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Difference between revisions of "Lorenz equations"

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''Lorenz system''
 
''Lorenz system''
  
 
The system of equations
 
The system of equations
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110180/l1101801.png" /></td> </tr></table>
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$$\dot x=-\sigma x+\sigma y,$$
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110180/l1101802.png" /></td> </tr></table>
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$$\dot y=rx-y-xz,$$
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110180/l1101803.png" /></td> </tr></table>
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$$\dot z=-bz+xy.$$
  
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It arises as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110180/l1101804.png" />-mode truncation of the two-dimensional [[Convection equations|convection equations]] for parallel horizontal walls at constant, but different, temperatures.
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It arises as the $3$-mode truncation of the two-dimensional [[Convection equations|convection equations]] for parallel horizontal walls at constant, but different, temperatures.
  
 
See [[Lorenz attractor|Lorenz attractor]] for more details and references; see, e.g., [[#References|[a2]]] for a picture.
 
See [[Lorenz attractor|Lorenz attractor]] for more details and references; see, e.g., [[#References|[a2]]] for a picture.

Revision as of 15:03, 30 July 2014

Lorenz system

The system of equations

$$\dot x=-\sigma x+\sigma y,$$

$$\dot y=rx-y-xz,$$

$$\dot z=-bz+xy.$$

It arises as the $3$-mode truncation of the two-dimensional convection equations for parallel horizontal walls at constant, but different, temperatures.

See Lorenz attractor for more details and references; see, e.g., [a2] for a picture.

References

[a1] Yu.I. Neimark, P.S. Landa, "Stochastic and chaotic oscillations" , Kluwer Acad. Publ. (1992) (In Russian)
[a2] H.G. Schuster, "Deterministic chaos. An introduction" , VCH (1988)
[a3] C. Sparrow, "The Lorenz equations: bifurcations, chaos, and strange attractors" , Springer (1982)
[a4] J.M.T. Thompson, H.B. Stewart, "Nonlinear dynamics and chaos" , Wiley (1986) pp. Chapt. 11
How to Cite This Entry:
Lorenz equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lorenz_equations&oldid=32570
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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