I- Few elementary notions

1. Definitions: dynamic system autonomic system, phases space, control space:

A dynamic system is a physical system of which the behaviour is described by an equation as:

                                     dX/dt = f

But the equation often involves other real parameters than X and t. mu for example, of which f is a function.
X is in E, phases space.
mu is in F, control space.
The system is autonomic if f is not a function of time. This is the case we are to focus on for this course.
 

2. Studied systems:

x' = mu + a x²
x' = x + a x3
z' = (mu+i w) + alpha |z|²z

Lorentz model:
            x' = -sigma x + sigma y      sigma > 0
            y' = -x z + r x - y                r > 0
            z' = x y - b z                        b > 0
 

3. Method of study:

Let us consider an autonomic dynamic system, of which the equation is : dX /dt ( t ) = f ( X ).

For each equation to solve, we proceed the same way:

1. Dimension of control space and phase space.

2. Equilibrium points:
                            X is an equilibrium point  f (X) = 0

3. We plot the ensemble of the points that satisfy this property. In the cordinate system, the ordonate is X and the abscise is one of the parameters f (mu for example). We get the curve of equilibrium points.

4. Different paths: We search for possible trajectories for a point depending on its initial position. The point is attracted by the "nearest" curve of equilibrium points.

5. Stability : we test the stability of an equilibrim point by imposing an infinitesimal deplacement around it, which means that we examine the behaviour of X = Xeq + U; Xeq being the comncerned equilibrium vector and U a vector satisying ||U|| << ||Xeq||,    (||.|| is the norm associated to the scalar product of phases space)
We are then lead to observe f (Xeq+U) which will be developped in series of Taylor.

f (Xeq+U) = f (Xeq) + U f'(Xeq) + U2/2! f''(Xeq) + ... m+ Un/n! f(n)(Xeq) + O(Un)
           and
U'= X' = f(X + U)

Generally, we stop the developpement on n = 1. As f(Xeq) = 0, the amplification U versas time will be:

U= A ef'(Xeq)t.      Thus, the discussion will be on f'(Xeq).

4. Classic examples


  Chaos theory is among the youngest of the sciences, and has rocketed from its obscure roots in the seventies to become one of the most fascinating fields in existence. At the forefront of much research on physical systems, and already being implemented in fields covering as diverse matter as arrhythmic pacemakers, image compression, and fluid dynamics, chaos science promises to continue to yield absorbing scientific information which may shape the face of science in the future.

Formally, chaos theory is defined as the study of complex nonlinear dynamic systems. Complex implies just that, nonlinear implies recursion and higher mathematical algorithms, and dynamic implies nonconstant and nonperiodic. Thus chaos theory is, very generally, the study of forever changing complex systems based on mathematical concepts of recursion, whether in the form of a recursive process or a set of differential equations modeling a physical system.

It is true that chaos theory dictates that minor changes can cause huge fluctuations. But one of the central concepts of chaos theory is that while it is impossible to exactly predict the state of a system, it is generally quite possible, even easy, to model the overall behavior of a system. Thus, chaos theory lays emphasis not on the disorder of the system--the inherent unpredictability of a system--but on the order inherent in the system--the universal behavior of similar systems.

Thus, it is incorrect to say that chaos theory is about disorder. To take an example, consider Lorenz's Attractor.
 
 

Lorenz's Attractor


The Lorenz Attractor is based on three differential equations, three constants, and three initial conditions. The attractor represents the behavior of gas at any given time, and its condition at any given time depends upon its condition at a previous time. If the initial conditions are changed by even a tiny amount, say as tiny as the inverse of Avogadro's number (a heinously small number with an order of 1E-24), checking the attractor at a later time will yield numbers totally different. This is because small differences will propagate themselves recursively until numbers are entirely dissimilar to the original system with the original initial.

Historically, Lorenz was looking for a way to model the action of the chaotic behavior of the gaseous system first mentioned above. Lorenz took a few "Navier-Stokes" equations, from the physics field of fluid dynamics. He simplified them and got as a result the following three-dimensional system:

Here delta represents the "Prandtl number." This number is the ratio of the fluid viscosity of a substance to its thermal conductivity (named after Ludwig Prandtl, a German physicist). The value Lorenz used was 10.

