, we can solve
for 
uniquely. If this is so we can define an inverse mapping
such that 
PH3160 Non-Linear Phenomena and Chaos
1-D MAPS (continued)
INVERTIBLE MAPS
It is important to distinguish between maps that are
noninvertible and maps that are invertible.
A map M is invertible if, given , we can solve
for uniquely.
If this is so we can
define an inverse mapping such that
It turns out that chaos is possible even in a 1D map provided
it is non-invertible.
If the map is invertible chaos can occur only if it is at
least 2D.
There is a close analogy between linear difference eqns and
linear differential eqns which does not exist when the eqns
are nonlinear. ( see eg. Chapt 7 of Rowlands Examples 1 and 2 )
The prototype 1D map that helped enormously to guide early work
concerning the discovery of details of chaotic behaviour is the :
LOGISTIC MAP
L a control parameter
-- illustrates features like attractors, repellors, period-doubling
and chaos.
It may be regarded as of normal form type since iteration of
any quadratic polynomial is equivalent to iteration of the
logistic map (with properly chosen parameters) ... see Ex 12.1
Clearly, it is a noninvertible map, since a sketch soon shows
that because of the parabolic shape there are two possible
values for any one choice of 
It is important to contrast the difference in behaviour of
solutions of the logistic map with those of the logistic ODE:
ie. dx/dt = L x(1 - x).
The latter are perfectly well-behaved and defined (see eg. the
discussion in Chapt 7 of Rowlands); whereas, as seen in Course
Work 1, the behaviour of the iterates of the logistic map are a
strong function of the control parameter L, and, unlike the
solutions of the differential eqn, the solutions for L > 3 become
unstable 
chaotic type.
OVERVIEW of the LOGISTIC MAP
Although the logistic map is commonly written:
the choice of symbols varies.
For 0 < L < 4 it defines a family of noninvertible mappings of the
interval from
1 into itself.
For L < 1 ... 
0 as 
for 1 < L < 3 ... non-zero steady state
There exists a cascade of bifurcations starting from the so-called
period-1 cycle ... which is just a constant value of x.
It illustrates period doubling bifurcations
starting with a bifurcation from
period-1 to period-2 when L = 3,
and
continuing with ...............period-2 to period-4 when
etc ... period -8, period-16 etc.
where convergence is essentially geometric
The values of L for which the behaviour is periodic form an infinite
number of finite intervals. In fact, between L=3 and L=4 periodic
regions of all periods can be found! The values of L for which it is
chaotic lie between 3.57 and 4.0 (note that the cobweb goes haywire
over this range) and form a Cantor set whose dimension appears to be 1.0.
If we substitute 
and 
into the
logistic map we obtain:
which is another convenient form of the logistic map eqn.
The first two period doubling bifurcations occur at C = -3/4 and -5/4.
The values 
where bifurcations from period 
to 
occur
converge to
while the ratios
converge 4.6692
a number discovered
by Feigenbaum to be characteristic of bifurcations occurring in a wide
class of mappings and flows.
Solutions of period-3 first appear at a saddle-node bifurcation at
C = -7/4.
When C and z are taken to be complex, where C = a + ib and
z = x + iy, the eqn can be written as:
These eqns generate the Mandelbrot set -- if one iterates 
with
z starting at 0 and vary C -- ie. the point (a,b) is in the set
if the sequence 
starting from (0,0) does not
These eqns also generate the Julia set -- if one iterates for a
fixed C and varies the z-values- ie. if display (x,y) for fixed C.
QUALITATIVE UNIVERSALITY
One of the most astonishing and unexpected results in all of nonlinear
dynamics is that:
Other unimodal maps (ie. those which are smooth, concave down and have
a single maximum) display the same qualitative features as those
of the logistic map -- although it is important to note that there are
quantitative differences.
As an example of this aspect compare the bifurcation diagram of the
sine map with that of the logistic map. The horizontal scale of the
sine map diagram needs scaling (renormalizing) by a factor of 4 since
the max of 
is r whereas that of r x(1-x) is r/4.
The qualitative features are identical: undergoing period-doubling
routes to chaos, with periodic windows interwoven with chaotic bands.
This behaviour forms part of what is generally true for unimodal maps,
for which the detailed algebraic form of f(x) is irrelevant.
The fact that the periodic attractors occur in the same sequence gives
rise to the name ... the universal or U-sequence.
BERNOULLI SHIFT MAPS
Perhaps the simplest deterministic system to go chaotic ( or exhibit
statistical mechanical behaviour, as it is sometimes referred, in
connection with this map ) is the 1-D Bernoulli shift map.
Consider 
(mod 1)
with intial condition 
e.g. a discrete map on the circle of unit circumference.
The solution is 
(mod 1)
where 
the initial condition must be specified.
Since 
as
for 
the origin here is a stable fixed point.
For D = 2, 3, 4 ... we have what are referred to as Bernoulli shifts.
The case D = 2, the binary shift, 
(mod 1)
has wide use in number theory and is given by:
If is rational then the orbit is periodic.
If is irrational,
there is no periodicity and we obtain a chaotic orbit.
Since there is an irrational number close to every rational number
(and vice versa), the map exhibits sensitive dependence on initial
conditions.
Question: What is the attractor for the Bernoulli shift -- with irrational
Assignment 12.
Review what was suggested/discussed/done for Assignment 6
The different Views allow you to examine the detailed paths
taken by the iteration on cobweb diagrams ... which makes clear
what is happening when the parameter is such that period-doubling
occurs. It also provides an overview of the fixed points and the
parameter L vs 
plot illustrates the Feigenbaum fig-tree type
bifurcations and regions of chaotic behaviour and a plot of the
Lyapunov exponents vs L value.
Note also that the Chaos Demos 1-D maps contain, in addition
to the logistic map, the following:
Sine , Sine-Squared, Cubic, Tent and Shift Maps.
Exercises and Examples 12.
1. Show that the Verhulst type eqn: 
is equivalent to: 
where 
and a = r + 1.
2. Consider the logistic map eqn in the form
By considering the second-iterate map (ie. f(f(x))) show
why a 2-cycle dosn't exist for L < 3.
3. Consider the tent map defined by the function:
for 0 < r < 2 and 0 < x < 1 .
Show that for r < 1 the only fixed point is an attractor
at the origin.
Show that for r > 1 there are two unstable fixed points.
Show that for r = 2 these two repellors are at 
and 2/3.
Show that the Lyapunov exponent 
(ie. that it is
independent of the initial condition ).
§§5.9 and 5.10 of Hilborn discuss the tent and Bernoulli maps.
MW - 15/1/99