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Introduction to Topology: Pure and Applied

Colin Adams, Robert Franzosa

Chapter 8

Dynamical Systems and Chaos - all with Video Answers

Educators

Chapter Questions

Problem 1

For each of the following functions $f: \mathbb{R} \rightarrow \mathbb{R}$, find all fixed points and periodic points and sketch a phase diagram for the dynamical system:
(a) $f(x)=x^3$
(g) $f(x)=\frac{1}{2} \sin (x)$
(b) $f(x)=-x^3$
(h) $f(x)=\frac{\pi}{2} \sin (x)$
(c) $f(x)=-x^{1 / 3}$
(i) $f(x)=e^x$
(d) $f(x)=x-x^2$
(j) $f(x)=2\left(x-x^2\right)$
(e) $f(x)=\frac{4}{\pi} \tan ^{-1}(x)$
(k) $f(x)=x+\sin (x)$
(f) $f(x)=1-x^2$

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Problem 2

(a) Consider the linear systems $L_a: \mathbb{R} \rightarrow \mathbb{R}$ defined by $L_a(x)=a x$ where $a \in \mathbb{R}$. Through a collection of phase diagrams, classify the different dynamic behaviors seen in $L_a$, as $a$ ranges over the real numbers.
(b) Show that if $L_a$ and $L_b$ have the same dynamic behavior (as identified in part (a)), then $L_a$ and $L_b$ are topologically conjugate.

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01:44

Problem 3

Show that if $x$ is a period- $m$ point for a function $f: X \rightarrow X$, then the points $x, f(x), \ldots, f^{m-1}(x)$ are all distinct and are all period- $m$ points.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:46

Problem 4

Find as many eventual fixed points as you can for the tent function $T$. (In Section 8.3 we show how to identify all of them.)

John Gehad
John Gehad
Numerade Educator
04:46

Problem 5

Find a period-3 point for the tent function $T$.

Lucas Finney
Lucas Finney
Numerade Educator
00:49

Problem 6

Consider the second savings-account model in Example 8.3:
$$
g(x)= \begin{cases}1.05 x-1000 & \text { if } 1.05 x \geq 1000 \\ 0 & \text { if } 1.05 x \leq 1000\end{cases}
$$
(a) Determine $g^{-1}(0)$. That is, determine the interval of amounts that result in a balance of 0 the following year.
(b) Determine the interval of amounts that result in a nonzero balance after $n-1$ years but a balance of 0 after $n$ years.

James Kiss
James Kiss
Numerade Educator

Problem 7

In this exercise we ask you to explore, by explicit computation, orbits associated with the functions $f_\alpha:[0,1] \rightarrow[0,1]$, defined by $f_\alpha(x)=\alpha x(1-x)$. We examine this family of functions further in Section 8.4.
(a) For $f_{0.6}:[0,1] \rightarrow[0,1], f_{0.6}(x)=0.6 x(1-x)$, explore and compare the orbits determined by the initial values $0,0.1$, and 0.7 .
(b) For $f_{1.6}:[0,1] \rightarrow[0,1], f_{1.6}(x)=1.6 x(1-x)$, explore and compare the orbits determined by the initial values $0,0.1$, and 0.7 .
(c) For $f_{3.2}:[0,1] \rightarrow[0,1], f_{3.2}(x)=3.2 x(1-x)$, explore and compare the orbits determined by the initial values 0.25 and 0.75 .
(d) For $f_4:[0,1] \rightarrow[0,1], f_4(x)=4 x(1-x)$, explore and compare the orbits determined by the initial values 0.261 and 0.262 .

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01:40

Problem 8

Find the fixed points of $g_a: \mathbb{R} \rightarrow \mathbb{R}, g_a(x)=a\left(x+x^2\right)$, for $a>0$. On an $x$ versus $a$ coordinate system, plot the fixed points, demonstrating how their location changes as the parameter $a$ changes.

Zachary Watson
Zachary Watson
Numerade Educator

Problem 9

Prove Corollary 8.5: Let $h$ be a topological conjugacy between $f: X \rightarrow X$ and $g: Y \rightarrow Y$, and assume that $x \in X$.
(a) If $x$ is a fixed point of $f$, then $f(x)$ is a fixed point of $g$.
(b) If $x$ is a period-m point of $f$, then $f(x)$ is a period-m point of $g$.
(c) If $x$ is an eventual fixed point of $f$, then $f(x)$ is an eventual fixed point of $g$.
(d) If $x$ is an eventual periodic point of $f$, then $f(x)$ is an eventual periodic point of $g$.

