https://sites.google.com/site/matteorichiardi/teaching/complexity-economics

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Matteo Richiardi

Complexity Economics

University of Turin

February-April 2024

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Lecture 3:�Chaos

Matteo Richiardi

Complexity Economics

University of Turin

February-April 2024

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Sources:

• Mitchell, ch. 2

Aristotle

Earth: Objects move in a straight line, either up or down.

Heavens: Objects move in perfect circles around the earth.

Copernicus

The sun is stationary and the planets orbit around it

Galileo

Pioneer of the experimental method. He proved experimentally that most of Aristotle's laws of motion were false.

Newton

Founder of the modern science of dynamics, discovers the law of gravity (which applies to both earth and heavens). Invents calculus (with Leibniz).

Laplace

Big proponent of Newtonian reductionism and determinism.

🡪 🡪 🡪

Laplace’s determinism

“We may regard the present state of the universe as the effect of its past and the cause of its future, an intellect which at a certain moment would know all forces that set nature in motion and all positions of items of which nature is composed.

If this intellect were also vast enough to submit these data to analysis it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atoms.

For such an intellect nothing would be uncertain, and the future just like the past would be present before its eyes.”

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• This view of the possibility of complete prediction was widely accepted until the late 19th or early 20th century.

Poincare’ and chaos

“If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment.”

“But even if it were the case that the natural laws no longer had any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that's all we require and we should say that the phenomenon has been predicted, that is it governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible…”

Sensitivity to initial conditions: The butterfly effect

• Lorenz (1972), “Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?”, American Association for the Advancement of Sciences; 139th meeting.

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Edward Norton Lorenz

What is the difference between chaos and randomness? 🡪 🡪 🡪

Non-linearity and chaos: Population growth

Linear map: n(t+1)=(birthrate-deathrate)*n(t)

Logistic map

n(t+1) = (birthrate-deathrate)(n(t)-n(t)²/max) (‘max’ generally called ‘k’)

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n(t+1) = (birthrate-deathrate)[max⋅n(t)-n(t)²)/max]

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Let:

x(t) = n(t)/max 🡪 (fraction of carrying capacity)

R = (birthrate – deathrate)

Then:

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x(t+1) = Rx(t)[1-x(t)]

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Non-linearity and chaos: Population growth

Linear map: n(t+1)=(birthrate-deathrate)*n(t)

Logistic map: n(t+1) = (birthrate-deathrate)(n(t)-n(t)²/max)

Logistic map: cobweb diagram

🡪 Draw a cobweb diagram for the linear map

Logistic map: Period-doubling route to chaos (*)

(*) valid for all unimodal maps

🡪 SimpleLogisticMap.nlogo

Logistic map: bifurcation diagram

Feigenbaum’s constant: �Each new bifurcation appears about 4.6692016 faster than the previous one �(same for all unimodal maps)

R ≈ 3.0 period 2

R ≈ 3.44949 period 4

R ≈ 3.54409 period 8

R ≈ 3.564407 period 16

R ≈ 3.568759 period 32

…

R ≈ 3.569946 period ∞ (chaos)

Period-doubling route to chaos (valid for all unimodal maps)

Lessons from chaos

• Seemingly random behavior can emerge from deterministic systems, with no external source of randomness.
• The behavior of some simple, deterministic systems can be impossible, even in principle, to predict in the long term, due to sensitive dependence on initial conditions.
• Although the detailed behavior of a chaotic system cannot be predicted, there is some “order in chaos” (e.g. Feigenbaum’s constant) 🡪 There are some higher-level aspects of chaotic systems that are indeed predictable.

Self-organised criticality

• A concept that describes how complex systems �can naturally evolve to a state �on the edge between order and chaos.
• Think of a sandpile. As you drop grains of sand onto it, the pile grows, and avalanches (small, medium, or large) may occur.
• In simpler terms, self-organized criticality is a way that complex systems, like the sandpile, naturally arrange themselves to be in a delicate balance where small events can have big consequences.
• It's a state where the system is always ready to shift from order to disorder in a spontaneous and unpredictable manner.