PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 9- S C Chapman
Maps and chaos
So far have discussed continuous 1st, 2nd order DE
- found trajectories in phase space are "well behaved"
dq
dt
trajectories cannot cross
q
only possibility for divergence of trajectories.
.
dq
dt
.
eg: at saddle
q
This does not permit stochastic (mixing up/shuffling) diffusion in phase space. This is only possible
(in plane of paper) if we add another dimension(out of plane of paper) ie in 3 or more dimensions.
1
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 9- S C Chapman
Now look at simplest systems that show route to chaos:
Difference equations
i)
ii)
iii)
iv)
Show how they "work" – what are their properties; example of chaos.
Show how they are fundamentally different to the analogous differential equations.
Quantify the route to chaos – Lyapunov exponents.
Introduce Universality – the bifurcation route to chaos, Feigenbaum numbers.
M(x)
1D Noninvertible maps – the easiest place to start.
Piecewise linear 1D maps
xn+1
- the tent map
x
xn+1
1
= 1 ! 2 xn !
2
1
! M(x)
Noninvertible
2 values of xn+1
per value of
xn + 1
- tells us how x + 1th step depends on
xth step
1
2
Write as:
xn+1 =
2xn
2(1! xn )
1
2
1
xn #
2
xn "
xn
What happens when we iterate the map?
If (i)
xn < 0
xn+1 < 0 and xn+1 > xn
ie: x ! "#
If (ii)
xn > 1
xn+1 < 0, so xn+2 < 0
as in (i) ie: x ! "# .
xn+2 > xn+1
So, just consider the interval x = !" 0,1 #$ - iterates of x are bounded in this range.
We can still look for fixed points, linearise to examine stability as before → local behaviour.
Interesting difference will be in global behaviour of maps.
2
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 9- S C Chapman
Fixed points are where xn doesn't change as n ! " so
xn+1 = xn is fixed point
M (x)= x
=x
Graphically:
xn+1
xn+1 = xn line
•1
•
x
•
•
xn
1
2
x
1
ie:
! xn ! 1
2
x = 2 (1 ! x )
Fixed point is in range
so fixed point is
also in range 0 ! xn !
1
2
xn+1 = 2 (1 ! xn )
x = 2 ! 2x
2
x=
3
fixed point
x = 0 here
xn = x + ! xn
xn+1 = x + ! xn+1
Stability/classification. Can still write
discontinuous.
xn+1 = 2( 1! xn )
sub in to
(
x + !xn+1 = 2 1! x ! !xn
x=
2
3
! xn+1 = 2 ! 3 x ! 2! xn
= !2 ! xn
3
xn+1 = 2 xn
)
ie: ! x is small but now
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 9- S C Chapman
So
! xn+1 = !2 ! xn
also
! xn = !2! xn!1
so
!xn+1 = !2.2 !xn!1 = !2.2.2 !xn!2
= "# !2 $%
j+1
!xn! j = "# !2 $%
n+1
!x0 where ! x0 is the initial condition
Hence, the fixed point is unstable – oscillates.
x = 0,!xn+1 = 2 !xn - unstable
Similarly,
[ xN +1 = 2 xN ]
Consider global behaviour.
Iterate many times
xn+1 = M ( xn )
Look at one interate:
(some notation here)
two iterates:
xn+2 = M 2 ( xn )
= M ( M ( xn ))
xn+ p = M p ( xn )
p iterates
2x
M(x) =
2( 1! x )
where
1
2
1
x#
2
x"
Fixed points – notice that
M
4
M2
2
•
•
•
No of fixed points doubles each iterate.
4
•
•
•
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 9- S C Chapman
Another way of looking at M p ( x ) graphically:
M (x)
xn+1 = xn
•
•
•
•
•
.
•
•
•
•
1
•
2
4
3
0 5
vertical lines are M ( xn )
xn+1 = M ( xn )
horizontal
note that the iterates are ‘shuffled’
5
4
2
0
1
3