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03:28

Problem 10

Define the function $g:[0,1] \rightarrow[0,1]$ by
$$
g(x)= \begin{cases}3 x & \text { for } 0 \leq x \leq \frac{1}{3} \\ 2-3 x & \text { for } \frac{1}{3} \leq x \leq \frac{2}{3} \\ 3 x-2 & \text { for } \frac{2}{3} \leq x \leq 1 .\end{cases}
$$

Show that $g$ is not topologically conjugate to the tent function.

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
01:13

Problem 11

Let $f: X \rightarrow X$ be continuous. Show that if $x \in X$ and $y=\lim _{n \rightarrow \infty} f^n(x)$, then $y$ is a fixed point of $f$. That is, if the orbit of a point converges, then it converges to a fixed point. (Hint: Use Theorem 4.7.)

Carson Merrill
Carson Merrill
Numerade Educator

Problem 12

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous, and assume that $I$ and $J$ are disjoint closed and bounded intervals such that $f(I) \subset J$ and $f(J) \subset I$. Prove that there is a period-2 point of $f$ in $I$.

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Problem 13

(a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a homeomorphism. Prove that $f$ has no periodic points with period greater than 2 .
(b) Show that for every $n \in \mathbb{Z}_{+}$there exists a homeomorphism $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with a period- $n$ point.

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Problem 14

Assume that $f: X \rightarrow X$ is injective.
(a) Prove that $f^n: X \rightarrow X$ is injective for each $n \in \mathbb{Z}_{+}$.
(b) Prove that $f$ has no eventual periodic points.

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Problem 15

Consider $f(x)=\frac{1}{1-x}$. Let $X$ be the maximal subspace of $\mathbb{R}$ on which $f$ defines a dynamical system. Show that $X$ has three components, and prove that each $x \in X$ is a period-3 point with an orbit having one point in each component of $X$.

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01:13

Problem 16

For each of the following functions determine the fixed and periodic points and for each such point use Theorem 8.9 to determine its stability:
(a) $f(x)=x^3$
(f) $f(x)=\frac{1}{2} \sin (x)$
(b) $f(x)=-x^3$
(g) $f(x)=\frac{\pi}{2} \sin (x)$
(c) $f(x)=-x^{1 / 3}$
(h) $f(x)=2\left(x-x^2\right)$
(d) $f(x)=\frac{4}{\pi} \tan ^{-1}(x)$
(i) $f(x)=x+\sin (x)$
(e) $f(x)=1-x^2$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 17

(a) Prove that if $x^*$ is a stable period-m point of a function $f$, then so are $f\left(x^*\right), f^2\left(x^*\right), \ldots, f^{m-1}\left(x^*\right)$.
(b) Prove that if $x^*$ is an asymptotically stable period-m point of a function $f$, then so are $f\left(x^*\right), f^2\left(x^*\right), \ldots, f^{m-1}\left(x^*\right)$.
(c) Prove that if $x^*$ is a neutrally stable period-m point of a function $f$, then so are $f\left(x^*\right), f^2\left(x^*\right), \ldots, f^{m-1}\left(x^*\right)$.
(d) Prove that if $x^*$ is an unstable period-m point of a function $f$, then so are $f\left(x^*\right), f^2\left(x^*\right), \ldots, f^{m-1}\left(x^*\right)$.

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01:55

Problem 18

Draw web diagrams for the dynamical systems in Examples 8.3, 8.4, and 8.8, illustrating the stability properties discussed in Example 8.9.

Robert Huber
Robert Huber
Numerade Educator

Problem 19

Prove Theorem 8.8: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the linear function $f(x)=m x+b$.
(a) If $m \neq 1$, then $f$ has a unique fixed point, and that fixed point is neutrally stable if $m=-1$, asymptotically stable if $|m|<1$, and unstable if $|m|>1$.
(b) If $m=1$, then $f$ has no fixed points if $b \neq 0$, and every $x \in \mathbb{R}$ is a neutrally stable fixed point if $b=0$.

