Sharkovsky's theorem

Sharkovsky's theorem orders positive integers so that a continuous map of an interval with a periodic point of one period must also have points of every period later in that order.

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Sharkovsky ordering: A linear ordering of positive integers in which odd numbers come first, followed by their even multiples, powers of two, and finally one. The theorem’s exact forcing rule is defined by this ordering.
Oleksandr Sharkovsky: A Ukrainian mathematician whose work established the ordering of periods for continuous interval maps. He proved the theorem and introduced the ordering that bears his name.
Logistic map: The family of maps x ↦ r x(1 − x), used to model population growth and exhibit bifurcations and chaos. Its period-doubling dynamics provide familiar examples of periodic orbits within interval-map theory.
Sharkovsky's theorem for graphs: Extensions of period-forcing results from intervals to maps on finite graphs, where the possible orbit structures differ. Graph domains show how changing the underlying space alters the interval theorem’s forcing patterns.
Periodic point: A point that returns to itself after a finite number of iterations of a function. Its least period is the input that determines which other periods are forced.
Topological dynamics: The study of dynamical systems using topological properties such as continuity and recurrence. The theorem belongs to this field’s study of orbit structure without requiring smoothness.
Period-doubling bifurcation: A bifurcation in which a periodic orbit loses stability as a new orbit with twice its period appears. The logistic map’s route through doubled periods contrasts with the theorem’s full forcing order.
Poincaré–Bendixson theorem: A theorem restricting the limit sets of continuous flows in the plane under suitable conditions. It is another low-dimensional restriction on dynamics, but concerns continuous-time planar flows.
Continuous interval map: A continuous function from an interval of the real line into itself. Continuity and the interval domain are the hypotheses that make the forcing theorem hold.
Discrete dynamical system: A system whose state evolves through repeated application of a map at discrete time steps. The theorem concerns successive iterates of a map rather than continuous-time flow.