Iteration of complex maps

Author:
a.zampa
This applet shows the iteration of complex maps of type , were , as a generalization of the concepts discussed in the previous chapters. Complex numbers , for , may be introduced as numbers needed to solve equations of degree 2, , when their discriminant is negative: there exist no real number whose square is negative, so we "invent" a new number, , such that its square equals , i.e. . Using this definition and algebraic properties, it is easy to find rules for mathematical operations on complex numbers (conjugation and modulus are used to define the reciprocal number and, therefore, division):
  • Addition:
  • Subtraction:
  • Multiplication:
  • Conjugation:
  • Modulus:
  • Reciprocal number:
  • Division:
Since a complex number is defined in terms of two real numbers, it can be represented by a point on the plane: its abscissa is the real part of and its ordinate is the immaginary coefficient (the immaginary part being ).
The Graphics view displays the first 56 iterations of the map starting with . Interesting questions are:
  • What happens changing values of and ?
  • Given the initial value , for which value of the set of iterates is bounded?
Explore.