# chaos

W
Shared by:
Categories
Tags
-
Stats
views:
7
posted:
1/23/2012
language:
English
pages:
51
Document Sample

Chaos and the Logistic Map

PHYS220 2004
by Lesa Moore
DEPARTMENT OF PHYSICS

Macquarie University   1
Different Types of Growth
Arithmetic Growth                 y = 2x + 10

70

60

50

40
y

30

20

10

0
0   5    10           15             20     25          30
x
Macquarie University                           2
y = x 2 - 2E-13x

1200

1000

800
y(x)

600

400

200

0
0   5   10       15         20    25       30       35
x

Macquarie University                            3
Cyclic Growth & Decay

1.5

1

0.5
y(x)

0
0   1      2         3            4   5   6   7
-0.5

-1

-1.5
x

Macquarie University                   4
Geometric Growth                  0.4055x
y=e
45

40
35
30

25
y(x)

20
15
10
5
0
0   2         4                 6    8              10
x

Macquarie University                            5
A Population Example
   Every generation, the population of fish
in a lake grows by 10%.
   Nn is the population of generation n.
   r=1.1 is the constant growth rate.
   The difference equation is: Nn+1=rNn.
   The population sequence for N1=100 is:
100, 110, 121, 133, 146, 161, …

Macquarie University      6
The Analytical Solution
   The rate of change of population N is:
dN
 N
dt
dN
   Separating the variables:                   dt
N
   And integrating both sides:                N
dN '
t

 N '   dt '
N0       0

 N 
 N   t
ln    
 0
Macquarie University                        7
Final Steps
   Exponentiating both sides:
 N  t

N e
 0
t
   Yields:        N  N 0e

   This example exhibits geometric growth and
the analytic solution is an exponential
function.
Macquarie University              8
These Systems are Predictable
growth, and cyclic growth and decay
are predictable systems with analytical
solutions.
   The state x(t) at time t may be
predicted from the state at time t=0
using an analytical formula.
   Predictable for bank loans, filling a
water tank, a “simple” pendulum.
Macquarie University       9
Linearity
   Linear systems are easy to understand:
double the input yields double the
output.

Macquarie University   10
Unpredictability
   Not all systems are predictable.
   Some systems have no analytical
solutions.
   We now consider a different type of
growth, known as “logistic” growth,
which we will see is not predictable.
   This system is an example of nonlinear
dynamics.
Macquarie University       11
Logistic Growth
   Describes the behaviour of a population
that has limited resources
(food, water, space).
   Growth of the population is limited by a
carrying capacity K.
   The population increases, but becomes
saturated as it gets closer to the
carrying capacity forcing the rate of
growth to decrease.
Macquarie University     12
Effect of the Limit
   We want to know how the population N
behaves when it gets “close” to the
carrying capacity K.
   Will it level off and stabilise at N=K ?
N<K ?
   Will it overshoot and settle back down?
   Will it go into an oscillation?
   Will it do something else?
Macquarie University     13
Logistic Growth Variables
   How can we model this in Excel®?
   Consider a population N and saturation level K
such that 0 ≤ N ≤ K.
   Also introduce a variable x where:
N
x
K
   Think of x as a “fraction of possible population”.
Macquarie University        14
e.g.
   Suppose that for Australia, K = 100,000,000.
   If the current population is Nn = 20,000,000
then:
Nn 1
xn      0.2
K 5

   Of course, 0 ≤ xn ≤ 1 always
and the remaining capacity is 1 - xn.
Macquarie University      15
The Logistic
Difference Equation
   Assume that the growth rate is not constant
but proportional to the
remaining capacity:             r  1  xn
   Growth rate term is now r (1-xn).
   For small xn growth rate is ~r.
   For large xn growth rate is ~0.
   Population from generation n to generation
n+1 is given by: xn+1 = r (1-xn)xn .

