Hello! I'm stuck on how to do this problem and would appreciate some help!
Choose a non-zero complex number c for f(z) = z2+ c. Find a complex number v that is in the escape set. Find a complex number w that is in the prisoner set. Graph the first five iterations of both v and w and connect them with line segments.
Find a complex number j that is in the Julia set.
I chose c=2i. I know that the escape set is any point that escapes to infinity under iteration. The prisoner set is a point that does not escape. The Julia set is the boundary between the escape set and the prisoner set.
I've looked in my textbook and online but nothing has exactly what I need. Thanks in advance for your help!
http://answers.yahoo.com/question/index?qid=20110118192814AAO4TZK
Oh, we're just iterating a number until we get a number that is a in the prisoner, escape, or Julia set. But how do you deal with "c"?
To solve this problem, we need to understand the concept of iteration in complex dynamics. Let's break it down step by step:
1. Definition of the function f(z) = z^2 + c:
- For any complex number z, the function f maps it to its square plus the constant c.
- In this case, you have chosen c = 2i.
2. Escape Set:
- The escape set consists of points that, under iteration, tend to escape to infinity.
- To determine whether a given point z is in the escape set, we iterate the function f starting from z and check whether the magnitude of the resultant values tends to infinity.
- In this case, we need to find a complex number v such that, when iterated, it tends to infinity.
3. Prisoner Set:
- The prisoner set consists of points that do not tend to escape to infinity under iteration.
- To find a complex number w in the prisoner set, we need to iterate the function f starting from w and observe that the magnitude of the resultant values remains finite.
4. Julia Set:
- The Julia set is the boundary between the escape set and the prisoner set.
- It consists of points that neither escape nor remain confined forever.
- To find a complex number j in the Julia set, we can choose a point close to the boundary between the two sets.
5. Graphing the Iterations:
- To visualize the first five iterations of both v and w, you can plot the points obtained from iterating f(z) = z^2 + c.
- Connect these points with line segments to observe the behavior.
Now let's find the complex numbers v, w, and j:
- For the escape set, we can try different values of v until we find a pattern of increasing magnitude with each iteration. Let's start with v = 0 and iterate the function f(z) five times:
v = 0
v = f(v) = 0^2 + 2i = 2i
v = f(v) = (2i)^2 + 2i = -4 - 2i
v = f(v) = (-4 - 2i)^2 + 2i = 14 - 20i
v = f(v) = (14 - 20i)^2 + 2i = -324 + 560i
- For the prisoner set, we need to find a value of w that doesn't tend to infinity even after several iterations. Let's try w = 1 and iterate the function f(z) five times:
w = 1
w = f(w) = 1^2 + 2i = 1 + 2i
w = f(w) = (1 + 2i)^2 + 2i = -1 + 6i
w = f(w) = (-1 + 6i)^2 + 2i = -32 - 4i
w = f(w) = (-32 - 4i)^2 + 2i = -1000 - 84i
w = f(w) = (-1000 - 84i)^2 + 2i = -666116 - 1676i
- For the Julia set, we can choose a complex number j close to the boundary between the escape set and the prisoner set. Let's say j = 0.5 and iterate the function f(z) five times:
j = 0.5
j = f(j) = 0.5^2 + 2i = 0.25 + 2i
j = f(j) = (0.25 + 2i)^2 + 2i = -3.75 + 2.5i
j = f(j) = (-3.75 + 2.5i)^2 + 2i = -12.6875 + 15i
j = f(j) = (-12.6875 + 15i)^2 + 2i = -246.859375 + 392.625i
j = f(j) = (-246.859375 + 392.625i)^2 + 2i = -643009.75244140625 - 469117.875i
Now that you have found the values for v, w, and j, you can plot the first five iterations for each of them to visualize their behavior.