main course non linear dynamics

8/2/2019 Main Course non linear dynamics

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From linear

to non-linear

I) Description of Boussinesq equations system.Let us consider a fluid layer placed horizontally in a gravity field. This layer is supposed to be

infinite in this plan and its thickness is noted d. The fluid can be a gas or a liquid; It is characterized

by its viscosity and its heat-storage capacity . We define its PRANDTL number=

For ordinary liquids (water, alcohol,) 7= , for viscous liquids (Oil, honey, ) isgreater : It reaches 1010 for the earth inner layer (manteau terrestre ??) ; Gas have diffusion

coefficients quite identical giving 7.0 . Mercury, which is a good heat conductor, has a coefficient025.0= . The inner part of the sun has a PRANDTL number very small.

The borders of this layer consist of two free or rigid surfaces which one fixes the temperature

or the heat flux. We consider here, the simplest case:

- Free wall: Vvertical =vertical

vertical

x

V

= 0 on the walls

- Free-free wall:2

2

vertical

vertical

x

V

= 0 on the walls (unrealistic conditions in experiments)

and infinitely conducting of heat the temperature is fixed on the wall.

"As soon as" the temperature field is not uniform any more (small disturbance), it is prone toa Archimedes force because its density is related to its temperature (state equation of the fluid).These effects have as a characteristic time:

21

1

=d

gA

These forces are destabilizing and can induced fluid motions. But, this fluid is also prone to

two types of effects which are stabilizing:

The diffusion of a temperature disturbance with a characteristic time :

2dC=


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The viscosity damping of a velocity disturbance with a characteristic time :

2d

V=

The ratio2

A

VC

compares the stabilizing effects over the destabilizing effects. This

dimensionless number is called RAYLEIGH number and it can be define by:

3dg

Ra

=

A great Rayleigh number means that characteristic times of damping (c et v) are large andthus that the disturbances are deadened weakly: They have time to grow and are at the origin of the

convective state. On the other hand, when Ra is small, the stabilizing effects can "kill" the

disturbances very quickly: the fluid remains at rest in the conductive state.

The Rayleigh number is proportional to the temperature difference applied to the fluid layer,

its a dimensionless measurement of the constraint which is applied to it.

The equations which describe the state of this system called Rayleigh-Benard System can

be written:

With:

)(. += GradVtdt

d

=

),,,(),,,(),,,(

tzyxwtzyxvtzyxu

V

),,,( tzyxp and ),,,( tzyxT

- Continuity equation : 0)( =+ VDivdt

d r

[1]

- Energy equation : += TdtdTCV [2]

- WithVC

= and the heat flux created by the viscous friction.

- Momentum Equation (Navier-Stokes) : VFpGraddtVd

A

rrr

++= )( [3]

- State equation of the fluid : ( )( )00 1 TT= [4]where is thermal dilatation coefficient.


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The Boussinesq approximations consists to neglect:

- The fluid compressibility : 0=dt

d

[1] becomes 0)( =VDivr

[1]

- The heat production from viscous effects : 0= [2] becomes TTGradV

tT =+ )(.

v

[2]

- The Archimedes force can be written as : zgFA rr = and we can assume that dtVddtVdrr

0

(density variations only for the gravity forces).

[3] becomes VzgpGradVGradVtV

rr

rrr

+=

+ )()(.0 [3]

Theses three equations, with their boundary conditions, are the Boussinesq equations. A

stable solution is trivial solution:

0rr

=V and ( ) bazzT +=

if ( ) 10 TzT == and ( ) 2TdzT == we obtain ( ) 112 TzdTT

zT +

=

Generally, we note : 021 >

=d

TT

So ( ) zTzT = 1

We also obtain the hydrostatic pressure gradient: ( )( )010 1 TTzgzP +=

We assume now that we disturb this stable state:

pp= ( )tzyx ,,,+

( )zTT = 1 ( )tzyx ,,,+

0rr

=V ( )tzyxu ,,,r

+


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, and ur

are small quantities introduced in the equations:

