Period-doubling bifurcation
- Period-doubling bifurcation
In mathematics, a Period doubling bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system. The hallmark of this is a Floquet multiplier of -1.
Example
Consider the following logistical map for a modified Phillips curve:
where is the actual inflation, is the expected inflation, u is the level of unemployment, and is the money supply growth rate. Keeping and varying , the system undergoes period doubling bifurcations, and after a point becomes chaotic, as illustrated in the bifurcation diagram on the right.
Period-halving bifurcation
A Period halving bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with half the period of the original system. A series of period-halving bifurcations leads the system from chaos to order.
ee also
*Feigenbaum constants
External links
* [http://www.egwald.com/nonlineardynamics/onedimensionaldynamics_1.php#flipbifurcationconditions The Flip (Period Doubling) Bifurcation] in Discrete Time, Dynamic Processes
Wikimedia Foundation.
2010.
Look at other dictionaries:
Doubling — may refer to: in maths: multiplication by 2 doubling the cube, a geometric problem doubling time, the period of time required for a quantity to double in size or value doubling map period doubling bifurcation in music, the composition or… … Wikipedia
Bifurcation diagram — In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system. It is usual to represent … Wikipedia
Bifurcation theory — is the mathematical study of changes in the qualitative or topological structure of a given family. Examples of such families are the integral curves of a family of vector fields or, the solutions of a family of differential equations. Most… … Wikipedia
bifurcation — When a non linear dynamical system ( dynamical systems) develops twice the possible solutions that it had before it passed its critical level. A bifurcation cascade is often called the period doubling route to chaos because the transition from an … Financial and business terms
List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia
Chaos theory — This article is about chaos theory in Mathematics. For other uses of Chaos theory, see Chaos Theory (disambiguation). For other uses of Chaos, see Chaos (disambiguation). A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 … Wikipedia
Dynamical system — This article is about the general aspects of dynamical systems. For technical details, see Dynamical system (definition). For the study, see Dynamical systems theory. Dynamical redirects here. For other uses, see Dynamics (disambiguation). The… … Wikipedia
Numerical continuation — is a method of computing approximate solutions of a system of parameterized nonlinear equations, The parameter λ is usually a real scalar, and the solution an n vector. For a fixed parameter value λ,, maps Euclidean n space into itself. Often the … Wikipedia
Logistic map — The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non linear dynamical equations. The map was popularized in a seminal 1976 paper by the… … Wikipedia
Blue sky catastrophe — The blue sky catastrophe is a type of bifurcation of a periodic orbit. In other words, it describes a sort of behaviour stable solutions of a set of differential equations can undergo as the equations are gradually changed. This type of… … Wikipedia