Macquarie University      16
What is r ?
   r remains as a parameter in the growth
rate term {r (1-xn) }, but r itself is a
variable.
   Its lower bound is zero (if r=0,
population goes straight to zero; r<0 as
cannot have a “negative” population).

Macquarie University     17
The Growth Rate Term
   If you multiply existing population xn by
1, you get back the same population
(stable).
   If r (1-xn) < 1, the population will
decrease.
   If r (1-xn) > 1, the population will
increase.
   Is there an upper bound to r ?

Macquarie University     18
Let’s try r=1.5
Growth rate is 1.5(1-xn)
r = 1.5, N(0)=0.1, x(0)=0.1

4
3.5
3
Population

2.5
Geometric (N)
2
Logistic (x=N/K)
1.5
1
0.5
0
0   1   2    3     4    5    6   7      8   9
Generations

Macquarie University                              19
Population reaches equilibrium
   When the growth rate is equal to 1.5
times the remaining population,
saturation pushes the population into
equilibrium at x=0.33.
   Is equilibrium a normal condition for all
values of r ?
   We have used an initial population
fraction of x0=0.1. What if we change
the initial population?
Macquarie University     20
Next try r=2.8
Growth rate is 2.8(1-xn)
r = 2.8

0.8
0.7
0.6
0.5                                        x(0)=0.1
x(n)

0.4                                        x(0)=0.2
0.3                                        x(0)=0.3
0.2
0.1
0
0   5   10        15       20   25
n

Macquarie University                    21
An Attractor
   It appears that no matter what initial
population reaches the same
equilibrium value (after transients die
out) for r=2.8.
   When a population settles like this, for
any starting value, the eventual
behaviour is known as an attractor.
Macquarie University        22
r=3.14
Growth rate is 3.14(1-xn)
r = 3.14

0.9
0.8
0.7
0.6
x(0)=0.1
0.5
x(n)

x(0)=0.2
0.4
x(0)=0.3
0.3
0.2
0.1
0
0   5   10        15       20   25
n

Macquarie University                    23
r=3.45
Growth rate is 3.45(1-xn)
r = 3.45

1
0.9
0.8
0.7
0.6
x(0)=0.2
x(n)

0.5
x(0)=0.3
0.4
0.3
0.2
0.1
0
0   5 10 15 20 25 30 35 40 45 50 55
n

Macquarie University                24
r=3.45
4-cycle
r = 3.45

1
0.9
0.8
0.7
0.6
x(n)

0.5                                           x=0.2
0.4
0.3
0.2
0.1
0
0   5   10 15 20 25 30 35 40 45 50 55
n

Macquarie University               25
r=3.8
Growth rate is 3.8(1-xn)
r = 3.8

1
0.9
0.8
0.7
0.6
x(0)=0.2
x(n)

0.5
x(0)=0.3
0.4
0.3
0.2
0.1
0
0   20   40        60       80   100
n

Macquarie University                     26
Attractors
   Attractors have different behaviours and
values depending on value of r.

r = 2.8     r = 3.14          r = 3.45       r = 3.8

equilibrium 2-cycle           4-cycle        aperiodic
constant    oscillates        oscillates     appears
random
Macquarie University               27
Mapping the Attractor
   It can be shown mathematically that
r=4 is a limit for this model.
   Can we create a map in Excel® that
displays the long-term behaviour of the
attractor for 0 ≤ r ≤ 4 ?
   For each r, we can plot a sequence of
values of xn for large n (after transients
have died out).
Macquarie University      28

A       B             C                D         E         F
1    n              r           0               0.1       0.2       0.3
2        0       x(0)         0.1               0.1       0.1       0.1
3        1              =C\$1*(1-C2)*C2
4        2
5        3
6        4

Macquarie University                       29

A       B          C             D        E      F
1   n              r         0        0.1      0.2      0.3
2       0       x(0)       0.1        0.1      0.1      0.1
3       1                    0     0.009    0.018    0.027
4       2                    0 0.000892 0.003535 0.007881
5       3                    0 8.91E-05 0.000705 0.002346
6       4                    0 8.91E-06 0.000141 0.000702
7       5                    0 8.91E-07 2.82E-05 0.00021
8       6                    0 8.91E-08 5.63E-06 6.31E-05

