Chaos and the Logistic Map

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Chaos and the Logistic Map

• PHYS220 2004
• by Lesa Moore
• DEPARTMENT OF PHYSICS

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Different Types of Growth
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A Population Example

• Every generation, the population of fish in a
  lake grows by 10.
• Nn is the population of generation n.
• r1.1 is the constant growth rate.
• The difference equation is Nn1rNn.
• The population sequence for N1100 is 100, 110,
  121, 133, 146, 161,

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The Analytical Solution

• The rate of change of population N is
• Separating the variables
• And integrating both sides

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Final Steps

• Exponentiating both sides
• Yields
• This example exhibits geometric growth and the
  analytic solution is an exponential function.

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These Systems are Predictable

• Arithmetic, quadratic and geometric 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 t0 using an analytical
  formula.
• Predictable for bank loans, filling a water tank,
  a simple pendulum.

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Linearity

• Linear systems are easy to understand double the
  input yields double the output.

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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.

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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.

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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 NK ? NltK ?
• Will it overshoot and settle back down?
• Will it go into an oscillation?
• Will it do something else?

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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
• Think of x as a fraction of possible
  population.

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e.g.

• Suppose that for Australia, K 100,000,000.
• If the current population is Nn 20,000,000
  then
• Of course, 0 xn 1 always
• and the remaining capacity is 1 - xn.

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The Logistic Difference Equation

• Assume that the growth rate is not constant but
  proportional to the
  remaining capacity
• 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 n1 is
  given by xn1 r (1-xn)xn .

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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 r0, population goes
  straight to zero rlt0 as cannot have a negative
  population).

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The Growth Rate Term

• If you multiply existing population xn by 1, you
  get back the same population (stable).
• If r (1-xn) lt 1, the population will decrease.
• If r (1-xn) gt 1, the population will increase.
• Is there an upper bound to r ?

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Lets try r1.5 Growth rate is
1.5(1-xn)
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Population reaches equilibrium

• When the growth rate is equal to 1.5 times the
  remaining population, saturation pushes the
  population into equilibrium at x0.33.
• Is equilibrium a normal condition for all values
  of r ?
• We have used an initial population fraction of
  x00.1. What if we change the initial population?

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Next try r2.8 Growth rate is 2.8(1-xn)
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An Attractor

• It appears that no matter what initial population
  x0 we start with, the population reaches the same
  equilibrium value (after transients die out) for
  r2.8.
• When a population settles like this, for any
  starting value, the eventual behaviour is known
  as an attractor.

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r3.14 Growth rate is 3.14(1-xn)
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r3.45 Growth rate is 3.45(1-xn)
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r3.45 4-cycle
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r3.8 Growth rate is 3.8(1-xn)
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Attractors

• Attractors have different behaviours and values
  depending on value of r.

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Mapping the Attractor

• It can be shown mathematically that r4 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).

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The Spreadsheet Formula
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Fill the Spreadsheet
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Only plot data after transients have died out
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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 r3.8, for example, the values
  for x00.1 and x00.2 are very different at later
  times.

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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!!

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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.

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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.

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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).

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Period-Doubling

• Constant gt period-two gt period-4 gt period-8 gt gt
  chaos gt
• Bifurcations mark the transition from order into
  chaos.
• Bifurcations follow a pattern, occurring closer
  and closer together, ad infinitum.
• Look at their relative separations

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this length this length
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Feigenbaums Constant

• Feigenbaums constant is
• The Feigenbaum point is at r3.5699456

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Universality in Chaos

• Feigenbaums number is observed in all chaotic
  systems.
• Measured in physical systems
• Dripping taps.
• Oscillation of liquid helium.
• Fluctuation of gypsy moth populations.

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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.

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Attractor of Hénon

• Make two columns, one for x and one for y values.
• Can choose (0,0)
  as starting point.
• Generate subsequent
  rows using formulae
• Changing parameters
  a and b will generate
  different attractors.

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Attractor with parametersa7/5, b3/10
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The 3-lane feature
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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.

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Examples of Chaos

• Laser instabilities.
• Fluid turbulence.
• Progression to heart attack.
• Population biology.
• Weather.

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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.

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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.

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Acknowledgements

• This presentation was based on lecture material
  for PHYS220 presented by Prof. Barry Sanders,
  2000-2003.
• Additional References
• Peitgen, Jürgens Saupe, Chaos and Fractals New
  Frontiers of Science, 1992.
• Gleick, Chaos Making a New Science, 1987.