[1] 0)( =uDivr

[1]

[2] ( ) =++ )(. 1 zTGradut

r

[2]

[3] ( ) uzgGraduGradutu rrrrr

+++=+ 00 )()(. [3]

Let now rewrite these equations in a dimensionless form using as scales:

2

0

dt = for the time

d =0 for the temperature

du =0 for the velocity

2

2

00 dp

= for the pressure

and d for the length scale

We obtain:

[1"] 0)( =uDivr

[1]

[2"] +=+ zuGradu

t

rrr

.)(. [2]

[3"] { }uzRaGraduGradutu rrrrr

++=+ )()(. [3]

where is PRANDTL number :

=

Ra is the Rayleigh number:

4gdRa =

With the boundary conditions:

0. =zurr

at z=0 and z=1

=0 at z=0 and z=1


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II) Linear equations and marginal stability (linear stability)

We now assume that the quantities , and ur

are and will stay small. Then the transportterms )(. Gradu

r

and )(. uGradurr

can be neglected for the linear stability study:

[1] 0)( =uDivr

[1]

[2] += zu

t

rr

. [2]

[3] { }uzRaGradt

u rrr

++=

)( [3]

Taking the curl of the last equation, the pressure will disappear. On the other side we now

obtain an equation for =rr

)(utRo , the vorticity. Usually its easier to take the curl twice because:

( )( ) uuDivGradutRotRorrr

=))((

The equations then become:

[1] 0)( =uDivr

(unchanged)

[2] += zu

t

rr

. (unchanged)

[3] ( ) ( )( ) ( ){ }uzRotRotRatu rrr

+=

. [3]

We try to find a solution for these equations as linear rolls type:

Ifwvu

u=r

, we find ur

as: 0=v and 0=

y

The system is then reduced to:


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[1] 0=+

zw

xu

[2] += w

t

[3] ( ) ( )

=

zxRau

t

u

2

. [3u]

( ) ( )

+=

2

2

.x

Rawt

w [3w]

(L System)

According the experiments, we look at solutions as straight rolls type:

( ) tikxzuu ee .=

( ) tikxzww ee .=

( ) tikxz ee .= (Normal modes)

where is the eigenvalue of a linear operator.

If we try to solve the problem for w and , we obtain:

[2]

+=

2

2

2

k

z

w [II]


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[3w]

+

=

.2 242

22

4

42

2

2

Rakwkz

wk

z

wwk

z

w [IIIw]

The functions w(z) and (z) must satisfy the boundary conditions (w(0)=0 ; w(1)=0 ; (0)=0and (1)=0). If we take ( ) ( )znSinWzw 0= and ( ) ( )znSinz 0= , we then obtain:

[II] ( ){ } 000222 =++ Wkn [II]

[IIIw] ( ) ( ){ } 0. 0202222222 =+++ kRaWknkn [IIIw]

The marginal stability is given by =0 :

[II] ( ) 000222 =+ Wkn

[IIIw] ( ) 0. 0202222 =+ kRaWkn

To obtain a non-trivial solution ( 00 et 00 W ), the determinant of this system must vanish.

So:

( ) 0. 23222 =+ kRakn or( )

2

3222

k

knRa

+=

We can plot this set of curves in the (r,Ra) plan. They are parameterized by n :

The lower curve is obtain for n=1. The minimum of this curve corresponds to the convectionthreshold. Numerically, we obtain for the critical values:


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8

2214.22= Ck et

( )51.657

4

27 4

2

322

=+

=

C

C

C k

kRa

Calculus details?

In fact, when the surfaces are not free but rigid, the functions w(z) and (z) are different andthe marginal stability curve has for minimums :

1708CRa and 117.3Ck

III) Lorenz attractor (non-linear approach of the problem).

