Transition to turbulence; Viscous incompressible fluid flows; Flow of fluids; Energy-conserving nonlinearity

International Standard Serial Number (ISSN)

0170-4214; 1099-1476

Document Type

Article - Journal

Document Version


File Type





© 2017 John Wiley & Sons, All rights reserved.

Publication Date

01 May 2017