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Complexity: Ch. 2

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Dynamical Systems

•Merely means systems that evolve with time• not intrinsically interesting in our context

•What is interesting are certain non-linear dynamical systems• So lets work our way up to those

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What’s coming in this chapter

•Dynamical systems•Nonlinear dynamical systems•Nonlinear chaotic dynamical systems•Order in nonlinear chaotic dynamical systems

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Dynamical systems

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Newtonian Mechanics

•A triumph of reductionism•Led to Laplace’s description of a predictable, clockwork universe•Famous 19th century complacency that physics was essentially complete except for some details 5

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Worms in the apple of certainty•Quantum mechanics and Heisenberg’s Principle•not sure this is relevant•Discovery of presumably well-posed problems that are pathologically dependent on initial conditions

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On initial conditions

The state of a dynamical system, such as the Solar System, depends on• the position and velocity of all of its components at some particular time, called initial conditions, and• the application of its equations of motion (i.e., Newton’s laws, updated). 7

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The logistic map

The logistic map is the name of a class of dynamical systems that play a large role in complexity theory.

Think of it as thelogistic model of population growth

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The logistic map

“Logistic” may come from the French word meaning “to house” but that’s only a guess.

“Map” just means function.

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The logistic population model

Here it is:

is in [0, 1] and is the population at time n relative to the maximum possible population

R is in [0, 4] is a measure of the replacement rate

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Show us pictures!

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Same R, different starting value

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Larger R

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Slightly larger R

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R = 4: SDIC and chaos

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Bifurcation map

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First bifurcation

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Bifurcation map

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Onset of chaos (period doubling)

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Mitchell Feigenbaum

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Feigenbaum’s Constant

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4.6692016

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Example: logistic map

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δ = 4.669 201 609 102 990 671 853 203 821 578

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Takeaway

• Simple, deterministic systems can generate apparent random behavior• Long term prediction for such systems may be impossible in principle• Such systems may show surprising regularities: period-doubling and Feigenbaum’s Constant 24