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01:13

Problem 20

In the transition cases that are not addressed by Theorem 8.9 , where $\left|f^{\prime}\left(x_0\right)\right|=$ 1 at a fixed point $x_0$ of $f$, it is possible that $x_0$ could be neutrally stable, asymptotically stable, or unstable.
(a) Find an example of a function $f$ with a fixed point $x_0$ where $\left|f^{\prime}\left(x_0\right)\right|=1$ and $x_0$ is neutrally stable.
(b) Find an example of a function $f$ with a fixed point $x_0$ where $\left|f^{\prime}\left(x_0\right)\right|=1$ and $x_0$ is asymptotically stable.
(c) Find an example of a function $f$ with a fixed point $x_0$ where $\left|f^{\prime}\left(x_0\right)\right|=1$ and $x_0$ is unstable.

Carson Merrill
Carson Merrill
Numerade Educator
01:07

Problem 21

Assume $X \subset \mathbb{R}$ and $f: X \rightarrow X$ is differentiable. Show that if $x_1, \ldots, x_m$ constitute a period- $m$ orbit of $f$, then for each $i$,
$$
\left(f^m\right)^{\prime}\left(x_i\right)=f^{\prime}\left(x_1\right) f^{\prime}\left(x_2\right) \ldots f^{\prime}\left(x_m\right) .
$$

Carson Merrill
Carson Merrill
Numerade Educator
01:44

Problem 22

Newton's method is an iterative process for approximating the zeros of a function $f: D \rightarrow \mathbb{R}$ with $D \subset \mathbb{R}$. We assume that $f$ is twice differentiable. For such an $f$ define $g(x)=x-\frac{f(x)}{f^{\prime}(x)}$. The idea behind Newton's method is to make an initial guess $x_0$ of a zero of $f$, and then iterate $g$ on $x_0$ to obtain a sequence of points, $x_n=g^n\left(x_0\right)$, that (we hope) converges to a zero of $f$.
(a) Show that if $x$ is a fixed point of $g$ then $x$ is a zero of $f$.
(b) Show that every fixed point of $g$ is asymptotically stable.
If $x$ is such that $f^{\prime}(x)=0$ then $g$ is not defined at $x$. Such an $x$ could be a zero of $f$. If $z$ is a zero of $f$ for which $f^{\prime}(z) \neq 0$, then the results of parts (a) and (b) imply that if we make an initial guess $x_0$ sufficiently close to $z$, then the orbit of $x_0$ under $g$ converges to $z$.
Now, consider the situation where we have two fixed points $z_1$ and $z_2$ of $g$ with no fixed point between them. By (b) there are neighborhoods $U_1$ and $U_2$ of $z_1$ and $z_2$, respectively, such that the orbit of each point in $U_i$ converges to $z_i$. Something must happen between $z_1$ and $z_2$ to separate the points whose orbits converge to $z_1$ from those whose orbits converge to $z_2$. At the very least, we can do the following:
(c) Prove that between every pair of fixed points of $g$ there is a point at which $g$ is not defined.

Carson Merrill
Carson Merrill
Numerade Educator

Problem 23

Prove that $x \in[0,1]$ is an eventual fixed point of the tent function if and only if the binary expansion of $x$ ends in a sequence of all 0 s , all 1 s , or alternating 0 s and 1 s .

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Problem 24

(a) Show that if $X$ has the trivial topology, then $f: X \rightarrow X$ is chaotic if and only if $f$ has a periodic point.
(b) Show that if $X$ has the discrete topology and $f: X \rightarrow X$ is chaotic, then $X$ is finite and $f$ is a cyclic permutation of the elements of $X$.

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Problem 25

Prove Theorem 8.14: Let $X$ be an infinite Hausdorff space. If $f: X \rightarrow X$ is chaotic then every periodic point of $f$ is unstable.

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Problem 26

Prove Theorem 8.15: If $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos.

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Problem 27

Verify that the homeomorphism $h:[0,1] \rightarrow[0,1]$, defined by $h(x)=$ $\sin ^2\left(\frac{\pi}{2} x\right)$, is a topological conjugacy between the tent function and the function $Q:[0,1] \rightarrow[0,1]$, defined by $Q(x)=4 x(1-x)$.