This calculation, published in 1963 by LORENZ, makes it possible to reach the slightly non-

linear behaviour of the convection. In fact, this one does not behave exactly like the Lorenz model

predicts it. In reality, the non-linear effects act even with the high orders. This model however gave

birth of the deterministic chaotic system theory. It is a very good calculation exercise of the first

non-linear terms and it allows to correctly describing the bifurcation between the conductive and the

convective state.

Rewrite now the non-linear equations of the convection:

[1] 0)( =uDivr

[2] +=+ zuGradu

t

rrr

.)(.

[3] { }uzRaGraduGradutu rrrrr

++=+ )()(.

where is the PRANDTL number : =

Ra is the Rayleigh number:

4gdRa =

As during the linear stability analysis, we take the curl twice and we assume for the solution a

straight roll type (w

uu 0=r

et 0=

y).


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Calculus details:

If w

uu 0=r

( )

0

0

=

=

=

==

yu

xv x

w

z

uzv

yw

uRotr

r

u

wu

= 0r

r

Reminder: ( ) ( )

+=

2.. WWGradWWRotWGradW

rrrrrr

so ( ) ( )( ) ( )WRotWWRotRotWGradWRotrrrrrr

==.

then: ( ) ( )( )WRotRotWGradWRotRotrrrr

=.

( )( ) ( )

0

0

Az

u

z

wuRot =

+

=

rr

( )( )

( ) ( ) ( )

( ) ( ) ( )2

2

2

2

12

22

0B

zx

w

x

u

x

A

Bz

w

zx

u

z

A

uRotRot=

+

=

=

=

=

rr

Normal modes:

We then can express the equations using the normal modes. We chose for u, w and to takethe following functions:

( ) ( )zCoskxSinutzxu ..),,( 0=

( ) ( )zSinkxCoswtzxw ..),,( 0=

and ( ) ( )zSinkxCostzx ..),,( 0=

Remark:

The flow is bi-dimensional : Introducing the LAGRANGE stream function ( )tzx ,, , thecontinuity equation is automatically verified if :

ztzxu

=),,( and

xtzxw

+=),,(

Boundary conditions:

No flux through the surfaces 0),,( =+=x

tzxw

Free surface 0),,( 22

== zztzxu


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10

So a function like ( ) ( )zSinkxSinttzx .).(),,( 0= is appropriate

Then ( ) ( ) ( ) ( )zCoskxSinuzCoskxSintz

tzxu

u

....)(),,( 00

0

===

43421

and ( ) ( ) ( ) ( )zSinkxCoswzSinkxCostkx

tzxw

w

....)(),,( 00

0

==

+=321

0)( =uDivr

( ) ( ) ( ) ( ) ( ) ( ) ( ) 0..... 0000 =+=+=+

zCoskxCoswkuzCoskxCoswzCoskxCosku

zw

xu 000 =+ wku

for the curl : ( )0

0=

==

xw

zuuRot

rr

we obtain ( ) ( ) ( ) ( ) ( )zSinkxSinzSinkxSinkwuxw

zu .. 000

0

=+=

=

43421

( ) ( ) ( )

0

0

Az

u

z

wuRot =

+

=

rr

with( ) ( )

( ) ( )zSinkxSinww

zSinkxSinu

u

200

200

.22

2.2=

=

then

( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) 02.2.2

.222

2.22

0000000 =

+=+= zSinkxSinwkuzSinzCoskxSinwzSinkxCoskxkSinuA !!!

The non-linear disappears: It gives no contribution. We then obtain the same system as the

previous case (Linear analysis).

[3u] ( ) ( )

=

zxRau

t

u

2

.

