PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
Logistic map
-
Feigenbaum
1970’s
period doubling – bifurcation route to chaos
a universal behaviour
x N +1 = ! x N ( 1 " x N
May 1976
1)
)
-
insect population dynamics
!
-
birth rate
! " x N2
-
saturation/competition
!>0
Why not use corresponding differential equation instead?
- construct analogous continuous 1st order DE and investigate the dynamics
assume that x N has a continuous limit x(t)
time
t = N!
so
x N = x(N !)
Taylor expand
N = 0,1,2....
x N +1 = x(N ! + !) , etc.
!" is small – for continuous x(t) :
x(N ! + !) = x(N !) +
d
x(N !).! + ......
dt
(A)
but this is also coincident with x N +1 etc., so for map:
x(N ! + !) = M ( x(N !) ) .
= ! x(N ") ( 1 # x(N ") )
( A ) ! ( B ) and write x(N !) = x(t) = x
"
dx
= ( ! # 1 ) x # ! x2
dt
this is the continuous 1st order DE analogue to M ( x ) .
Look at fixed points, classify for continuous case.
Fixed points
1
( ! " 1) x " ! x 2 = 0
x [( ! " 1) " ! x ] = 0
(B)
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
so
x = 0,
x=
! "1
.
!
Classify by sketching the phase plane
H (x) = "
dx
= ( ! # 1 ) x # ! x2
dt
! does not matter
sketch
H (x) vz x
!>0
so H ( x ) ! "# x ! ±#
t.p at
dH
= 0 = ( ! " 1 ) " 2! x
dx
ie: at x =
! "1
2!
or
x=
x
2
Sketch of phase plane
●
x
Then for the continuous DE:
●
x
! >1
0 < ! <1
2
H(x)
0 < ! <1
H(x)
! >1
●
●
x
x
x =0
! "1
x=
!
repellor
x =0
attractor
attractor
x
x
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
x=
! "1
!
repellor
Now look at the difference equation (map)
x N +1 = ! x N ( 1 " x N
)
x N +1 = x N = x
fixed points at
x = ! x (1 " x )
( ! " 1) x " ! x 2 = 0
ie:
Classify:
-
same expression as continuous DE.
Linearize the map
xN = x + ! xN
x N +1 = x + ! x N +1
Sub into map
x + ! xN +1 = " ( x + ! xN ) (1 # x # ! xN )
= " $& x (1 # x ) # x! xN + (1 # x )! xN # ! xN2 %'
but at the fixed points x = ! x (1 " x )
N +1
= [ ! (1 # 2 x ) ]
and at x = 0
x=
so
+0 (! xN2
! xN +1 = "! xN (1 # 2 x )
so
! "1
!
x =0
x=
! "1
!
" x0
! xN +1 = " N +1! x0
" xN +1
! # 2! + 2 & '
= $* ! %(
)+
!
, ./
N +1
" x0
" xN +1 = ( 2 # ! )N +1 " x0
! >1
0 < ! <1
x =0
x =0
repellor
attractor
-same as continuous DE
- for
0 < ! < 1 is a repellor- same as continuous DE
however, no longer an attractor for all ! > 1
different to continuous DE
3
)
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
2! " <1
instead, for
1 < ! < 3 map has attractor at x =
i.e
! "1
!
What happens for ! > 3 ?
! >1
Sketch of map:
M(x)
M ( x ) = !x (1 " x )
intercepts at x = 0,1
dM
= ! " 2! x
dx
1
= 0 at x =
2
!#
1& !
M 12 = % 1 " ( =
2$
2' 4
( )
x is bounded
ie: x ! [ 0,1 ]
if ! " 4
!
4
●
0
1
2
●
1
We will look in the range 3 < ! < 4 (this is where all the interesting behaviour is).
First we can do ! = 4
in this case
M p ( x ) = [ 0,1 ] (like tent map).
Topologically, same as tent map as well – see this by change of variables
" !y% 1
x = sin 2 $
= [ 1 ( cos ( ! y ) ]
# 2 '& 2
x = [ 0,1 ] as y = [ 0,1 ]
sub into
xN +1 = 4 xN (1 ! xN )
"!y
%
1 1
"
%
sin 2 $ N +1 ' = 2 ( 1 ( cos ! y N ) $ 1 ( + cos ! y N '
#
&
2 2
# 2 &
= ( 1 ! cos " y N
4
) ( 1 + cos " yN ) = 1 ! cos2 ( " yN ) = sin 2 ( " yN )
x
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
ie:
"!y
%
sin 2 $ N +1 ' = sin 2 ( ! y N
# 2 &
so
! y N +1
= ±! y N + S !
