What are the Julia sets?
What are the Julia sets?
In general terms, a Julia set is the boundary between points in the complex number plane or the Riemann sphere (the complex number plane plus the point at infinity) that diverge to infinity and those that remain finite under repeated iteration of some mapping (function).
Are Julia sets chaotic?
Thus the behavior of the function on the Fatou set is “regular”, while on the Julia set its behavior is “chaotic”.
What is the Fatou set?
The Fatou set F is an open set which is completely invariant. That is, if z ∈ F, then f(z) ∈ F and f−1(z) ⊂ F. The Julia set J is a completely invariant and compact set in ̂C. A few examples of J and F for polynomials with attracting periodic points are shown in Fig- ure 5.8.
What is the difference between Julia set and Mandelbrot set?
The Mandelbrot set is the set of all c for which the iteration z → z2 + c, starting from z = 0, does not diverge to infinity. Julia sets are either connected (one piece) or a dust of infinitely many points. The Mandelbrot set is those c for which the Julia set is connected.
Why are Julia sets fractals?
For Julia sets, c is the same complex number for all pixels, and there are many different Julia sets based on different values of c. By smoothly changing c we can transform from one Julia set to another over time, creating animated fractal shapes.
How is a Julia set generated?
Julia set fractals are normally generated by initializing a complex number z = x + yi where i2 = -1 and x and y are image pixel coordinates in the range of about -2 to 2. Then, z is repeatedly updated using: z = z2 + c where c is another complex number that gives a specific Julia set.
What is the equation for the Mandelbrot set?
The Mandelbrot set can be explained with the equation zn+1 = zn2 + c. In that equation, c and z are complex numbers and n is zero or a positive integer (natural number).