Math 690 - Analyzing Maps, Fixed Points, and Orbits (Fall 2018)

This assignment solution covers several problems related to dynamical systems, specifically focusing on maps, fixed points, and orbits. The first question involves computing the first five points of an orbit for the map f(x) = x^2 and finding f2 and f3 for the map f(x) = x^2 - 1. The second question focuses on identifying real fixed points for the maps f(x) = |x| and f(x) = x sin x. The third question examines the orbits of specific points under a given map f: [0, 1) -> [0, 1), discussing whether these points are eventually periodic and determining their cycles. It also explores the fixed points of fn. The fourth question discusses the conditions for non-hyperbolic maps regarding the stability of fixed points based on the derivatives of the function. Finally, the fifth question examines cycles of period n for specific points under a given map.

Q.1)
(a) f(x) = x2 .
F1(1/2)=F(1/2)=1/4
F2(1/2)=F(1/4)=1/16
F3(1/2)=F(1/16)=1/256
F4(1/2)=F(1/256)=1/65536
F5(1/2)=F(1/65536)=1/4294967296
(b) f(x) = x2 − 1. Find f2 and f3
f2(x) = f(f(x))
= f(x2 - 1)
= (x2 - 1)2 -1
= x4 −2 x2 +1−1
= x4 −2 x2
f 3 ( x ) =f ( f 2 ( x ) )
¿ f ( x4 −2 x2 )
¿ ( x4 −2 x2 )2
−1
¿ x8−4 x6+4 x4
Q.2)
a) f(x) = |x|
For fixed real points,
f(x) = x
|x| = x
Therefore, x ∈ [0,∞)
b) f(x) = x sin x
For fixed rea points,
f(x) = x
x sin x = x
sin x = 1
x = 2n π, where n = 0,1,2,3,…..
Q.3)
a) map f : [0, 1) → [0, 1) given by

f() =
{ 2 x if 0 ≤ x< 1
2
2 x−1 if 1
2 ≤ x<1
I. x0 = 0.3
Is eventually periodic, with period 4. Cycle is {0.6, 0.2, 0.4, 0.8}.
II. x0 = 0.7
Is eventually periodic, with period 4. Cycle is {0.6, 0.2, 0.4, 0.8}.
III. x0 = 1/8
Is eventually fixed at x = 0. (d) x0 = 1/7
Is periodic with period 3. Cycle is {1/7, 2/7, 4/7}.
IV. x0 = 3/11
Is periodic with period 10. Cycle is 3xi/11, where xi is the ith element of {3, 6, 1, 2, 4, 8, 5, 10, 9,
7}.
b) fixed points of fn
The graph of T n (x) is made up of 2n “tents” each of which has width (½)n. The line y = x will cross each
of the tents exactly twice — so there will be 2n+1 fixed points.
Q.4)
For non-hyperbolic maps following statements hold true for a fixed point x∗ of f where f is C3,
If f’(x)=1 we have 3 cases,
1. if f’’(x*) ≠0, then x* is semi asymptotically stable from left if f’’(x*)>0 and from right f’’(x*)<0
2. if f’’(x*) =0 and f’’’(x)<0, then x* is asymptotically stable.
3. if f’’(x*) =0 and f’’’(x)>0, then x* is unstable.
Q.5)
f() =
{ 2 x if 0 ≤ x< 1
2
2 x−1 if 1
2 ≤ x<1
Suppose that a point x0 lies on a cycle of period n (n-cycle).
I. x0 = 0.3, Cycle is {0.6, 0.2, 0.4, 0.8}.
II. x0 = 0.7, Cycle is {0.6, 0.2, 0.4, 0.8}.
III. x0= 1/8, Cycle is {1/7, 2/7, 4/7}.
X0=3/11, Cycle is 3xi/11, where xi is the ith element of {3, 6, 1, 2, 4, 8, 5, 10, 9, 7}.
Suppose that a point x0 lies on a cycle of period n (n-cycle) then the cycle is fixed at appoint x=0.

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