I want to find if I give an initial value, what value it will converge to after iterating a lot of time.
(1) I set r<1, here r=0.5, iterate_num=1000, we only consider interval [0, 1]:
This image means no matter where I begin my initial value, it will converge to 0 after iterating a lot of time. This also means 0 is the only fixed point and is globally stable. (Here globally means [0,1]).
(2) I set r=1:
In this case, 0 is still the globally stable fixed point even though it does not satisfy the theorem we always use.
(3) r=2,
Two fixed points, 0 and 1/2. 0 is the unstable one while 1/2 is the stable one.
If the initial value is 0 and 1, it will converge to 0, otherwise 1/2.
Some study of the logistic map:
I want to find if I give an initial value, what value it will converge to after iterating a lot of time.
(1) I set r<1, here r=0.5, iterate_num=1000, we only consider interval [0, 1]:
This image means no matter where I begin my initial value, it will converge to 0 after iterating a lot of time. This also means 0 is the only fixed point and is globally stable. (Here globally means [0,1]).
(2) I set r=1:
In this case, 0 is still the globally stable fixed point even though it does not satisfy the theorem we always use.
(3) r=2,
Two fixed points, 0 and 1/2. 0 is the unstable one while 1/2 is the stable one. If the initial value is 0 and 1, it will converge to 0, otherwise 1/2.
(4)r=3
Two fixed points 0 and 2/3, both unstable.
(5)r=3.2
Two corresponding period-2 orbit, stable.
(6) r=4
chaotic