Macquarie University              30
A    B      C       D          E          F         G        H         I       J        K
1    n       r     0        0.5          1        1.5        2      2.5          3    3.5         4
2     0   x(0)   0.1        0.1        0.1        0.1      0.1      0.1        0.1    0.1       0.1
3     1            0      0.045       0.09      0.135     0.18    0.225       0.27  0.315      0.36
4     2            0   0.021488     0.0819   0.175163   0.2952 0.435938   0.5913 0.755213   0.9216
5     3            0   0.010513   0.075192   0.216721 0.416114 0.61474 0.724993 0.647033 0.289014
6     4            0   0.005201   0.069538   0.254629 0.485926 0.592087 0.598135 0.799335 0.821939
7     5            0   0.002587   0.064703    0.28469 0.499604   0.6038 0.721109 0.561396 0.585421
8     6            0    0.00129   0.060516   0.305462      0.5 0.598064 0.603333 0.861807 0.970813
9     7            0   0.000644   0.056854   0.318233      0.5 0.600959 0.717967 0.416835 0.113339
10    8            0   0.000322   0.053622   0.325441      0.5 0.599518 0.607471 0.850793 0.401974
11    9            0   0.000161   0.050746   0.329294      0.5 0.60024 0.71535 0.444306 0.961563
12   10            0   8.04E-05   0.048171   0.331289      0.5 0.59988 0.610873 0.864144 0.147837
13   11            0   4.02E-05   0.045851   0.332305      0.5 0.60006 0.713121 0.410898 0.503924
14   12            0   2.01E-05   0.043749   0.332818      0.5 0.59997 0.613738 0.847213 0.999938
15   13            0   1.01E-05   0.041835   0.333075      0.5 0.600015 0.711191 0.453051 0.000246
16   14            0   5.03E-06   0.040084   0.333204      0.5 0.599992 0.616195 0.867285 0.000985
17   15            0   2.51E-06   0.038478   0.333269      0.5 0.600004 0.709496 0.402856 0.003936
18   16            0   1.26E-06   0.036997   0.333301      0.5 0.599998 0.618334 0.84197 0.015682
19   17            0   6.28E-07   0.035628   0.333317      0.5 0.600001 0.707991 0.465697 0.061745
20   18            0   3.14E-07   0.034359   0.333325      0.5      0.6 0.620219 0.870882 0.23173
21   19            0   1.57E-07   0.033178   0.333329      0.5      0.6 0.706642 0.393564 0.712124
22   20            0   7.85E-08   0.032078   0.333331      0.5      0.6 0.621897 0.83535 0.820014
23   21            0   3.93E-08   0.031049   0.333332      0.5      0.6 0.705423 0.481392 0.590364
24   22            0   1.96E-08   0.030085   0.333333      0.5      0.6 0.623404 0.873788 0.967337
25   23            0   9.82E-09    0.02918   0.333333      0.5      0.6 0.704315 0.385989 0.126384
26   24            0   4.91E-09   0.028328   0.333333      0.5      0.6 0.624767 0.829505 0.441645
27   25            0   2.45E-09   0.027526   0.333333      0.5      0.6   0.7033 0.494993 0.986379
28   26            0   1.23E-09   0.026768   0.333333      0.5      0.6 0.626008 0.874912 0.053742
29   27            0   6.14E-10   0.026051   0.333333      0.5      0.6 0.702366 0.383043 0.203415
30   28            0   3.07E-10   0.025373   0.333333      0.5      0.6 0.627144 0.827124 0.64815
31   29            0   1.53E-10   0.024729   0.333333      0.5      0.6 0.701503 0.500466 0.912207
32   30            0   7.67E-11   0.024117   0.333333      0.5      0.6 0.628189 0.874999 0.320342
33   31            0   3.84E-11   0.023536   0.333333      0.5      0.6 0.700703 0.382814 0.870893
34   32            0   1.92E-11   0.022982   0.333333      0.5      0.6 0.629155 0.826936 0.449754
Only plot   35   33            0   9.59E-12   0.022454   0.333333      0.5      0.6 0.699957 0.500894 0.989902

