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Difference between revisions of "Lorenz equations"

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The system of equations
 
The system of equations
  
$$\dot x=-\sigma x+\sigma y,$$
+
$$  
 +
\begin{eqnarray*}
 +
\dot x &=& -\sigma x+\sigma y, \\
  
$$\dot y=rx-y-xz,$$
+
\dot y &=& rx-y-xz, \\
  
$$\dot z=-bz+xy.$$
+
\dot z &=& -bz+xy.
 +
\end{eqnarray*}
 +
$$
  
 
It arises as the $3$-mode truncation of the two-dimensional [[Convection equations|convection equations]] for parallel horizontal walls at constant, but different, temperatures.
 
It arises as the $3$-mode truncation of the two-dimensional [[Convection equations|convection equations]] for parallel horizontal walls at constant, but different, temperatures.

Latest revision as of 03:05, 22 June 2022

Lorenz system

The system of equations

$$ \begin{eqnarray*} \dot x &=& -\sigma x+\sigma y, \\ \dot y &=& rx-y-xz, \\ \dot z &=& -bz+xy. \end{eqnarray*} $$

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=52470
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article