E. Applied Systems
E.3 Complex Systems
Course description
Some systems can't be understood by understanding their parts: aggregate behavior depends on how the parts interact, not just on what each one is. The logistic map, a one-line recurrence for bounded population growth, produces stable points, cycles, and chaos depending on a single parameter, and its bifurcation diagram is one of the most striking pictures in applied mathematics. Edward Lorenz stumbled on the same phenomenon, sensitive dependence on initial conditions, around 1961, when he restarted a simple weather simulation from a printout that had rounded 0.506127 to 0.506; his 1963 paper showed it in a three-variable convection model. Steven Strogatz's Cornell lectures cover both examples, free.
Networks are the second piece: a graph's structure, random versus scale-free, with or without small-world shortcuts, shapes what spreads across it or how it fails. Agent-based models are the third: simple local rules produce aggregate patterns none of the individuals intended. Thomas Schelling's segregation model is the classic case, where mild individual preferences about neighbors produce strong aggregate segregation with nobody an extremist. Chaos limits prediction even in a deterministic system, which is why a weather forecast holds for days but not past roughly one to two weeks, however much computing power is available.
Melanie Mitchell's Complexity: A Guided Tour is the reading; the Santa Fe Institute's Complexity Explorer offers free courses with exercises. Donella Meadows's Thinking in Systems covers stocks-and-flows thinking, the habit of tracing feedback loops before intervening in a system. The unit draws on E.1's feedback, E.2's markets, B.3's information measures, C.3's evolution and ecology, and C.4's climate.
Checkpoints
Each checkpoint is tagged with the stage of the cycle it tests.| Stage | Checkpoint | Attempts |
|---|---|---|
naive realism | Before integrating anything, guess how long two runs of the Lorenz equations that differ in the sixth decimal place will stay close, and write the guess down. | Post the first attempt |
formalism | Implement the logistic map, plot its bifurcation diagram, and locate the first few period doublings. | Post the first attempt |
skepticism | Integrate the Lorenz equations from two starting points differing in the sixth decimal place, and measure how long the trajectories stay close. | Post the first attempt |
formalism | Implement Schelling's segregation model and find the smallest preference threshold that produces strong segregation on your grid. | Post the first attempt |
abstraction | Grow a network by preferential attachment, compare its degree distribution against a random graph of the same size, and name one real network each resembles. | Post the first attempt |
intuition | Explain to a non-specialist why a deterministic system can still be unpredictable in practice, without using "chaos" as a stand-in for the explanation. | Post the first attempt |
Discussion
Proposed edits
Found an error, a missing idea, or a better source? Propose a change. A moderator reviews each proposal.Sources
- Lorenz, Edward N. "Deterministic Nonperiodic Flow." Journal of the Atmospheric Sciences 20, 130–141 (1963).
- Strogatz, Steven. Nonlinear Dynamics and Chaos, Cornell lectures (2014), free.
- Schelling, Thomas C. Micromotives and Macrobehavior (W. W. Norton, 1978).
- Mitchell, Melanie. Complexity: A Guided Tour (Oxford University Press, 2009).
- Santa Fe Institute. Complexity Explorer (free courses, incl. Introduction to Complexity).
- Meadows, Donella H. Thinking in Systems: A Primer (Chelsea Green, 2008).