Foundational Papers in Complexity Science pp. 1685–1734
DOI: 10.37911/9781947864542.58
Universality in Nonlinear Dynamical Systems
Author: Elizabeth Bradley, University of Colorado Boulder and Santa Fe Institute
Excerpt
Dynamical systems—those that evolve with time—are all around us, and the vast majority of them are nonlinear. (Indeed, there is an old chestnut in the field that likens the study of nonlinear systems to the study of nonelephant animals.) Many of these interesting and important systems undergo what are called bifurcations: qualitative changes in their behavior. The population of foxes and rabbits in your backyard might be stable, for instance, but an increase in the rabbits’ birth rate could cause that population to go into an every-other-year alternation: lots of rabbits one year, then lots of foxes then next, and so on. In other words, changes in some parameter of the system—the birth rate, in this example—can cause systemic shifts in the long-term dynamics.
Mitchell Feigenbaum was interested in exactly that situation: what is called a period-doubling bifurcation. He pared the problem down to its bare bones, focusing on a simple quadratic equation called the logistic map that, given the state of a system at some point in time (e.g., the fox–rabbit ratio in 2022), tells you what that state will be at the following time step. This equation—number (15) in the paper in this collection—has a single bifurcation parameter, λ. For low values of λ, the system reaches a steady state. As λ is raised, the dynamics bifurcate into a “two-cycle” like the every-other-year oscillation in the example above. Further increases in λ cause more bifurcations: to a four-cycle, then an eight-cycle, and so on.
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