Talk:Hénon map

From Wikipedia, the free encyclopedia

WikiProject Systems
Systems science Portal
This article is within the scope of WikiProject Systems, which collaborates on articles related to Systems science.
Systems rating: Start Class Mid importance  Field: Chaos theory
Please update this rating as the article progresses, or if the rating is inaccurate. Please also add comments to suggest improvements to the article.

Contents

[edit] Stub??

This is a stub? Seems too comprehensive to be a stub. 24.147.122.101 04:38, 27 January 2006 (UTC)

It's missing at least two important points: some mention of the rigorous results and some discussion of the homoclinic tangencies. But you may be right, no longer a stub. XaosBits 11:58, 27 January 2006 (UTC)

[edit] Status of rigorous proof

For the Hénon map at the canonical values one knows:

  • Except for orbits of length 3 and 5, orbits of all periods exist (Szymczak).
  • The 25th iterate of the map has horseshoe dynamics (Stoffer and Palmer).
  • Topological entropy bigger than 0.3381 (Galias and Zgliczyński).

Shouldn't that be sufficient to call it chaotic?  — XaosBits 03:27, 28 February 2006 (UTC)

[edit] References

  • A. Szymczak (1997). "A combinatorial procedure for finding isolating neighbourhoods and index pairs". Proc. Royal Soc. Edin. 127: 1075–1088. 
  • Piotr Zgliczyński (1997). "Computer assisted proof of the horseshoe dynamics in the Hénon map". Random & Computational Dynamics 5: 1–17. 
  • D. Stoffer and K. Palmer (1999). "Rigorous verification of chaotic behavior of maps using validated shadowing". Nonlinearity 12: 1683–1698. 
  • Zbigniew Galias and Piotr Zgliczyński (2001). "Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map". Nonlinearity 14: 909–932. 

[edit] Bifurcation Diagram?

Is the bifurcation diagram really a bifurcation diagram? Or is it an orbit diagram? I was under the impression that bifurcation diagrams plotted both stable and unstable fixed points, while orbit diagrams plotted just stable fixed points. The image looks like it only plots stable fixed points...

I did not produce the plot, but it seems that for each parameter value of the map the orbit points are plotted. I would prefer the expression orbit diagram, but bifurcation diagram is common usage. Many of the plotted points are close to unstable periodic orbits, so in some sense there are unstable points in the diagram. — XaosBits 14:40, 29 April 2006 (UTC)