PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 1- S C Chapman
Concepts
1.
Landau theory – a macroscopic model for a microscopic system.
- few relevant parameters – many dof system
- bifurcation and hysteresis - phase transitions
2.
1st order nonlinear DEs
fixed points, stability (get dynamics without actually solving).
3.
2nd order nonlinear DEs - phase plane analysis.
4.
Limit cycles – predator prey, etc.
5.
Difference equations – chaos in 1D maps
Lyapunov exponents.
Bifurcation sequence to chaos and RG
Feigenbaum nos and universality in chaos.
6.
7.
Many dof systems on computers – order-disorder transitions and self organisation – scale
free behaviour.
8.
Buckingham Pi theorem – dimensional analysis approach to finding the few relevant
parameters.
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 1- S C Chapman
Introduction – universality (linear systems)
Most of what you have seen so far is linear:
An example: pendulum
d 2!
+ " 2 sin ! = 0
dt 2
!
!
- Nonlinear since sin ! term –
d! d 2!
not e.g. !, , 2 ( ! linear )
dt dt
d 2!
Immediately assume ! small than sin ! ! ! and 2 + " 2! = 0 .
dt
- now linear – solution of the form ! ! Acos( "t + # )
A, ! from initial conditions.
- this is an example of general description of particle motion in potential V
"V ( x )
md 2 x
=
!
"x
dt 2
dV
here
…
dx
- function of x
V (x)
only
x
x0
V has a minimum – expect small oscillations
about x0 - find frequency ω of oscillations.
Lets (do properly) linearize about x = x0 - Taylor expand
x = x0 + ! x
dV ( x )
dx
V ( x0 ) is a minimum,
=
dV ( x0 + !x )
dx
=
dV
d ! dV
+ ##
dx dx #" dx
$&
d 2 ! dV
&&!x + 2 ##
%
dx #" dx
$& !x 2
+ ...
&&
% 2
r.h.s. evaluated at x0
d 2V ( x0 )
dV
! x + 0 (! x 2 )
(x)=
2
dx
dx
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 1- S C Chapman
So,
d 2V ( x0 )
d 2! x
=
!
! x + 0 (! x 2 )
dt 2
dx 2
d 2x
=m 2
dt
m
d 2V ( x0 )
dx 2
!
d2
V (x)
dx 2
x = x0
‘Linearize’ means terms 0 (! x 2 ) ! 0 ( ! x )
ie:
all functions are linear in ! x
equivalent to ! x ! 1 small displacements from x0 .
etc.
V !!( x0 ) d 2"x
V !!( x0 )"x
, 2 ="
m
m
dt
* hence this linear equation is universal – works for any (conservative) system with a minimum in
V ( x ) provided ! x is small enough.
(V !! > 0 ) " ! is real
- recover linear pendulum equation with ! 2 =
[intuitively obvious – anything with potential ⇒
!x
V (x)
- oscillations
and know ! x ( t ) - oscillates about x0
"roughly" sinusoidal ]
x
Other advantage of linear equations – principle of superposition
ie: if you have the simplest situation – ie: one oscillator ! = Acos( "t + # )
any system can be obtained by
! ( t ) = ! Aj cos( " j t + # j )
which is why we have Fourier theory.
Here we have j linear coupled oscillators (normal modes) each one has two constants ( Aj , ! j ) and
one frequency ! j .
Another equation you will have seen –
wave equation
! 2!
1 ! 2!
=
!x 2
c 2 !t 2
c constant
Nonlinear of c ( ! ) - dispersion, diffusion.
Linear equation 'works' for any waves, ie: water waves, light waves… universality.
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 1- S C Chapman
!!
!!
=
"
!t
!x 2
Diffusion equations
! constant linear.
So linear equations – good news was
- universality – (works for anything)
- superposition – (only need simplest solution – just add them up)
- Not many parameters, eg:
oscillator
– just ω
waves
- just c
diffusion
!
- Not many variables
- eg: !, "....
Now real life is Nonlinear
- a problem!
- superposition fails (try it!) - bad news
Good news:
- still get universal equations (so don't need to learn many)
- only need few order parameters
- only need a few variables.
Last point → few variables – often this is because system appears to have many variables but
relevant physics can be described by few
eg: Magnet – many spins – but we can understand it by just following M (total
magnetisation).
Why this is so is a current hot topic in physics (self organisation, critical phenomena, complexity).
Normalisation and Dimensional Analysis – (more later)
Normalisation (boring but important)
Have written down universal equations eg:
! 2!
1 ! 2!
=
!x 2
c 2 !t 2
wave equation
experimentally you have displacementtimewrite ! = !*!0
x=x L
*
then sub in
t = t *T
Normalised units!
x (meters)
t (seconds)
T seconds
L meters
!0 - whatever ! is measured in
! 2 !*!0
1 ! 2 !* !0
=
!x *2 L2
c 2 !t *2 T 2
! 2 !* ! 2 !* 1 L2
= *2 2 " 2
!x *2
!t c T
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 1- S C Chapman
L2
c is a velocity
T2
! is anything (this is why it is universal).
this is dimensionless if c 2 =
So, to do the maths you can work in (*) units – or to solve on a computer.
[Also found out what c meant – and if equation is correct!]
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