World Library  
Flag as Inappropriate
Email this Article

Period-doubling bifurcation

Article Id: WHEBN0004457659
Reproduction Date:

Title: Period-doubling bifurcation  
Author: World Heritage Encyclopedia
Language: English
Subject: Oscar Lanford, Logistic map, Chaos theory, Dynamical system, Index of physics articles (P)
Collection: Bifurcation Theory, Nonlinear Systems
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Period-doubling bifurcation

In mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which a slight change in a parameter value in the system's equations leads to the system switching to a new behavior with twice the period of the original system. With the doubled period, it takes twice as many iterations as before for the numerical values visited by the system to repeat themselves.

A period doubling cascade is a sequence of doublings and further doublings of the repeating period, as the parameter is adjusted further and further.

Period doubling bifurcations can also occur in continuous dynamical systems, namely when a new limit cycle emerges from an existing limit cycle, and the period of the new limit cycle is twice that of the old one.

Contents

  • Examples 1
  • Period-halving bifurcation 2
  • See also 3
  • References 4
  • External links 5

Examples

Bifurcation diagram for the modified Phillips curve.

Consider the following logistical map for a modified Phillips curve:

\pi_{t} = f(u_{t}) + a \pi_{t}^e

\pi_{t+1} = \pi_{t}^e + c (\pi_{t} - \pi_{t}^e)

f(u) = \beta_{1} + \beta_{2} e^{-u} \,

b > 0, 0 \leq c \leq 1, \frac {df} {du} < 0

where :

  • \pi is the actual inflation
  • \pi^e is the expected inflation,
  • u is the level of unemployment,
  • m - \pi is the money supply growth rate.

Keeping \beta_{1} = -2.5, \ \beta_{2} = 20, \ c = 0.75 and varying b, the system undergoes period doubling bifurcations, and after a point becomes chaotic, as illustrated in the bifurcation diagram on the right.

Bifurcation from period 1 to 2 for complex quadratic map

Period-halving bifurcation

Period-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos.

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.

See also

References

  • Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory. Applied Mathematical Sciences 112 (3rd ed.).  

External links

  • The Flip (Period Doubling) Bifurcation in Discrete Time, Dynamic Processes
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.