[3w] ( ) ( )

+=

2

2

.x

Rawt

w

On the other side, for the temperature we obtain: +=+

+

w

zw

xu

t

Using sine functions for the normal modes, we compute at first the non-linear contributions:

( ) ( )zSinwz

wx

u 22 00=

+

In fact

( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]zCoskxCoszSinkxCoswzSinkxSinkzCoskxSinuz

wx

u ........ 0000 +=+


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{( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( )zSinwzCoszSinwzCoszSinkxCoswzSinzCoskxSinku

w

22

.... 00

002

002

00

0

==+

=

The term +=+ zuGradu

t

rrr

.)(. generates a second harmonic in the vertical direction. Its this

harmonic which initiates the distortion of the temperature profile:

This harmonic is the only contribution of the normal modes non-linearity product. For taking

account in the equations, we have to counterbalance by a second harmonic in the temperature

expression:

( ) ( ) ( )zSinzSinkxCostzx 2...),,( 10 =

Let us recomputed the non-linear term:

( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ].2.2........ 10000 zCoszCoskxCoszSinkxCoswzSinkxSinkzCoskxSinuzwxu +=+

{

( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( )zCoszSinkxCoswzCoszSinkxCoszSinzCoskxSinkuw

2..2.... 0120200

0

+

=

[ ] ( ) ( ) ( ) ( ) ( )zCoszSinkxCoswzCoszSinw 2..2 0100 =

( ) ( ) ( ) ( )zCoszSinkxCoswzSinw 2..222 01

00

=

( ) ( ) ( ) ( )[ ]zSinzSinkxCoswzSinw 3..22 01

00 +

=

Because ( ) ( ) ( ) ( ) ( ) ( )aSinaSinaSinaSinaCosaSin 32 2212 == and ( ) ( ) ( )aSinaSinaSin 334 3 =


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12

The system is then the following:

[NL-I] 0=+

zw

xu

[NL-II] +=+

+

w

zw

xu

t

[NL-IIIu] ( ) ( )

=

zxRau

t

u

2

.

[NL-IIIw] ( ) ( )

+=

2

2

.x

Rawt

w

By injecting the expressions of the terms resulting from the normal modes in the velocity and

temperature equations then projecting on each harmonic in z (forgetting the third harmonic), weobtain :

( ) ( )zCoskxSinutzxu ..),,( 0= ( ) ( ) ( )zCoskxSinkuu .. 220 +=

( ) ( )zSinkxCoswtzxw ..),,( 0= ( ) ( ) ( )zSinkxCoskww .. 220 +=

( ) ( ) ( )zSinzSinkxCostzx 2...),,( 10 = ( ( ) ( ) ( ( )zSinzSinkxCosk 2.4... 21220 ++=

[NL-I] ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0..... 0000 =+=+=+

zCoskxCoswkuzCoskxCoswzCoskxCosku

zw

xu 000 =+ wku

[NL-II] +=+

+

w

zw

xu

t

( ) ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ .2.2........2. 1000010 zCoszCoskxCoszSinkxCoswzSinkxSinkzCoskxSinuzSintzSinkxCost

++

( ) ( ) ) ( ) ( ) ) ( )zSinzSinkxCoskzSinkxCosw 2.4..... 212200 ++=

( ) ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]zSinzSinkxCoswzSinwzSint

zSinkxCost

3..22

2. 010010 +

+

( ) ( ) ( ) ( ) ( ) ( ) ( )zSinzSinkxCoskzSinkxCosw 2.4..... 212200 ++=

( ) ( )( ) ( ) ( ) ( ) ( )[ ] 034.2

2.. 012

100122

00010 =+

++

+++

wzSinw

tzSinkww

tkxCoszSin

We then obtain for the first harmonic 1 : ( ) 0. 2200010 =+++

kwwt

so ( )2200010 . ++=

kwwt

And for the second: ( ) 04.22

1

001

=+

w

t so 12001

42

=

wt


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13

[NL-IIIu] ( ) ( )

=

zxRau

t

u

2

.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ){ }zCoskxSinkRazCoskxSinkuzCoskxSinkt

u ...... 0

2220

220 +=+

( ) ( ){ }02220022 . Rakkutuk ++=+ so ( ) ( )

+

++=

02222

00 .

kRakku

tu

[NL-IIIw] ( ) ( )

+=

2

2

.x

Rawtw

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ){ }zSinkxCoskRazSinkxCoskwzSinkxCoskt

w ...... 02

2220

220 ++=+

( ) ( ){ }022220022 . Rakkwtw

k +=+ so ( ) ( )

+++=

022

2

220

0 .

k

Rakkw

tw

We now only interested by the variables 0w , 0 et 1 .

