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Elliptic Fixed Point

An elliptic fixed point of a two-dimensional autonomous system of ordinary differential equations is a fixed point for which the stability matrix has purely imaginary eigenvalues lambda_+/-=+/-iomega with omega>0 (Tabor 1989, pp. 22-25).

In the classification of nonidentity linear fractional transformations lambda|->(alphalambda+beta)/(gammalambda+delta) with real coefficients and alphadelta-betagamma>0, the elliptic case is characterized by the inequality (Walkden 2017, pp. 58-59)

(delta-alpha)^2+4betagamma<0.

See also

Differential Equation, Fixed Point, Hyperbolic Fixed Point, Linear Fractional Transformation, Linear Transformation, Parabolic Fixed Point, Stable Improper Node, Stable Node, Stable Spiral Point, Stable Star, Unstable Improper Node, Unstable Node, Unstable Spiral Point, Unstable Star

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References

Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 22-25, 1989.Walkden, C. Hyperbolic Geometry. Lecture notes, Sep. 4, 2017, pp. 58-59. https://personalpages.manchester.ac.uk/staff/charles.walkden/hyperbolic-geometry/hyperbolic_geometry_old.pdf.

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Elliptic Fixed Point

Cite this as:

Weisstein, Eric W. "Elliptic Fixed Point." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticFixedPoint.html

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