Onset of Chaos in Fluid Dynamics

Abstract

Forced nonlinear systems exhibit deterministic behaviour when the applied stress is small enough. Let us consider a driven anharmonic oscillation with an x3 restoring force. If the driving force is small, the motion is almost sinusoidal. The spectrum is composed of a sharp peak at the driving frequency and, eventually, smaller peaks at the higher harmonics. As the force is increased, the motion becomes more complicated and additional sharp peaks are introduced in the spectrum. One of course might wonder whether by making the driving force arbitrarily large, the spectrum will continue to be in the form of instrumentally narrow peaks. The answer is no, and this holds true for the simple case we are considering as well as for a large variety of more complex nonlinear systems. Above a given threshold the spectrum will show a continuum in between the remnants of the sharp peaks and one says that the system has reached a “chaotic” state.