2
)
S integer
y = [ 0,1 ] so can only have certain s.
y N +1
= ± yN + S
2
If
S =0
y N +1 = 2 y N
S =1
y N +1 = 2 ( 1 ! y N
0 ! yN !
)
1
2
1
2
! yN ! 1
+ sign
- sign
this is just the tent map.
So, have shown there is global chaos at ! = 4
Problem – what happens as we go from ! < 3 attractor to ! = 4
We will need to find the pth iterated map M p (or its essential properties).
5
- chaos?
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
Iterates of the logistic map
(use graphics)
Sketch
M
!
4
M(x) = x
●
x=
slope < 1
stable
1< ! < 3
! "1
!
●
x
●
0
1
2
!
4
M(x) = x
●
3< ! < 4
●
x=
! "1
!
0
goes
1
2
stable → unstable
attractor → repellor
as ! goes through 3.
6
●
1
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
Look at M 2 ( x )
M 2 ( x ) = M ( M ( x ))
= ! M (1 " M )
= ! 2 x (1 " x ) (1 " ! x (1 " x ) )
!
= ! 2 #% x " (1 + ! ) x 2 + 2! x3 " ! x 4 $&
fixed points of M 2 ( x ) are M 2 ( x ) = x
or
M2(x )! x = 0
or
! 2 "% x $ (1 + ! ) x 2 + 2! x 3 $ ! x 4 #& $ x = 0
x = 0 and 3 others.
so roots are
- 3 real or 1 real, 2 imaginary
- all depends on ! .
Sketch M 2 ( x )
-
it is a quadratic
-
it is symmetric about x =
(because M is)
asymptotes
M 2 ( x ) ! "#
intercepts at x = 0,1
7
x ! ±#
1
2
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
M2(x)
Sketch
M2( x) = x
!<3
M2( x) = x
!>3
2
1
●
●
●
●
●
0
●
0
1
1
2
1
2
Fixed points:
x =0
x =0
+1
Now if x is a fixed point of M
must be fixed point of M
call this x*
2
(converse is not time)
!<3
!>3
-
8
! "1
x =
is attractor
!
*
x * is repellor
(
d M2
dx
)
<1
x*
d (M 2 )
> 1 and
dx
but two new fixed points appear
they may (may not) be attractive.
dM
>1
dx
+3
1
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
3< ! < 4
x* =
! "1
!
is repulsive fixed point of M , M 2 ....
but two new fixed points of M 2 appear at ! > 3
not fixed points of M .
If these are attractive this is period 2 cycle of M .
M
(x)
Period 2 in M is fixed point of M 2 .
Local gradients to x , ie: ! must be "just right".
See handout for the full sequence….
9
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
Logistic map and Feigenbaum numbers
As ! increases there are successive period doublings
x
x*
!1 = 3
Pairs of bifurcated fixed points
Separated by !x p appear at ! p
- looks "self similar".
10
!2
!3
!"
!
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
Self – similar?
A piece of the sequence
(a)
!" p
•
! "x p
enlarge
x
•
(b)
!
!p
! p"1
!" p = " p # " p#1
looks the same
!x p = x p(A) " x p( B )
If self similar then
!" p
!" p+1
!x p
!x p+1
= const
ratio independent of p
= const
indepenent of p
Feigenbaum's result (numerical) – found by playing on calculator!
!" p
!" p+1
!x p
!x p+1
# $ F = 4.66920....
" # F = 2.59029.....
- these are "universal" numbers ! F , " F
- same for many such maps – originally thought for all maps.
actually turn out to be for all quadratic maps. (Find why this is so.)
11
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 11-12 - S C Chapman
- this is a 'universality class' (show this next).
First note since ! F , " F are constants
show next
there is a termination to the sequence at !" .
beyond this – global chaos.
( ! bifurcations, ! close together)
+ some interesting islands of periodic 'superstable' behaviour.
What is !" ?
We have
(just depends on ! F and a const.)
$! p
! p # ! p #1
=
= "F
$! p +1 ! p +1 # ! p
and !" = lim ! p
p#"
now
!" p =
So we write
!" p#1
$F
=
!" p#2
$ F2
, etc.
!# = ! p + $! p +1 + $! p + 2 + ....
= !p +
$! p $! p
+ 2 + ....
"F
"F
but ! F > 1 so roughly (lowest order in ! F )
#! p
#
!" ! ! p +
= ! p + p0
$F
$F
or
! p = !" #
C
C = some const, ie: ! 0
$ Fp
thus !" depends on details of the maps, ie: C- Not universal.
Universal behaviour is the period doubling ( ! F ," F ) , sequence and the existence of some !" .
Other universality classes:
There is a family of maps
f ( x) = 1! a x
- different ! F , " F one for each q.
12
q