In complex dynamics, the bifurcation locus of a family of holomorphic functions informally consists of those maps for which the dynamical behavior changes drastically under a small perturbation of the parameter. Thus the bifurcation locus can be thought of as an analog of the Julia set in parameter space. Without doubt, the most famous example of a bifurcation locus is the boundary of the Mandelbrot set. Complex dynamics is the study of dynamical systems for which the phase space is a complex manifold. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. ... A rendering of the Mandelbrot set In mathematics, the Mandelbrot set is defined as the connectedness locus of the family of complex quadratic polynomials. ...

Parameters in the complement of the bifurcation locus are called J-stable.

References

[EL]

Alexandre E. Eremenko and Mikhail Yu. Lyubich Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020, http://www.numdam.org/item?id=AIF_1992__42_4_989_0.

[L]

Mikhail Yu. Lyubich, Some typical properties of the dynamics of rational mappings (Russian), Uspekhi Mat. Nauk 38 (1983), no. 5(233), 197–198.

[MSS]

Ricardo Mañé, Paulo Sad and Dennis Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217, http://www.numdam.org/item?id=ASENS_1983_4_16_2_193_0.

[McM]

Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994. ISBN 0-691-02982-2.