Dynamics, Chaos, and Prediction
Aristotle, 384 – 322 BC
Nicolaus Copernicus, 1473 – 1543
Galileo Galilei, 1564 – 1642
Johannes Kepler, 1571 – 1630
Isaac Newton, 1643 – 1727
Pierre- Simon Laplace, 1749 – 1827
Henri Poincaré, 1854 – 1912
Werner Heisenberg, 1901 – 1976
• Dynamical Systems Theory:
– The general study of how systems change over time
• Calculus
• Differential equations
• Discrete maps
• Algebraic topology
• Vocabulary of change
Isaac Newton
1643 – 1727
• The dynamics of a system: the manner in which
the system changes
• Dynamical systems theory gives us a vocabulary
and set of tools for describing dynamics
• Chaos:
– One particular type of dynamics of a system
– Defined as “sensitive dependence on initial
conditions”
– Poincaré: Many-body problem in the solar system
Henri Poincaré
1854 – 1912
Dr. Ian Malcolm
“You've never heard of Chaos theory? Non-linear equations?
Strange attractors?”
Dr. Ian Malcolm
“You've never heard of Chaos theory? Non-linear equations?
Strange attractors?”
Chaos in Nature
• Dripping faucets
• Heart activity (EKG)
• Electrical circuits
• Computer networks
• Solar system orbits
• Population growth and
dynamics
• Weather and climate (the
“butterfly effect”)
• Brain activity (EEG)
• Financial data
What is the difference between chaos and randomness?
What is the difference between chaos and randomness?
Notion of “deterministic chaos”
A simple example of deterministic chaos:
Exponential versus logistic models for population growth
nt 1  2nt

Exponential model: Each year each pair of parents mates,
creates four offspring, and then parents die.
Linear Behavior
nt 1  2nt

Linear Behavior:
The whole is the sum of the parts
Linear Behavior:
The whole is the sum of the parts
Linear: No interaction among the offspring, except pair-wise mating.
Linear Behavior:
The whole is the sum of the parts
Linear: No interaction among the offspring, except pair-wise mating.
More realistic: Introduce limits to population growth.
Logistic model
• Notions of:
– birth rate
– death rate
– maximum carrying capacity k
(upper limit of the population that the habitat will support,
due to limited resources)
Logistic model
• Notions of:
– birth rate

– death rate
n t 1  birthrate  n t  deathrate n t
 (b  d)n t
– maximum carrying capacity k
(upper limit of the population that the habitat will support due to
limited resources)
k  n t 
n t 1  (b  d)n t 

 k 
kn  n 2 
t
 (b  d) t

 k

interactions between offspring make this model nonlinear
Logistic model
• Notions of:
– birth rate

– death rate
n t 1  birthrate  n t  deathrate n t
 (b  d)n t
– maximum carrying capacity k
(upper limit of the population that the habitat will support due to
limited resources)
k  n t 
n t 1  (b  d)n t 

 k 
kn  n 2 
t
 (b  d) t

 k

interactions between offspring make this model nonlinear
Nonlinear Behavior
nt 1  (birthrate  deathrate)[knt  nt 2 ]/k

Nonlinear behavior of logistic model
birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)
Nonlinear behavior of logistic model
Nonlinear: The whole is different than the sum of the parts
birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)
Logistic map
xt 1  Raaa
xt (1 xt )
Lord Robert May
b. 1936