}
36   34            0   4.79E-12    0.02195   0.333333      0.5      0.6 0.630052 0.874997 0.039986

data after   37
38
39
35
36
37
0
0
0
2.4E-12
1.2E-12
5.99E-13
0.021468
0.021007
0.020566
0.333333
0.333333
0.333333
0.5
0.5
0.5
0.6 0.69926 0.38282 0.153548
0.6 0.630887 0.826941 0.519885
0.6 0.698606 0.500884 0.998418

transients   40
41
42
38
39
40
0
0
0
3E-13
1.5E-13
7.49E-14
0.020143
0.019737
0.019347
0.333333
0.333333
0.333333
0.5
0.5
0.5
0.6 0.631667 0.874997 0.006317
0.6 0.697991 0.38282 0.025107
0.6 0.632398 0.826941 0.097905

have    43
44
45
41
42
43
0
0
0
3.75E-14
1.87E-14
9.36E-15
0.018973
0.018613
0.018267
0.333333
0.333333
0.333333
0.5
0.5
0.5
0.6 0.697412 0.500884 0.353278
0.6 0.633086 0.874997 0.913891
0.6 0.696865 0.38282 0.314778

died out   46
47
44
45
0
0
4.68E-15
2.34E-15
0.017933
0.017611
0.333333
0.333333
0.5
0.5
0.6 0.633733 0.826941 0.862771
0.6 0.696347 0.500884 0.473588

Macquarie University                                                   31
The Logistic Map

1.2

1

0.8
The Attractor

0.6

0.4

0.2

0
0   1   2   3   3.1      3.2     3.48      3.553   3.59   3.69   3.79   3.89        3.99
r

Macquarie University                                     32
The Logistic Map

1

0.9

0.8

0.7

0.6
The Attractor

0.5

0.4

0.3

0.2

0.1

0
0   1   2   3   3.1     3.2     3.48      3.553   3.59   3.69   3.79   3.89        3.99
r

Macquarie University                                    33
The Logistic Map
   The Logistic Map looks the same for all
values of starting population fraction x0
(because the whole map is an attractor,
and we are looking at the long-term
behaviour).
   But if we look at r=3.8, for example,
the values for x0=0.1 and x0=0.2 are
very different at later times.
Macquarie University     34
Sensitive Dependence on
Initial Conditions (SDOIC)
   A small difference in the value of r or x0
can make a huge difference in the
outcome of the system at generation n
(“butterfly effect”).
   No formula can tell us what x will be at
some specified generation n even if we
know the initial conditions.
   The system is unpredictable!!

Macquarie University          35
Stephen Hawking:
   “We already know the physical laws
that govern everything we experience
in everyday life … It is a tribute to how
far we have come in theoretical physics
that it now takes enormous machines
and a great deal of money to perform
an experiment whose results we cannot
predict.”
Macquarie University     36
CHAOS
   The attractor branches into two, then four,
then eight and so on. The sequence follows a
geometric progression, but soon looks like a
mess.
   Messy regions are cyclically interspersed with
clear “windows”.
   Existence of period-3 windows implies chaos.

Macquarie University    37
Features of Chaos
   Period 3 region.
   Chaotic systems show self-similarity or
fractal behaviour.
   SDOIC – points that start off close
together can be widely separated at a
later time (also referred to as “mixing”).