The system becomes:

( ) ( )

+++=

022

2

220

0 .

k

Rakkw

tw

( )2200010 . ++=

kwwt

12001 4

2

= w

t

We use: 22 += kA

tAtk ..22 =+=

=

20

AwX

=

23

2

0A

RakY

=

3

2

1 A

RakZ

The equations become:

1) ( ) ( )

+++=

022

2

220

0 .

k

Rakkw

tw ( )YXAAY

Rak

A

A

RakAAXXAA +

=

+

=

.2

22..22

32

so YXX

+=

2) ( )22

0001

0

.

++=

kwwt YRak

A

AX

A

XZ

A

Rak

AYARak

A

+

=

2

3

2

3

2

3 222

.

2


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YXA

RakXZY

+=

3

2

so YrXXZY +=

with( )

+=

322

2

k

Rakr

3) 12001 42

= w

t Z

Rak

AXYA

Rak

AZARak

A

=

2

3

22

3

2

3

422

2.

ZA

XYZ

=

24

so bZXYZ =

with ( )2224

kb

+=

At the end, the system is the following:

YXX

+=

YrXXZY +=

bZXYZ =

avec( )

+=

322

2

k

Rakr

et ( )22

24

kb

+=

LORENZ System

Remarks:

The linear stability theory has shown that the bifurcation between conductive-convective state

have for main characteristics:

2

=Ck and( )

2

322

C

C

C k

kRa

+=

for these same conditions we obtain ( ) 38

2

442

2

2

22

2

=

+

=+

=

Ckb et

=CRa

Rar . Adding 10=

(PRANDTL number for air) we obtain a set of parameters called canonical parameters .

CHAPTER II: Dynamical System Analysis

LORENZ dynamical system Analysis:

YXX

+=

YrXXZY +=

bZXYZ =


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15

1) Critical points finding :The first part of the analysis consists to find the stationary solutions of the equations. We are then

finding the points like 0==

=

ZYX , so :

0=+ YX

YX=

0=+ YrXXZ ( ) 01 =+ rZX

0=bZXY bZX =2

The second equation impose 0=X or ( )1= rZ .

If 0=X , we obtain 0==XY and 02==b

XZ . The first critical point is then A(0 ;0 ;0) : Steady

state.

If ( )1= rZ , we obtain ( ) == 1rbX (if and only if r1) ( ) === 1rbXY . We then obtaintwo new points B(+ ;+ ;r-1) and C(- ;- ;r-1) if r1.

Note: The first one corresponds to the steady state, the differences with the static case are logically

null. The two others cases correspond to the convective mode, the sign of indicates the directionof rotation of the cells (or rather of a cell in particular).

2) Critical point nature.Point A(0 ;0 ;0) : The steady state X=Y=Z=0 is a trivial solution of the system. It is of course oneof the critical points of this dynamic system with three degrees of freedom. Let us study its linear

stability. We write:

xX +=0 YXX

+= becomes yxx

+=

and with the linearization : yxx

+

yY +=0 YrXXZY +=

becomes yrxxzy +=

and with the linearization : yrxy

zZ +=0 bZXYZ =

becomes bzxyz =

and with the linearization : bzz

The associated matrix of the linear system id then : [A]=

br

00010

. Let us calculate the eigenvalues

of this matrix:

Det(A-I)= ( ) ( ) ( )rb ++++ 112 =0

The first eigenvalue is b= (1, this product

is negative. That means one of these two eigenvalues is positive and the other is negative. Thesteady state is then unstable for r>1 and this, according the definition of CRaRar= , corresponds to

the case where the convection appears Ra>RaC.


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We have p=-tr[A]= ( )+1 >0 and q=Det[A]= ( )r1 .