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Problem 28

Show that the functions $Q:[0,1] \rightarrow[0,1]$, defined by $Q(x)=4 x(1-x)$, and $R:[-1,1] \rightarrow[-1,1]$, defined by $R(x)=1-2 x^2$, are topologically conjugate. (It follows $R$ is chaotic.)

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Problem 29

Show that the function $g:[0,1] \rightarrow[0,1]$, defined by
$$
g(x)= \begin{cases}3 x & \text { for } 0 \leq x \leq \frac{1}{3}, \\ 2-3 x & \text { for } \frac{1}{3} \leq x \leq \frac{2}{3}, \\ 3 x-2 & \text { for } \frac{2}{3} \leq x \leq 1,\end{cases}
$$
is chaotic, by using ternary expansions to show each of the following:
(a) Periodic points are dense.
(b) $g$ is topologically transitive.

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Problem 30

Show that if $f: X \rightarrow X$ has a point $x$ whose orbit is dense in $X$ then $f$ is topologically transitive. (The converse of this theorem is also true if $X$ is a compact subset of either $\mathbb{R}$ or $S^1$, but the proof requires tools not introduced in this text.)

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Problem 31

On $S^1 \subset \mathbb{R}^2$, let $\theta$ represent the point at angle $\theta$ measured counterclockwise from the positive $x$-axis. Define $f: S^1 \rightarrow S^1$ by $f(\theta)=2 \theta$. In (a) and (b) we prove that $f$ is chaotic.
(a) Prove that for $n, j \in \mathbb{Z}_{+}$, the points with angular representation $\frac{2 \pi j}{2^n-1}$ are fixed points of $f^n$, and thereby show that periodic points of $f$ are dense in $S^1$.
(b) Prove that every interval of the form $\left[\theta, \theta+\frac{\pi}{2^{n-1}}\right] \subset S^1$ is mapped onto $S^1$ by $f^n$, and then use this result to show that $f$ is topologically transitive.

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Problem 32

Prove Lemma 8.17. (Hint: Use induction; the $n=1$ case holds by the definition of $T$.)

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Problem 33

Prove that there exist $n, j \in \mathbb{Z}_{+}$such that the interval $\left[\frac{j-1}{2^{n-1}}, \frac{j}{2^{n-1}}\right]$ lies in $U$.

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Problem 34

With $n$ as in the previous exercise, prove that $T^n$ has a fixed point in $U$ and therefore that there is a periodic point of $T$ in $U$.

Next we establish topological transitivity. Let $U$ and $V$ be arbitrary open sets in $[0,1]$.

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Problem 35

Use Exercise SE 8.33 and Lemma 8.17 to show that there exists $n \in \mathbb{Z}_{+}$such that $V \subset T^n(U)$, and thereby conclude that $T$ is topologically transitive.

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Problem 37

Over the domain $x \in[-1,1]$ sketch separate web diagrams for $f_\alpha$ with $\alpha$ slightly less than 1 , equal to 1 , and slightly greater than 1 . Discuss the changes you observe in the fixed points and their stability as the parameter $\alpha$ passes through 1. The bifurcation observed here is known as a transeritical bifurcation.

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Problem 38

Determine the period-2 points of $f_\alpha$, and use Theorem 8.9 to deduce their stability. (Hint: Finding the period-2 points involves solving a degree-4 polynomial; the fixed points of $f_\alpha$ are two of the solutions to the polynomial.)

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Problem 39

Using a computer graphing program, examine the graphs of the function $f_\alpha^2$ and the line $y=x$ for $\alpha$ slightly less than 3 , equal to 3 , and slightly greater than 3 in order to approximate the fixed points of $f_\alpha^2$. Which fixed points of $f_\alpha^2$ that you observe are fixed points of $f_\alpha$ and which are period-2 points of $f_\alpha$ ? Discuss the changes you observe as the parameter $\alpha$ passes through 3 . The bifurcation observed here is known as a period-doubling bifurcation.

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Problem 40

Using a computer graphing program, examine the graphs of $f_\alpha^4$ and the line $y=$ $x$ in order to approximate the value of $\alpha$ where the period-doubling bifurcation, from an asymptotically stable period-2 orbit to an asymptotically stable period-4 orbit, takes place.

