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GeezerVison is showing mathworld.wolfram.com — Halvorsen System -- from Wolfram MathWorld

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Halvorsen System

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HalvorsenSystem

The Halvorsen system is a cyclically symmetric quadratic dynamical system given by

x^'=-ax-4y-4z-y^2
(1)
y^'=-ay-4z-4x-z^2
(2)
z^'=-az-4x-4y-x^2,
(3)

where a is a real parameter (Sprott 2010, pp. 83-85). The variables x(t), y(t), and z(t) depend on time, and primes denote derivatives with respect to t. Cycling the coordinates leaves the system unchanged. On the invariant line x=y=z=u, it reduces to u^'=-(a+8)u-u^2. The equilibrium point u=0 on this line is accompanied by a second one at u=-a-8 when a!=-8. The divergence is -3a, so for a>0 the flow contracts volume in phase space. Sprott (2010, pp. 83-85) gives a numerical example of a strange attractor for a=1.3. The figure shows the corresponding trajectory from (x(0),y(0),z(0))=(-6.4,0,0) after its initial transient.

See also

Dynamical System, Strange Attractor

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References

Sprott, J. C. "Circulant Systems." §3.6 in Elegant Chaos: Algebraically Simple Chaotic Flows. Singapore: World Scientific, pp. 83-85, 2010. https://doi.org/10.1142/9789812838827.

Cite this as:

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

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