Chapter 1, Phase-Plane Portraits Video Solutions, Computer Algebra Recipes

Problem 1

Recast each of the following second-order ODEs into a first-order ODE system and identify $P$ and $Q$. Which systems are nonlinear? For each autonomous system, locate all the stationary points. All parameters are real and positive.
(a) Airy equation: $y^{\prime \prime}-x y=0$;
(b) "soft" spring equation: $\ddot{x}+\left(1-x^2\right) x=0$;
(c) Hermite equation: $y^{\prime \prime}-x y^{\prime}+n y=0$;
(d) Rayleigh equation: $\ddot{x}-\epsilon\left(1-\dot{x}^2\right) \dot{x}+x=0$;
(e) confluent hypergeometric equation: $x y^{\prime \prime}+(\gamma-x) y^{\prime}-\alpha y=0$;
(f) plane pendulum equation: $\ddot{\theta}+\sin \theta=0$.

Narayan Hari

Numerade Educator

01:07

Problem 2

Display and evaluate the following derivatives:
(a) $\frac{d^5}{d x^5}\left(x^6 \ln (x) \cos (x) e^{-x^2}\right)$,
(b) $\frac{d^9}{d x^9}\left(\frac{x^{11} \tanh (2 x)}{\sqrt{1+x^4}}\right)$

Joseph Liao

Numerade Educator

05:27

Problem 3

For each of the following possible love affairs involving Romeo and Juliet:
- produce a tangent field plot using the dfieldplot command and identify each stationary point and its stability;
$\bullet$ produce a phase-plane portrait with the specified initial condition;
$\bullet$ use the phaseportrait command and scene options to plot $J(t)$ and $R(t)$;
$\bullet$ use the dsolve command to derive the analytic solution;
$\bullet$ discuss how the love affair evolves with time.
(a) $\dot{R}=-2 R+J, \quad \dot{J}=-R-2 J, \quad R(0)=-1, J(0)=1$.
(b) $\dot{R}=J, \quad \dot{J}=-R, \quad R(0)=5, J(0)=-2$.
(c) $\dot{R}=J, \quad \dot{J}=-R+J, \quad R(0)=0.1, J(0)=0$.
(d) $\dot{R}=2 R+J, \quad \dot{J}=R+2 J, \quad R(0)=0.2, J(0)=-0.1$.

Sachchidanand Prasad

Numerade Educator

01:46

Problem 4

Given $\omega=1$, what $\beta$ value corresponds to critical damping of the SHO? Make a phase-plane portrait for this case using the DEplot command and the initial condition $x(0)=1, \dot{x}(0)=0$. Use this command to plot $x(t)$. Then obtain the analytic solution.

Manik Pulyani

Numerade Educator

04:54

Problem 5

Two biological species competing for the same food supply are described by the following nonlinear population number equations:
$$
\dot{N}_1=\left(4-0.0002 N_1-0.0004 N_2\right) N_1, \quad \dot{N}_2=\left(2-0.00015 N_1-0.00005 N_2\right) N_2 \text {. }
$$
(a) Locate all the stationary points.
(b) Create a tangent field plot that includes all the stationary points. Identify the nature of these points.
(c) Create a phase-plane portrait that includes several representative trajectories that support your identification of the stationary points.
(d) Use the scene option to plot $N_1(t)$ and $N_2(t)$.
(e) Attempt to obtain an analytic solution of the ODE system.

Sana Riaz

Numerade Educator

01:28

Problem 6

With all other parameters as in the text recipe, show for a number of different initial conditions that all trajectories wind onto the limit cycle. Take the orientation that shows the $y$ versus $x$ phase plane.

Dan Rhoads

Numerade Educator

00:46

Problem 7

With all other parameters as in the text recipe, investigate the behavior of the Van der Pol equation as the resistance $R$ is varied. Choose an orientation that shows $x$ versus $\tau$. Discuss the results.

Sriram Soundarrajan

Numerade Educator

01:20

Problem 8

In the text recipe, use the phaseportrait command instead of DEplot3d to make a phase-plane portrait with the tangent field included.

Mohamed Mohamed

Numerade Educator

02:56

Problem 9

In the text example, keep all equation coefficients the same (leaving $\gamma=0.25$ ) but take the forcing amplitude $F$ to be zero.
(a) If the inverted spring system is initially at rest in the right potential well with total energy $E=+\frac{1}{4}$, to what stationary point does it asymptotically evolve? Make a phase-plane portrait and a plot of $x$ versus $t$.
(b) Identify the stationary point.
(c) How long does it take for the system to be within $1 \%$ of this point?
(d) If the spring system is initially at rest in the right potential well, what is the minimum value that the total energy must have so that it asymptotically approaches the stationary point in the left potential well?

