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 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 all 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 atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. —Pierre Simon Laplace, A Philosophical Essay on Probabilities [39] [39]

Henri Poincaré, 1854 – 1912 Text In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which energy is conserved. The theorem is commonly discussed in the context of ergodic theory,dynamical systems and statistical mechanics.mathematicsonservnamical systemal mechanics.

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 Poincaré 1854 – 1912 Isaac Newton 1643 – 1727

Iteration and a path to chaos: Laundry example: Wash once, clothes clean. Wash twice, cleaner still.

Iteration and a path to chaos: Some important operations don’t follow this pattern: Rabbit population next year.

“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?” 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?

Notion of “deterministic chaos”

A simple example of deterministic chaos: Exponential versus logistic models for population growth Exponential model: Each year each pair of parents mates, creates four offspring, and then parents die.

Linear Behavior

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. Linear Behavior: The whole is the sum of the parts

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) interactions between offspring make this model nonlinear

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) interactions between offspring make this model nonlinear

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 Logistic map Lord Robert May b Mitchell Feigenbaum b. 1944

LogisticMap.nlogo 1. R = 2 2. R = R = R = R = R = R = 4, look at sensitive dependence on initial conditions Notion of period doubling Notion of “attractors”

Bifurcation Diagram

R 1 ≈ 3.0: period 2 R 2 ≈ period 4 R 3 ≈ period 8 R 4 ≈ period 16 R 5 ≈ period 32 R ∞ ≈ period ∞ (chaos) Period Doubling and Universals in Chaos (Mitchell Feigenbaum)

R 1 ≈ 3.0: period 2 R 2 ≈ period 4 R 3 ≈ period 8 R 4 ≈ period 16 R 5 ≈ period 32 R ∞ ≈ 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) R 1 ≈ 3.0: period 2 R 2 ≈ period 4 R 3 ≈ period 8 R 4 ≈ period 16 R 5 ≈ period 32 R ∞ ≈ period ∞ (chaos) Rate at which distance between bifurcations is shrinking:

Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 ≈ 3.0: period 2 R 2 ≈ period 4 R 3 ≈ period 8 R 4 ≈ period 16 R 5 ≈ period 32 R ∞ ≈ period ∞ (chaos) Rate at which distance between bifurcations is shrinking:

Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 ≈ 3.0: period 2 R 2 ≈ period 4 R 3 ≈ period 8 R 4 ≈ period 16 R 5 ≈ period 32 R ∞ ≈ period ∞ (chaos) Rate at which distance between bifurcations is shrinking: In other words, each new bifurcation appears about times faster than the previous one.

Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 ≈ 3.0: period 2 R 2 ≈ period 4 R 3 ≈ period 8 R 4 ≈ period 16 R 5 ≈ period 32 R ∞ ≈ period ∞ (chaos) Rate at which distance between bifurcations is shrinking: In other words, each new bifurcation appears about times faster than the previous one. This same rate of occurs in any unimodal map.

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