Computer Algebra Recipes

Richard H Enns, George C McGuire

Chapter 8 Nonlinear Diagnostic Tools - all with Video Answers

Educators

Chapter Questions

Problem 1

Holding all other parameter values as in the text recipe, use the Poincaré section to determine the response of Duffings's ODE for each $F$ value when $\beta=2$.

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01:05

Problem 2

For each of the four $F_i$, explore the response of the Duffing ODE as the frequency $\omega$ is varied, all other parameters being the same as in the text recipe.

James Kiss

Numerade Educator

03:06

Problem 3

Determine the response of the Duffing ODE for each $F_i$ when all numerical values are the same as in the text recipe, but the signs of $\alpha$ and $\beta$ are interchanged.

Kyle Findley

Numerade Educator

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Using the Poincaré section approach, determine the periodicity of the steadystate solution of the following forced oscillator equations:
(a) $\ddot{x}+0.7 \dot{x}+x^3=0.75 \cos t$, with $x(0)=\dot{x}(0)=0$;
(b) $\ddot{x}+0.08 \dot{x}+x^3=0.2 \cos t$, with $x(0)=0.25, \dot{x}(0)=0$.

Victor Salazar

Numerade Educator

01:37

Problem 5

Using the Poincaré section approach, determine the periodicity of the steadystate solution of the following ODE for $F=0.357$ and $F=0.35797$ :
$$
\ddot{x}+0.5 \dot{x}-x+x^3=F \cos (t+1), \text { with } x(0)=0.09, \dot{x}(0)=0 .
$$

Manik Pulyani

Numerade Educator

03:02

Problem 6

Replace the potential in the text recipe with
$$
V=\frac{1}{2} q_1^2+\frac{1}{2} q_2^2+q_1^4 q_2-\frac{1}{4} q_2^3 .
$$
(a) Execute the modified recipe with $E=1 / 16$ and initial conditions as in the text recipe. Discuss the resulting plots.
(b) Explore other initial conditions for the same total energy as in part (a). Discuss the results.
(c) Explore what happens as the energy is increased with the same initial conditions as in part (a). Discuss the results.
(d) Explore other potential energy functions and discuss the results.

Satpal Satpal

Numerade Educator

09:21

Problem 7

The Toda potential [Jac90] is given by
$$
V=\frac{1}{3}\left(e^{\left(q_2+\sqrt{3} q_1\right)}+e^{\left(q_2-\sqrt{3} q_1\right)}+e^{\left(-2 q_2\right)}\right)-1 .
$$
(a) Create two- and three-dimensional contour plots of $V$, choosing suitable potential energy contours and viewing ranges.
(b) Locate and identify the nature of the stationary points.
(c) Generate the Hamiltonian equations.
(d) For $E=1.0, p_2(0)=-0.05, q_2(0)=-0.2, q_1(0)=-0.2$, determine $p_1(0)$.
(e) Plot the system's trajectory in the $q_1-q_2-p_2$ space and discuss the result.
(f) Generate the Poincaré section in the $q_2$ vs. $p_2$ plane. Interpret the result.
(g) Explore the Toda Hamiltonian for other $E$ values and discuss the results.

Ren Jie Tuieng

Numerade Educator

02:09

Problem 8

Suppose that Frank N. Stein's heartbeat also contains the term $0.1 \sin (6 \pi t)$. Plotting $\sqrt{S}$, show that the presence of the third harmonic is revealed.

Varsha Aggarwal

Numerade Educator

Problem 9

Holding all other parameter values as in the text, use the power spectrum to determine the response of the forced Duffing ODE for each $F$ value when $\beta=2$.

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01:22

Problem 10

For each of the four $F$ values, use the power spectrum to explore the response of the Duffing oscillator as $\omega$ is varied, all other parameters being unchanged.

Manish Jain

Numerade Educator

Problem 11

Use the power spectrum to determine the response of the forced Duffing oscillator for each $F$ value when all numerical values are the same as in the text recipe, but the signs of $\alpha$ and $\beta$ are interchanged.

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02:18

Problem 12

Using the power spectrum approach, determine the nature of the steady-state solution for the following forced oscillator equation:
$$
\ddot{x}+0.7 \dot{x}+x^3=0.75 \cos t, \quad x(0)=\dot{x}(0)=0 .
$$

James Kiss

Numerade Educator

Problem 13

Using the power spectrum approach, determine the nature of the steady-state solution for the following forced Duffing equation:
$$
\ddot{x}+0.08 \dot{x}+x^3=0.2 \cos t, \quad x(0)=0.25, \dot{x}(0)=0 .
$$

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Problem 14

Using the power spectrum approach, determine the nature of the steady-state solution for the following oscillator equation for $F=0.357$ and $F=0.35797$ :
$$
\ddot{x}+0.5 \dot{x}-x+x^3=0.357 \cos (t+1), \quad x(0)=0.09, \dot{x}(0)=0 .
$$

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02:28

Problem 15

The equations describing forced oscillations of the glycolytic oscillator are
$$
\dot{x}=-x+\alpha y+x^2 y, \quad \dot{y}=\beta-\alpha y-x^2 y+A+F \cos (\omega t) .
$$
Taking $\alpha=\beta=0, A=0.999, F=0.42, x(0)=2$, and $y(0)=1$, determine the periodicity of the response using the Poincaré section approach for (a) $\omega=2$ and (b) $\omega=1.75$. Explore the frequency range in between and identify any interesting solutions.