The variable r represents the difference in temperature between the top and bottom of the gaseous system. The value usually used in sample Lorenz attractors such as the one displayed here is 28.

The variable b is the width to height ratio of the box which is being used to hold the gas in the gaseous system. Lorenz happened to choose 8/3, which is now the most common number used to draw the attractor.

The resultant x of the equation represents the rate of rotation of the cylinder, y represents the difference in temperature at opposite sides of the cylinder, and the variable z represents the deviation of the system from a linear, vertical graphed line representing temperature.

Plotting the three differential equations requires the usage of a computer. Plotted on a three-dimensional plane, a shape unlike any other forms. Instead of a simple geometric structure or even a complex curve, the structure now known as the Lorenz Attractor weaves in and out of itself. Projected on the X-Z plane, the attractor looks like a butterfly; on the Y-Z plane, it resembles an owl mask. The X-Y projection is useful mainly for glimpsing the three-dimensionality of the attractor; it looks something like two paper plates, on parallel but different planes, connected by a strand of string. As the Lorenz Attractor is plotted, a strand will be drawn from one point, and will start weaving the outline of the right butterfly wing. Then it swirls over to the left wing and draws its center. The attractor will continue weaving back and forth between the two wings, its motion seemingly random, its very action mirroring the chaos which drives the process. conditions.
 

Rossler Attractor


Rossler's attractor is not a famous attractor, but is a rather nice attractor which draws a nifty picture. The attractor is formed with another bunch of Navier-Stokes equations, namely:

A, B, and C are constants.

The unique part of this attractor is that it displays banding. To understand banding, here is a little thought experiment, again. Start with the Cantor set. The Cantor set is simple to create; take a line, and trisect it; then cut out the middle third. You are left with two lines. Do the same for both lines; trisect them, and punch out the middle of those. With the four lines remaining, do the same thing. Now iterate this method forever. One will then have the "Cantor set," or "Cantor dust." This is an infinite number of infinitely small points arranged in a definite pattern.

Take the last iteration possible (the infinite one--just assume here that iteration five is infinite. I know it's not, but who's counting? Even my SuperVGA is incapable of showing that kind of infinitely small resolution). Find and mark the midpoint.

Rotate the iteration around the midpoint, and you will get what is known as the Cantor target. This is an infinite number of concentric circles, arranged so that if you took a diameter-slice out of it you would end up with the Cantor set dust. The arrangement of the circles is known as "banding."

Rossler's attractor displays a type of banding, which suggests that perhaps it is related to the Cantor set. Another interesting fact about Rossler's attractor is that it has a half-twist in it, which makes it look somewhat like a Möbius strip (what you get when you take a strip of paper, half-twist it once, and tape the ends together. A trick that almost everybody knows is to cut along the middle of the band, you will end up with a double-loop; and if you cut in the middle of that double-loop, you will end up with two separate, linked rings).


 

5.Opennings :How is chaos theory applicable to the real world?

First and foremost, chaos theory is a theory. As such, much of it is of use more as scientific background than as direct applicable knowledge. Chaos theory is great as a way of looking at events which happen in the world differently from the more traditional strictly deterministic view which has dominated science from Newtonian times. Moviegoers who watched Jurassic Park are surely aware that chaos theory can profoundly affect the way someone thinks about the world; and indeed, chaos theory is useful as a tool with which to interpret scientific data in new ways. Instead of a traditional X-Y plot, scientists can now interpret phase-space diagrams which--rather than describing the exact position of some variable with respect to time--represents the overall behavior of a system. Instead of looking for strict equations conforming to statistical data, we can now look for dynamic systems with behavior similar in nature to the statistical data--systems, that is, with similar attractors. Chaos theory provides a sound framework with which to develop scientific knowledge.

However, this is not to say that chaos theory has no applications in real life.

Chaos theory techniques have been used to model biological systems, which are of course some of the most chaotic systems imaginable. Systems of dynamic equations have been used to model everything from population growth to epidemics to arrhythmic heart palpitations.