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Problem 41

Consider the family of functions $g_\lambda: \mathbb{R} \rightarrow \mathbb{R}$ given by $g_\lambda(x)=\lambda x-x^3$ with $\lambda \in[0,2]$.
(a) Determine the fixed points and their stability.
(b) Discuss the bifurcation that takes place over the parameter domain, and include illustrations showing how the graph of $g_\lambda$ changes (in relation to the line $y=x$ ) as the bifurcation takes place. This bifurcation is called a pitchfork bifurcation. It is a pitchfork bifurcation that occurs at $\alpha=3$ in $f_\alpha^2$ that causes the first period-doubling bifurcation in the logistic family.

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Problem 42

Consider the family of functions $h_\lambda: \mathbb{R} \rightarrow \mathbb{R}$ given by $h_\lambda(x)=\lambda+x-x^2$ with $\lambda \in[-1,1]$.
(a) Determine the fixed points and their stability.
(b) Discuss the bifurcation that takes place over the parameter domain, and include illustrations showing how the graph of $h_\lambda$ changes (in relation to the line $y=x$ ) as the bifurcation takes place. This bifurcation is called a tangent bifurcation. Explain the name.

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Problem 43

Using a computer graphing program, examine the graphs of $f_\alpha^3$ and the line $y=x$ in order to approximate the value of $\alpha$ where the tangent bifurcation takes place that gives rise to the period-3 window shown in Figure 8.21.

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Problem 44

Here we show that neither topological transitivity nor the existence of a dense set of periodic points is by itself enough to imply sensitive dependence on initial conditions.
(a) Find a function $f: X \rightarrow X$ that have a dense set of periodic points in its domain but does not have sensitive dependence on initial conditions.
(b) Let $f: S^1 \rightarrow S^1$ be defined by $f(\theta)=\theta+1$; that is, $f$ is rotation of the circle by one radian. Prove that $f$ is topologically transitive but does not have sensitive dependence on initial conditions. (Hint: Given an interval $U$ in the circle, prove that there exists $n \in \mathbb{Z}_{+}$such that $f^n(U) \cap U \neq \varnothing$, but $f^n(U) \neq U$. Then consider the sets $f^{m n}(U)$ for $m \in \mathbb{Z}_{+}$.)

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Problem 45

In this problem, we demonstrate that sensitive dependence on initial conditions is not preserved under topological conjugacy. Let $f:(1, \infty) \rightarrow(1, \infty)$ be defined by $f(x)=2 x$, and let $g: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$be defined by $g(x)=1+x$.
(a) Prove that $f$ has sensitive dependence on initial conditions, but $g$ does not.
(b) By finding an explicit homeomorphism $h:(1, \infty) \rightarrow \mathbb{R}_{+}$, prove that $f$ and $g$ are topologically conjugate.

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Problem 46

Exercise 8.45 demonstrates that sensitive dependence on initial conditions is not generally preserved under topological conjugacy, but, as we show in this exercise, if we restrict to a compact domain, then it is.
Let $X$ and $Y$ be compact metric spaces with metrics $d_X$ and $d_Y$, respectively. Assume that $f: X \rightarrow X$ has sensitive dependence on initial conditions with sensitivity constant $\delta$, and assume that $g: Y \rightarrow Y$ is topologically conjugate to $f$ via a homeomorphism $h: X \rightarrow Y$.
(a) Prove that $C=\left\{\left(y_1, y_2\right) \in Y \times Y \mid d_X\left(h^{-1}\left(y_1\right), h^{-1}\left(y_2\right)\right)>\delta\right\}$ is a compact subset of $Y \times Y$.
(b) Prove that $d_Y\left(y_1, y_2\right)>0$ for all $\left(y_1, y_2\right) \in C$ and therefore $d_Y$ takes on a minimum positive value $\delta^*$ on $C$.
(c) Prove that $g$ has sensitive dependence on initial conditions with sensitivity constant $\delta^*$.

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Problem 47

Prove Theorem 8.20: A continuous function $f: X \rightarrow X$ is chaotic if and only if for every pair of open sets $U$ and $V$ in $X$ there is a periodic point $x \in U$ and $\mathrm{a} k \in \mathbb{Z}_{+}$such that $f^k(x) \in V$.

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