If r0 and q>0 : We obtain a stable node or focus.

If r>1, p>0 and q

rr the second term is positive if r>1 else it is negative. However in

the worst case (r=0), we obtain ( ) ( )2222 11241241 =+=++=+= >0 .

Then these two eigenvalues are: ( )21 +=

and the third being : b=

Figure =f(r) for the point A(0 ;0 ;0 )

It is noticed that two of the eigenvalues are always negative ( ( )21 += and b= ). On the

other hand, the third solution ( ( )21 ++= ) is positive for r higher than 1. The disturbance goes

developing and ends up "drowning" the stationary regime. r replaces Ra here.

Thus, for a certain temperature gap, the static solution is not stable anymore, the fluid starts

moving. We find the existence of a critical Rayleigh number, beyond which the movement is

maintained. For a sufficiently high temperature gap, the convective mode is established itself(computation shows indeed that the other solutions can be stable).

Thus when r 1, we pass from a stable state towards an unstable state: It is the passage of theconductive state to a convective state. This steady state "forks" towards the convective state given

by another critical point: (X=Y ; Z=X2/b).

Points B(+;+ ;r-1) : With ( )1= rb . This point corresponds to the state brYX 1== and

1=rZ

We pose :


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xX += YXX

+= becomes ( ) ( ) yxyxx

+=+++=

and with the linearization :

yxx

+

yY += YrXXZY +=

becomes ( )( ) ( ) ( ) yxzxzyxrzrxy +=+++++=

1 and with the

linearization : zyxy

( ) zrZ += 1 bZXYZ =

becomes ( )( ) ( ) bzxyyxzrbyxz ++=+++=

1 and with the

linearization : bzyxz +

The associated matrix of the linear system id then : [A]=

b

11

0. Let us calculate the eigenvalues

of this matrix :

Det(A-I)= [ ] ( )[ ] ( )[ ] ( ) 0121 2323 =+++=

++++++ CBArbrbb

CBA

434214342143421

Constants A, B and C are independent of: The second point C(- ;- ;r-1) will be thus asthe same nature as B. It is easier to study the eigenvalues graphically. For that we write :

[ ] ( )[ ] ( )[ ] 012123 =++++++ rbrbb [ ] [ ]bbrbbb 22123 +=++++ soit :

[ ]

[ ]

( )

f

bb

bbbr =

+

++++=

2

2123

Figure =f(r) for point B (and C).

We observe that this state is stable for r>1: (3 negatives eigenvalues) and unstable for r


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For r1 It is the opposite A becomes Unstable while B&C becomes Stable

So we observe a Stability exchange called bifurcation. Near the convection threshold we

obtain the bifurcation diagram:

CHAPTER III: Bifurcations

Note : This bifurcation is a pitchfork type bifurcation. Rayleigh-Bnard convection is asecond order phase transition, with as order parameter the velocity W. The symmetry which is

broken is the invariance with translation: The system is periodic in the plan (rolls) and no more


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19

uniform, we have now a periodic solution in space for the velocity (XW0) and the temperature(Y0).

We find the equation of the normal form of the bifurcation while eliminating from the Lorenz

system the slaves modes, i.e. those which do not have own dynamics. The linearized system at r=1

is:

[A]=

b00

011

0

Because b>>1: Z0 and of course dtdZ then dtdZ =0=XY-bZ bZ=Y2

Because >>1: XY

So Z and X (or Y) are the slave modes and Y (or X) are the master mode.

If we introduce the expression of two slaves modes X and Z, in the equation in Y [which is

the temperature mode due to the Archimedes force: Origin of the convection], we find:

( )b

YYr

dtdY

3

1 =

This equation is called the amplitude equation. It describes the behavior of the convection

in the vicinity of the threshold (i.e. in the vicinity of the critical value r=1). It can be obtain from a

more subtle means and even comprise terms of spatial variation which take account of the

modulations of the convective structures:

32

2

Ax

AA

dtdA

+=

When r, becomes >0 !! (for r25 (24.73) the eigenvalues become (-,+,+).