In fact, almost any chaotic system can be readily modeled--the stock market provides trends which can be analyzed with strange attractors more readily than with conventional explicit equations; a dripping faucet seems random to the untrained ear, but when plotted as a strange attractor, reveals an eerie order unexpected by conventional means.

Fractals have cropped up everywhere, most notably in graphic applications like the highly successful Fractal Design Painter series of products. Fractal image compression techniques are still under research, but promise such amazing results as 600:1 graphic compression ratios. The movie special effects industry would have much less realistic clouds, rocks, and shadows without fractal graphic technology.

And of course, chaos theory gives people a wonderfully interesting way to become more interested in mathematics, one of the more unpopular pursuits of the day.
 

II- The Taylor-Couette flow.

1. General presentation

It's Taylor who first analysed the stability of these kind of flows in 1923. It dealt with a flow between concentric cylinders rotating at different rates. The picture below show a brief installation of the phenomenon.
 
 







And it's shown below the different values used :
 
 


 








If the outer cylinder is at rest and if the speed of the inner cylinder is increased you go from a pure azimuthal movement to a more complex one described by the pictures below :
 
 


 



The photogrpahies of this results illustrate well this phenomenon :
 
 


 


If we increase also the outer cylinder, the flow become turbulent and the results are far more complex :
 



 



To well understand this phenomenon, we have to begin to the firts case : the pure azimuthal flow. The nature of this flow is shown shematically below :
 

Vortices of alternate senses circulate betweenthe two cylinders. Each vortex extends toroidally right round the annulus, so the overall flow is axisymmetric. These vortices are called Taylor cells. the nature of the instabillity producing Taylor cells may be understood by considering a toroidal element of fluid. Suppose it is displaced to a larger radius. It is now rotating faser than its new environement and the radial pressure gradient associated with the basic flow will be insufficient to balance the centrifugal force associated with the displaced element. The element will then tend to move further. Similarly, an element displaced to a smaller radius will tend to move further inwards.


2. Equations

According to the geometry of the problem, it seems to be natural to use cylindric Navier-Stokes equations which can be written using standard notations :
 


   

where 
 




3. Stability study

The original azimuthal flow is a solution of exact Navier-Stokes equations. This flow is entirely defined by the solution and the boundary condtions :
 


        with 
 


Begining with this flow, we can now use the classic way of small perturbations, supposing this perturbation is not fonction of teta. So we obatin :
 


     where lamba is a real and s a complex.
 


after several simplifications , supposing the two cylinders are quite similar,   the last equations can be written as follow
 


with the boundary conditions :


 


At this point of the study, we can have two different approach. A first one inspired by Taylor himself is an analytical method to analyse the precise case when the two velociies are virtually the same. The second one is a numerical method in which you have to find s in order that ûr and ûteta were non trivial solution of the last equations. The following graph shows how evolves the critical velocity of the inner cylinder vs the velocity of the outer cylinder.It's intersting to notice ta good compatibility between the two different methods, analytical and numerical.
 



 




The last graph shows the different regime you can find in the different instabilities region :
 



 





To conclude, we can speak about another way to describe such phenomeno linked to the use of a Rayleigh number to caracterize the phenomenon :
 



 


We can obviously draw a parallel with Rayleigh-Benard instabilities in which there is also a critical Rayleigh number. The present critical Rayleigh number for Taylor-Couette flows is given by the empiric formula : .

Thanks to a Fourier analysis we can determine the following graph :
 



4.Openings
 

One example of two-fluid Couette flow is the flow of two centrifugally stratified immiscible fluids in the annulus between horizontal coaxial cylinders. If the inner cylinder is then rotated at a critical rate above the outer cylinder rate, a centrifugally induced hydrodynamic instability, known as two-fluid Taylor-Couette flow, is produced.
 

Flow visualization of the vortices in the water phase is with dye and Kalliroscope.

The water phase contains blue dye to distinguish the two phases. The phases remain stratified without interface break-through to either cylinder (as seen in the interface view). At extreme differences between the inner and outer cylinder rotation rates, the two fluids begin to emulsify.