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Darboux-Halphen System

The Darboux-Halphen system is the cyclic system of three quadratic first-order ordinary differential equations

x^'(t)=y(t)z(t)-x(t)[y(t)+z(t)]
(1)
y^'(t)=z(t)x(t)-y(t)[z(t)+x(t)]
(2)
z^'(t)=x(t)y(t)-z(t)[x(t)+y(t)].
(3)

Primes denote derivatives with respect to t. The system arose in Darboux's study of triply orthogonal surfaces and was solved by Halphen (Darboux 1878, Halphen 1881, Ablowitz et al. 1998).

Summing the three equations gives (x+y+z)^'=-(xy+yz+zx). Differentiating again and substituting the system gives (x+y+z)^('')=6xyz. A further differentiation shows that u(t)=-2[x(t)+y(t)+z(t)] satisfies the Chazy equation

u^(''')=2uu^('')-3(u^')^2.
(4)

See also

Chazy Equation, Ordinary Differential Equation

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References

Ablowitz, M. J.; Chakravarty, S.; and Halburd, R. "The Generalized Chazy Equation and Schwarzian Triangle Functions." Asian J. Math. 2, 619-624, 1998. https://doi.org/10.4310/AJM.1998.v2.n4.a1.Darboux, G. "Mémoire sur la théorie des coordonnées curvilignes, et des systèmes orthogonaux." Ann. Sci. École Norm. Sup. 7, 101-150, 1878. https://doi.org/10.24033/asens.159.Halphen, G.-H. "Sur un système d'équations différentielles." C. R. Acad. Sci. Paris 92, 1101-1103, 1881.

Cite this as:

Weisstein, Eric W. "Darboux-Halphen System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Darboux-HalphenSystem.html

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