A result about one-dimensional systems in this field is Sharkovskii's theorem, and as two complex eigenvalues of a system emerge, a Hopf bifurcation can occur; the aforementioned theorem gives a result about systems with three-cycles.
The sum of certain quantities that arise in this field gives the entropy by Pesin's theorem. Those quantities are involved in a matrix defined as the limit as t increases without bound of the Jacobian times its transpose all raised to one-half t; that limit is guaranteed to exist by the Oseledec theorem and those quantities are Lyapunov exponents. A result about one-dimensional systems in this field is Sharkovskii's theorem, and as two complex eigenvalues of a system emerge, a Hopf bifurcation can occur; the aforementioned theorem gives a result about systems with three-cycles. Three-cycles appear briefly on a plot of the behavior of the logistic map, which experiences period-doubling with a frequency that approaches the Feigenbaum constant. This field studies systems like the Roessler map, the double pendulum, and a result of a weather model called the Lorenz attractor; each of those objects often traces out enormously different trajectories after only small changes in initial conditions. For 10 points, name this field colloquially described by the metaphor of the butterfly effect.
The sum of certain quantities that arise in this field gives the entropy by Pesin's theorem.
{chaos} theory