dynamics, chaos, and prediction

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• 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 Poincaré1854 – 1912

Isaac Newton1643 – 1727

“You've never heard of Chaos theory? Non-linear equations? Strange attractors?”

Dr. Ian Malcolm

• Dripping faucets

• Electrical circuits

• Solar system orbits

• Weather and climate (the “butterfly effect”)

• Brain activity (EEG)

• Heart activity (EKG)

• Computer networks

• Population growth and dynamics

• Financial data

Chaos in Nature

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: No interaction among the offspring, except pair-wise mating.


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

– maximum carrying capacity k (upper limit of the population that the habitat will support due to

limited resources)€

nt +1 = birthrate × nt − deathrate× nt

= (b − d)nt

€

nt +1 = (b − d)ntk − nt

k ⎛ ⎝ ⎜

⎞ ⎠ ⎟

= (b − d)knt − nt

2

k

⎛ ⎝ ⎜

⎞ ⎠ ⎟

interactions between offspring make this model nonlinear

€

nt +1 = (birthrate − deathrate)[knt − nt2]/k

Nonlinear Behavior

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

birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)

Nonlinear: The whole is different than the sum of the parts

aaa

€

x t +1 = R x t (1− x t )

Logistic map

Lord Robert May b. 1936

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nt +1 = (birthrate − deathrate)[knt − nt2]/k

Let x t = nt /k

Let R = birthrate − deathrate

Then x t +1 = Rx t (1− x t )

Mitchell Feigenbaumb. 1944

LogisticMap.nlogo

1. R = 2

2. R = 2.5

3. R = 2.8

4. R = 3.1

5. R = 3.49

6. R = 3.56

7. R = 4, look at sensitive dependence on initial conditions

Notion of period doubling

Notion of “attractors”

Bifurcation Diagram

R1 ≈ 3.0: period 2R2 ≈ 3.44949 period 4R3 ≈ 3.54409 period 8R4 ≈ 3.564407 period 16R5 ≈ 3.568759 period 32

R∞ ≈ 3.569946 period ∞ (chaos)

Period Doubling and Universals in Chaos(Mitchell Feigenbaum)




A similar “period doubling route” to chaos is seen in any “one-humped (unimodal) map.




Rate at which distance betweenbifurcations is shrinking:





€

R2 − R1

R3 − R2

=3.44949 − 3.0

3.54409 − 3.44949= 4.75147992

R3 − R2

R4 − R3

=3.54409 − 3.44949

3.564407 − 3.54409= 4.65619924

R4 − R3

R5 − R4

=3.564407 − 3.54409

3.568759 − 3.564407= 4.66842831

M

limn → ∞

Rn +1 − Rn

Rn +2 − Rn +1

⎛ ⎝ ⎜

⎞ ⎠ ⎟≈ 4.6692016





€

R2 − R1

R3 − R2

=3.44949 − 3.0

3.54409 − 3.44949= 4.75147992

R3 − R2

R4 − R3

=3.54409 − 3.44949

3.564407 − 3.54409= 4.65619924

R4 − R3

R5 − R4

=3.564407 − 3.54409

3.568759 − 3.564407= 4.66842831

M

limRn +1 − Rn

Rn +2 − Rn +1

⎛ ⎝ ⎜

⎞ ⎠ ⎟≈ 4.6692016

In other words, each new bifurcation appears about 4.6692016 times faster than the previous one.





€

R2 − R1

R3 − R2

=3.44949 − 3.0

3.54409 − 3.44949= 4.75147992

R3 − R2

R4 − R3

=3.54409 − 3.44949

3.564407 − 3.54409= 4.65619924

R4 − R3

R5 − R4

=3.564407 − 3.54409

3.568759 − 3.564407= 4.66842831

M

limRn +1 − Rn

Rn +2 − Rn +1

⎛ ⎝ ⎜

⎞ ⎠ ⎟≈ 4.6692016

In other words, each new bifurcation appears about 4.6692016 times faster than the previous one.

This same rate of 4.6692016 occurs in any unimodalmap.

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





• Limits to detailed prediction

• Universality

dynamics, chaos, and prediction

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