Mathematical Analysis Problems and Solutions

This text includes solutions to various mathematical analysis problems such as finding fixed points, periodic cycles, stability analysis for non-hyperbolic maps and more.

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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