Adriano Chikande

Numerade Educator

04:51

Problem 16

For the logistic map, produce the bifurcation diagram for the region $a=3.54$ to $a=3.6$, taking $x_0=0.2$, and dividing the $a$ interval into 100 steps. Summarize the various periodic solutions that you observe.

Melvin Adkins

Numerade Educator

01:29

Problem 17

Explore the periodic window in the vicinity of $a=3.8$ and report on what periodicities you observe at each $a$ value sampled.

Sneha Ravi

Numerade Educator

Problem 18

Produce a bifurcation diagram for the cubic map
$$
x_{n+1}=a x_n-x_n^3
$$
over a suitable range of the parameter $a$. Determine the values of $a$ in the diagram at which the periodicity changes.

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Problem 19

With $x_0=0.2$, produce the bifurcation diagram for the quartic map
$$
x_{n+1}=a x_n\left(1-x_n^3\right)
$$
over the range $a=1.5$ to $a=2.0$, taking as small an $a$ step size as you can. Summarize the behavior of the map as $a$ varies over the specified range.

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Problem 20

Produce the bifurcation diagram for the tent map
$$
x_{n+1}=2 a x_n, 0<x \leq \frac{1}{2} ; x_{n+1}=2 a\left(1-x_n\right), \frac{1}{2} \leq x<1
$$
with $0<a<1$. Take $x_0=0.2$ and $x_0=0.6$. Summarize the periodic solutions that you see in each case. Are there any differences in the bifurcation diagrams for the two inputs.

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01:03

Problem 21

Produce the bifurcation diagram for the sine map
$$
x_{n+1}=a \sin \left(\pi x_n\right)
$$
with $0 \leq a \leq 1$ and $0 \leq x \leq 1$. How does the bifurcation diagram qualitatively compare with that for the logistic map if only the range $a=0.7$ to 1 is plotted?

Carson Merrill

Numerade Educator

03:16

Problem 22

Produce bifurcation diagrams for the following maps over suitable ranges of $a>0$ and discuss the results:
(a) $x_{n+1}=x_n e^{a\left(1-x_n\right)}$
(b) $x_{n+1}=e^{-a x_n}$
(c) $x_{n+1}=a \cos \left(x_n\right)$
(d) $x_{n+1}=a+x_n^2$

Bobby Barnes

University of North Texas

Problem 23

With $x_0=0.2$, calculate the Lyapunov exponent for the quartic map
$$
x_{n+1}=a x_n\left(1-x_n^3\right)
$$
over the range $a=1.5$ to $a=2.0$, keeping all other parameters as in the text recipe. Over what regions of $a$ do periodic solutions occur?

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01:16

Problem 24

Plot the Lyapunov exponent $\lambda$ versus $a$ for the tent map
$$
x_{n+1}=2 a x_n, 0<x \leq \frac{1}{2} ; \quad x_{n+1}=2 a\left(1-x_n\right), \frac{1}{2} \leq x<1
$$
with $0<a<1$. Take $x_0=0.2$. Analytically show that $\lambda=\ln (2 a)$ and discuss your graph in terms of this result.

Nick Johnson

Numerade Educator

01:47

Problem 25

Plot the Lyapunov exponent versus $a$ for the sine map
$$
x_{n+1}=a \sin \left(\pi x_n\right)
$$
with $0 \leq a \leq 1$ and an $x$ value of your choosing from the range $0 \leq x \leq 1$. Discuss your graph.

Lucas Finney

Numerade Educator

01:03

Problem 26

Plot the Lyapunov exponent for the following map over the range $a=2.9$ to $a=6$ :
$$
x_{n+1}=a \sin \left(x_n\right)\left(1-\sin \left(x_n\right)\right) .
$$
Identify ranges of $a$ where periodic windows occur.

Carson Merrill

Numerade Educator

Problem 27

Taking $a=1.97$ and $x_0=0.9$, iterate the quartic map
$$
x_{n+1}=a x_n\left(1-x_n^3\right)
$$
to produce a time series with 200 points. Plot the time series with a line style. Attempt to reconstruct any possible underlying deterministic attractor by plotting $x_{n+1}$ versus $x_n$ using a point style. Show that these points lie on the curve $a X\left(1-X^3\right)$ with $a=1.97$. (This problem is a bit like a dog chasing its tail!)

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01:41

Problem 28

Consulting Maple's Help, produce a random set of 500 numbers from the positive integers one to six inclusive. This might simulate the rolling of an honest die. Make a plot of the "time series" and $x_{n+1}$ versus $x_n$ to show the randomness.

Nick Johnson

Numerade Educator

Problem 29

Apply the techniques of attractor reconstruction to some time series data (e.g., the Dow Jones index) extracted from newspapers, magazines, or whatever, and see whether a pattern emerges.

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Problem 30

Try reconstructing the butterfly attractor from the $z(t)$ time series, and explain why you do not obtain two "wings."

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Problem 31

The Rössler system is given by
$$
\dot{x}=-(y+z), \quad \dot{y}=x+a y, \quad \dot{z}=b+z(x-c) .
$$
Using the time series for $x(t)$ with $a=0.2, b=0.2, c=5.7, x(0)=y(0)=$ $z(0)=0.1$, construct the Rössler attractor. Hint: Your picture should resemble Figure 1.18.

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Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31