HYDRODYNAMIC INSTABILITIES REPORT

DELAUNAY Arnaud (mfn05)

Subject : The quadratic application

Summary

The quadratic application has played a fundamental role for its extreme wealth in the study of threads which are going to the determinist chaos. It is the prototyp of mechanism which have an infinity of "under harmonic" bifurcations.

For 1<r<3 the quadratic application has one fix point X* = 1-1/r.

When r reaches a first value of r1 = 3 a first bifurcation appears. Xn oscillates between two values X1 and X2. X* stays as a solution but is now unstable.. The freqencies f0 and f0/2 are now present in the spectrum. This phenomenon is called the "under harmonic stream".

Then a new bifurcation is present at the value r3 :

nd Xn tend towards a cycle with 4 points ( and the frequencies f0,f0/2, and f0/4 appear).

After k bifurcations, we obtain a cycle of points and the apparition of harmonics of frequencies . The set X(k) which form a cycle of points is called an attractor, because in this case, for every initial condition, the serie Xn converge to this set.

The bifurcations follow to a value

The limit set is not periodic, it is called a strange attractor.

For values of r that are superior of this value, we have choas :

The frequencies disappear and a noise appears. But the chaos for these values of r is not present everywhere: We have windows of three periodicity for the value of r as :

And it is followed by the chaos :

Here is the complete attractor of the quadratic application :

Note that A is the value of r.

Application : Rayleigh Benard Instability

Here we can see the "under harmonic stream" for the Rayleigh Benard convection. We can see that if we increase the ratio Ra/ Rac , where Rac is the critical Rayleigh number and Ra the Rayleigh Number, bifurcations arrive which correspond to the frequencies f0/2 , f0/4 ,f0/8. Here the segments show the length of a fluctuation period of the temperatures in one point of the fluid.