Macquarie University      38
Period-Doubling
   Constant > period-two > period-4 >
period-8 > … > chaos > …
   Bifurcations mark the transition from
order into chaos.
   Bifurcations follow a pattern, occurring
closer and closer together, ad infinitum.
   Look at their relative separations…

Macquarie University     39
Measuring Feigenbaum's Number

1
0.9
0.8
0.7            r1
r2
0.6
0.5                                                           r3
0.4
0.3
0.2
0.1                                        this length    this length
0
0       1                 2                 3                       4
r
Macquarie University                             40
Feigenbaum’s Constant
   Feigenbaum’s constant is:
rn  rn 1
lim              4.6692016609 10299097 ...
n 1  rn
n  r

   The Feigenbaum point is at
r=3.5699456…
Macquarie University   41
Universality in Chaos
   Feigenbaum’s number is observed
in all chaotic systems.
   Measured in physical systems:
   Dripping taps.
   Oscillation of liquid helium.
   Fluctuation of gypsy moth populations.

Macquarie University       42
Another Chaotic System
   The logistic map is a quadratic map in
one dimension – the one variable is
x(r).
   Chaos can involve multi-dimensional
systems.
   An example is the mapping that
generates the attractor of Hénon.

Macquarie University       43
Attractor of Hénon
   Make two columns, one for x and one for y
values.
   Can choose (0,0)
as starting point.    xn 1  yn  axn  1
2

Generate subsequent y
n 1  bxn

rows using formulae:
   Changing parameters a  7 / 5
a and b will generate
different attractors. b  3 / 10

Macquarie University          44
Attractor with parameters
a=7/5, b=3/10
Attractor of Henon

0.5

0.4

0.3

0.2

0.1

0
y

-1.5      -1   -0.5                0         0.5   1   1.5

-0.1

-0.2

-0.3

-0.4

-0.5
x
Macquarie University             45
The 3-lane feature
Attractor of Henon

0.3

0.25

0.2

0.15
y

0.1

0.05

0
0   0.1   0.2   0.3   0.4          0.5       0.6   0.7   0.8   0.9          1
x
Macquarie University                            46
Chaos is Everywhere
   Perfect systems may be easily modelled
according to the “laws of physics”: with
the massless ropes, frictionless surfaces
and perfect vacuum of physics text-
book problems.
   Real systems have friction, air-
resistance and physical variations that
make them unpredictable.
Macquarie University     47
Examples of Chaos
   Laser instabilities.
   Fluid turbulence.
   Progression to heart attack.
   Population biology.
   Weather.

Macquarie University   48
Bifurcation = Branching
   Branching is important for life:
   Trees, but also blood vessels, nerves.
   Clones are not identical:
   Branches are not pre-determined;
   DNA codes for branching capability;
   Makes the code economical.
   Non-living systems – lightning,
snowflakes.
Macquarie University       49
Landmark Publications
   Lorentz, Edward N., Deterministic Nonperiodic Flow,
J. Atmos. Sci. 20 (1963) 130-141.
   Li, Tien-Yien & Yorke, James A., Period 3 Implies
Chaos, American Mathematical Monthly 82 (1975)
343-344.
   Hénon, Michel, A two-dimensional mapping with a
strange attractor, Comm. Math. Phys. 50 (1976) 69-
77.
   May, Robert M., Simple mathematical models with
very complicated dynamics, Nature 261 (1976) 459-
467.
   Feigenbaum, Mitchell J., Quantitative universality for
a class of nonlinear transformations, J. Stat. Phys. 19
(1978) 25-52.
   Mandelbrot, Benoit B., Fractal aspects of the iteration
of z → lz(1-z) for complex l and z, Annals NY Acad.
Sciences 357 (1980) 249-257.
Macquarie University              50
Acknowledgements
   This presentation was based on lecture
material for PHYS220 presented by
Prof. Barry Sanders, 2000-2003.
   Peitgen, Jürgens & Saupe, Chaos and
Fractals: New Frontiers of Science, 1992.
   Gleick, Chaos: Making a New Science,
1987.
Macquarie University          51

Related docs
Other docs by huanghengdong
ME6105_Homework_4