Averell Hause

Carnegie Mellon University

05:13

Problem 10

In the text example, determine an approximate $F$ for which period 8 occurs.

Bobby Barnes

University of North Texas

01:35

Problem 11

In the text example, what effect does changing the driving frequency to $\omega=2$ have on the four graphs? Identify the period response for each $F$ value.

Ankit Gupta

Numerade Educator

01:28

Problem 12

With all other parameters the same as in the text, but with $F=0.42$, study the response of the inverted Duffing system as $\omega$ is varied over the range between zero and one. Interpret the results in each case.

James Kiss

Numerade Educator

02:53

Problem 13

Varying the damping coefficient
With all other parameters the same as in the text, investigate the effect on the four graphs when the damping coefficient is reduced to $\gamma=0.125$. Identify the period response for each $F$ value. Repeat with $\gamma=0.0625$.

Norman Atentar

Numerade Educator

Problem 14

Execute the text recipe with the $x^3$ term in the force law replaced with $x^5$ and discuss how this change affects the results. Then try some larger $F$ values (all other parameters remaining the same) and determine the period response of each solution.

Check back soon!

Problem 15

For the nonharmonic Duffing oscillator with $\alpha=0, \beta=1, \gamma=0.04, \omega=1$, $F=0.2, x(0)=0.25$, and $y(0) \equiv \dot{x}(0)=0$, determine the period response of the solution. Use both scene $=[x, y]$ and scene $=[t, x]$ before making your conclusion. What is the period response if $x(0)=0.2$, all other parameter values remaining the same?

Check back soon!

02:58

Problem 16

Determine the period response of the forced-oscillator equation
$$
\ddot{x}+0.7 \dot{x}+x^3=0.75 \cos t,
$$
subject to the initial condition $x(0)=\dot{x}(0)=0$. Explore the change in period response of the solution as the force amplitude is varied. What type of Duffing equation is the above equation?

Shoukat Ali

Other Schools

01:07

Problem 17

Instead of the planar plots presented in the text recipe, make use of the DEplot3d command to make three-dimensional plots in the $t$ versus $x$ versus $y$ space. Plot the trajectory for each $F$ value separately, choosing an orientation in each case that gives the best view.

Chris Trentman

Numerade Educator

Problem 18

Confirm that a limit cycle results in the Oregonator model, regardless of the initial (nonzero) concentrations.

Check back soon!

04:33

Problem 19

The rate equations for a certain chemical oscillator are
$$
\begin{array}{rll}
A & \stackrel{k_1}{\rightarrow} & X \\
B+X & \stackrel{k_2}{\rightarrow} & Y+* \\
2 X+Y & \stackrel{k_3}{\rightarrow} & 3 X \\
X & \stackrel{k_4}{\rightarrow} & *
\end{array}
$$
where the concentrations $A$ and $B$ of species A and B are held constant.
(a) Using the empirical rule for chemical reactions, write down the rate equations for $X$ and $Y$.
(b) Convert the rate equations into a normalized form by setting $\tau=k_4 t$, $x=\sqrt{\left(k_3 / k_4\right)} X, y=\sqrt{\left(k_3 / k_4\right)} Y, a=\sqrt{\left(k_3 / k_4\right)}\left(k_1 / k_4\right) A, b=\left(k_2 / k_4\right) B$.
(c) Taking $a=1, b=2.5, x(0)=y(0)=0.1$, produce a 3 -dimensional plot showing $x(t)$ vs. $y(t)$ vs. $t$. Choose an orientation that clearly shows a periodic orbit.
(d) Confirm the limit-cycle nature by trying a few different initial normalized concentrations.

Sana Riaz

Numerade Educator

00:55

Problem 20

In the text recipe for the Oregonator model, the fudge factor was taken to be $h=0.75$. Exploring the range $h=0.1$ to $h=1$, with all other conditions the same, determine whether a limit cycle occurs. Comment on the sensitivity of the model on $h$.

Dharmendra Jain

Numerade Educator

Problem 21

Run the text recipe with an initial condition near the second fixed point. What is the probable nature of this fixed point? What is the nature of the resulting trajectory as time progresses?

Check back soon!

Problem 22

Holding all other parameters as in the text recipe, explore the behavior of the Rössler system as the coefficient $c$ is varied. Interpret the results.

Check back soon!