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Concept
Period-doubling bifurcation
Summary
In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves.
A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system.
A period-doubling cascade is an infinite sequence of period-doubling bifurcations. Such cascades are a common route by which dynamical systems develop chaos. In hydrodynamics, they are one of the possible routes to turbulence.
The logistic map is
where is a function of the (discrete) time . The parameter is assumed to lie in the interval , in which case is bounded on .
For between 1 and 3, converges to the stable fixed point . Then, for between 3 and 3.44949, converges to a permanent oscillation between two values and that depend on . As grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period doublings culminate at , beyond which more complex regimes appear. As increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near .
In the interval where the period is for some positive integer , not all the points actually have period . These are single points, rather than intervals. These points are said to be in unstable orbits, since nearby points do not approach the same orbit as them.
Real version of complex quadratic map is related with real slice of the Mandelbrot set.
Feigenbaum stretch.png|Period-doubling cascade in an exponential mapping of the [[Mandelbrot set]]
Bifurcation diagram of complex quadratic map.png| 1D version with an exponential mapping
Bifurcation1-2.png|period doubling bifurcation
The Kuramoto–Sivashinsky equation is an example of a spatiotemporally continuous dynamical system that exhibits period doubling.
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Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.
Feigenbaum constants
In mathematics, specifically bifurcation theory, the Feigenbaum constants ˈfaɪɡənˌbaʊm are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate.
Bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.