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The numerical exploration of the system of the three equations, shows that for r=24.74, there

is a stability loss of the stationary regime. This instability occurs abruptly, and the system is

captured by a strange attractor which caused such an amount of passion: The LORENZ attractor.

CHAPTER IV : Transition to chaos


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LORENZ attractor

The fractal dimension of the attractor on which the trajectories are rolled up is about 2.06.

With higher r, the system finds a limit cycle formed by an inverse Hopf bifurcation, then a newstrange attractor...


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CHAPTER V : Chaos and dimension

Bifurcation diagram of LORENZ system.

Also let us note, that it is by exploring this dynamic system that Pomeau and Maneville

observed and understood the Type I intermittency concept, of which here some results: In the

vicinity of r=166, where a limit cycle loses its stability.

Iterated map showing the channel followed by the trajectories.

Finally let us notice, that it is because its "supercriticality" of the bifurcation, that the

Rayleigh-Bnard flow could be the subject to such detailed studies. The non-linear developmentsare based on the proper modes (modes propres) of the system which are first easily accessible and

on the other hand which is not very far away from the real solution.


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When the bifurcation is subcritical, the steady state solution abruptly loses its stability (by a

1st order transition) and the system is found in areas of the phase space very far from its basic state

and impossible to reach theoretically.

In addition, which made the Lorenz model so famous; it is its sensitivity to the initial

conditions. Lorenz had made its first simulations with five significant digits. Thinking of savingtime without harming the exactitude of its experiments, it remade them with only three digits. The

results, in spite of this tiny error, were completely different. Thus, some negligible variations of the

initial state of the system can lead to completely different states. This property is characteristic of

the chaotic states and is at the origin of all the theoretical conclusions of the study. We can illustrate

it by following simulation. Two hundred very close initial states are chosen and we look at their

evolutions in time in the phase space. If the points remain enough grouped initially, they tend to

move away more and more, so that the segment of origin is progressively transformed to a cloud

after fifteen seconds.

On the basis of these observations, Lorenz extrapolated its conclusions to the atmospheric

behaviors. The divergences between two initially close states are such that at the end of two weeksthat only the perfect knowledge of these initial states makes it possible to predict with certainty the

weather evolution. However the measurements are not sufficiently accurate to reach a such

precision, therefore weather is unpredictable beyond two weeks. In fact, the beat of a butterfly wing

can cause a sufficient disturbance to create a hurricane at the end of two weeks anywhere elsewhere

in the world. And we will never be able to know at one precise moment the location of all the

butterflies in the world. It is this image which gave its name for the "butterfly effect".

The Lorenz equations were discovered by Ed Lorenz in 1963 as a very simplified model of

convection rolls in the upper atmosphere. Later these same equations appeared in studies of lasers,

batteries, and in a simple chaotic waterwheel that can be easily built.Lorenz found that the trajectories of this system, for certain settings, never settle down to a fixed

point, never approach a stable limit cycle, yet never diverge to infinity. What Lorenz discovered

was at the time unheard of in the mathematical community, and was largely ignored for many years.

Now this beautiful attractor is the most well known strange attractor that chaos has to offer.

A simple physical model of the Lorenz equations at work is a leaky waterwheel. A waterwheel built

from paper cups with equal sized holes in the bottom of each cup is allowed to turn freely under the

force of a steady stream of water poured into the top cup. For a slow flow of water, the water leaks

out fast enough that friction keeps the waterwheel from moving. For just a little more flow the

waterwheel will pick a direction and spin in that direction forever. If the flow is increased further

the waterwheel does not settle into a stable cycle. Instead it spins in one direction for a bit, then

slows down and start to spin the other. The waterwheel will constantly change its direction of spin,

and never in a repeating predictable manner. Here is a picture of the waterwheel. Many more

sophisticated, but similar systems have been built, and they all show the same chaotic behavior.


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main course non linear dynamics

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