In complex dynamics

Complex dynamics

Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.-Techniques:*General** Montel's theorem...

, the bifurcation

Bifurcation theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...

 locus

Locus (mathematics)

In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

of a family of holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s informally is a locus

Locus (mathematics)

In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

 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

Julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...

 in parameter space. Without doubt, the most famous example of a bifurcation

Bifurcation theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...

 locus

Locus (mathematics)

In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

 is the boundary of the Mandelbrot set

Mandelbrot set

The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...

.

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