Homoclinic Bifurcations for Partial Differential Equations in Unbounded Domains

Abstract

The connection between low‐dimensional chaos in ordinary differential equations, and turbulence in fluids and other systems governed by partial differential equations, is one that is in many circumstances not clear. We discuss some examples of turbulent fluid flow, and consider ways in which they may be related to much simpler sets of ordinary differential equations, whose behavior can be reasonably well understood. (We are not advocating drastic Fourier truncation.) The generation of aperiodic solutions through the occurrence of homoclinic orbits is briefly analysed for ordinary differential equations, and the same kind of heuristic analysis is sketched for partial differential equations (in one space dimension). It is suggested that such an analysis can explain certain features of chaos, which have been observed in real fluids.