Mitchell Feigenbaum
b. 1944
2
n t 1  (birthrate  deathrate)[knt  n t ]/ k
Let x t  n t / k
Let R  birthrate  deathrate
Then x t 1  Rx t (1  x t )
LogisticMap.nlogo
1. R = 2
2. R = 2.5
Notion of period doubling
3. R = 2.8
Notion of “attractors”
4. R = 3.1
5. R = 3.49
6. R = 3.56
7. R = 4, look at sensitive dependence on initial conditions
Bifurcation Diagram
Period Doubling and Universals in Chaos
(Mitchell Feigenbaum)
R1 ≈ 3.0:
R2 ≈ 3.44949
R3 ≈ 3.54409
R4 ≈ 3.564407
R5 ≈ 3.568759
period 2
period 4
period 8
period 16
period 32
R∞ ≈ 3.569946 period ∞
(chaos)
Period Doubling and Universals in Chaos
(Mitchell Feigenbaum)
R1 ≈ 3.0:
R2 ≈ 3.44949
R3 ≈ 3.54409
R4 ≈ 3.564407
R5 ≈ 3.568759
period 2
period 4
period 8
period 16
period 32
R∞ ≈ 3.569946 period ∞
(chaos)
A similar “period doubling
route” to chaos is seen in any
“one-humped (unimodal)
map.
Period Doubling and Universals in Chaos
(Mitchell Feigenbaum)
R1 ≈ 3.0:
R2 ≈ 3.44949
R3 ≈ 3.54409
R4 ≈ 3.564407
R5 ≈ 3.568759
period 2
period 4
period 8
period 16
period 32
R∞ ≈ 3.569946 period ∞
(chaos)
Rate at which distance between
bifurcations is shrinking:
Period Doubling and Universals in Chaos
(Mitchell Feigenbaum)
R1 ≈ 3.0:
R2 ≈ 3.44949
R3 ≈ 3.54409
R4 ≈ 3.564407
R5 ≈ 3.568759
period 2
period 4
period 8
period 16
period 32
R∞ ≈ 3.569946 period ∞
(chaos)
Rate at which distance between
bifurcations is shrinking:
R2  R1
3.44949  3.0

 4.75147992
R3  R2 3.54409  3.44949
R3  R2
3.54409  3.44949

 4.65619924
R4  R3 3.564407  3.54409
R4  R3
3.564407  3.54409

 4.66842831
R5  R4 3.568759  3.564407
M
 R  R 
n
lim  n 1
 4.6692016
n   R
n 2  Rn 1 
Period Doubling and Universals in Chaos
(Mitchell Feigenbaum)
In other words, eachRate
new
at bifurcation
which distanceappears
betweenabout
R1 ≈ 3.0: 4.6692016
periodtimes
2
faster
than the
previous one.
bifurcations
is shrinking:
R2 ≈ 3.44949 period 4
R2  R1
3.44949  3.0

 4.75147992
R3 ≈ 3.54409 period 8
R3  R2 3.54409  3.44949
R4 ≈ 3.564407 period 16
R5 ≈ 3.568759 period 32
R3  R2
3.54409  3.44949
R4  R3
R∞ ≈ 3.569946 period ∞
(chaos)

3.564407  3.54409
 4.65619924
R4  R3
3.564407  3.54409

 4.66842831
R5  R4 3.568759  3.564407
M
 R  R 
n
lim  n 1
 4.6692016
Rn 2  Rn 1 
Period Doubling and Universals in Chaos
(Mitchell Feigenbaum)
In other words, eachRate
new
at bifurcation
which distanceappears
betweenabout
R1 ≈ 3.0: 4.6692016
periodtimes
2
faster
than the
previous one.
bifurcations
is shrinking:
R2 ≈ 3.44949 period 4
R2  R1
3.44949  3.0

 4.75147992
R3 ≈ 3.54409 period 8
R3  R2 3.54409  3.44949
R4 ≈ 3.564407 period 16
R5 ≈ 3.568759
period
32of 4.6692016
R3  R2 occurs
3.54409
any
3.44949
This same
rate
in
unimodal

 4.65619924
R4  R3 3.564407  3.54409
map.
R∞ ≈ 3.569946 period ∞
(chaos)
R4  R3
3.564407  3.54409

 4.66842831
R5  R4 3.568759  3.564407
M
 R  R 
n
lim  n 1
 4.6692016
Rn 2  Rn 1 
Significance of dynamics and chaos
for complex systems
Significance of dynamics and chaos
for complex systems
• Apparent random behavior from deterministic rules
Significance of dynamics and chaos
for complex systems
• Apparent random behavior from deterministic rules
• Complexity from simple rules
Significance of dynamics and chaos
for complex systems
• Apparent random behavior from deterministic rules
• Complexity from simple rules
• Vocabulary of complex behavior
Significance of dynamics and chaos
for complex systems
• Apparent random behavior from deterministic rules
• Complexity from simple rules
• Vocabulary of complex behavior
• Limits to detailed prediction
Significance of dynamics and chaos
for complex systems
• Apparent random behavior from deterministic rules
• Complexity from simple rules
• Vocabulary of complex behavior
• Limits to detailed prediction
• Universality