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Chapter 7: Problem 16

When you portray sufficient solution trajectories in the phase plane so as to determine all of the important behavior of a planar, autonomous system, you have created what is called a "phase portrait." In Exercises 15 - 22 , use pplane 6 to create a phase portrait for the indicated system on the prescribed display window. Take special notice of where solution curves end, as reported in the message window. $$ \begin{aligned} &x^{\prime}=y+\left(y^{2}-x^{2}+0.5 x^{4}\right)\left(1-x^{2}\right) \\ &y^{\prime}=x\left(1-x^{2}\right)-y\left(y^{2}-x^{2}+0.5 x^{4}\right) \\ &-3 \leq x \leq 3,-2 \leq y \leq 2 \end{aligned} $$

Short Answer

Expert verified
Use PPlane 6 to plot and analyze the phase portrait within the given window.

Step by step solution

01

Understand the System of Equations

Identify the system of differential equations given in the problem. We have \( x' = y + (y^2 - x^2 + 0.5x^4)(1 - x^2) \) and \( y' = x(1 - x^2) - y(y^2 - x^2 + 0.5x^4) \). These define a planar autonomous system, which means they do not explicitly depend on time.
02

Set Up the Display Window

Define the range of values for \( x \) and \( y \): \(-3 \leq x \leq 3\) and \(-2 \leq y \leq 2\). This specifies the part of the plane where you will plot the phase portrait.
03

Use PPlane 6 to Plot the Phase Portrait

Open the PPlane 6 tool, input the system \( x' = y + (y^2 - x^2 + 0.5x^4)(1 - x^2) \) and \( y' = x(1 - x^2) - y(y^2 - x^2 + 0.5x^4) \). Set the display window to \(-3 \leq x \leq 3\) and \(-2 \leq y \leq 2\). Use the software to generate the phase portrait.
04

Analyze the Phase Portrait

Observe the trajectories in the phase plane. Look for equilibrium points, cycles, or any recurring patterns. Focus on where solution curves originate and end, taking note of any points where the system tends to stabilize or become chaotic.
05

Report Findings from the Message Window

Once the simulation completes, review the message window in PPlane 6. This report often includes information about where solution trajectories begin and end, describing the system's stability areas, and any special characteristics noted.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Autonomous Systems
An autonomous system is a set of differential equations that does not explicitly depend on an independent variable, often time. In simpler terms, these systems maintain their form regardless of any external changes in time.
  • Autonomous systems in mathematics often appear as models in physics, biology, and ecology, where the state of the system can describe things like population dynamics, chemical reactions, or mechanical movements.
  • One key feature of being autonomous is that you can understand the system's behavior by studying its current state without needing an explicit time component. This makes them simpler to analyze in the context of phase portraits, which only need the system's current variables rather than their evolution over time.
Autonomous systems are described by equations such as \(x' = f(x, y)\) and \(y' = g(x, y)\), demonstrating their reliance on variables other than time.
Differential Equations
Differential equations are mathematical tools used to model systems where quantities change continuously, often to reflect natural phenomena. They express relationships between a function and its derivatives, capturing how a system evolves.
  • In our given system, \(x' = y + (y^2 - x^2 + 0.5x^4)(1 - x^2)\) and \(y' = x(1 - x^2) - y(y^2 - x^2 + 0.5x^4)\), these equations represent a dynamic system in a plane, meaning each change in one variable affects the other.
  • Understanding these differential equations is crucial for developing phase portraits that reveal the system’s behavior.
They serve as the foundation for analyzing systems in mathematical modeling, helping both visually and analytically to predict system behavior.
PPlane 6 Tool
The PPlane 6 tool is a graphical software application designed to visualize solutions to differential equations. It helps in plotting trajectories by creating phase portraits which are key for comprehensive system analysis.
  • The tool allows users to input a system of equations, define a display window, and observe real-time behavior of phases.
  • For beginners, PPlane 6 is intuitive and provides instant feedback on how solutions behave as parameters change.
In this exercise, using PPlane 6 involves inputting our system of differential equations and observing the dynamics over the given interval. It's incredibly valuable in revealing solutions that may not be evident through equations alone.
Equilibrium Points
Equilibrium points in a dynamical system are where the system remains constant, i.e., both derivatives in our differential equations become zero. This means no progression or regression at these points.
  • To find equilibrium points in our system, solve \(x' = 0\) and \(y' = 0\).
  • These points provide critical insight into the behavior and stability of the system.
  • Analyzing equilibrium points helps determine areas of stability and instability, revealing patterns such as sinks, sources, or nodes.
Equilibrium points assist in understanding the full picture of the system's dynamics, serving as anchors in phase portrait analyses.

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Most popular questions from this chapter

In the predator-prey system $$ \begin{aligned} &L^{\prime}=-L+L P \\ &P^{\prime}=P-L P \end{aligned} $$ \(L\) represents a lady bug population and \(P\) represents a pest that lady bugs like to eat. Enter the system in the PPLANE6 Setup window, set the display window so that \(0 \leq L \leq 2\) and \(0 \leq P \leq 3\), then select Arrows for the direction field. a) Use the Keyboard input window to start a solution trajectory with initial condition ( \(0.5,1.0)\). Note that the lady bug-pest population is periodic. As the lady bug population grows, their food supply dwindles and the lady bug population begins to decay. Of course, this gives the pest population time to flourish, and the resulting increase in the food supply means that the lady bug population begins to grow. Pretty soon, the populations come full cycle to their initial starting position. b) Suppose that the pest population is harmful to a farmer's crop and he decides to use a poison spray to reduce the pest population. Of course, this will also kill the lady bugs. Question: Is this a wise idea? Adjust the system as follows: $$ \begin{aligned} &L^{\prime}=-L+L P-H L \\ &P^{\prime}=P-L P-H P \end{aligned} $$ Note that this model assumes that the growth rate of each population is reduced by a fixed percentage of the population. Enter this system in the PPLANE6 Setup window, but keep the original display window settings. Create and set a parameter \(\mathrm{H}=0.2\). Start a solution trajectory with initial condition \((0.5,1.0)\). c) Repeat part b) with \(H=0.4,0.6\), and 0.8. Is this an effective way to control the pests? Why? Describe what happens to each population for each value of the parameter \(H\).

The equation \(m y^{\prime \prime}+d y^{\prime}+k y=0\) represents a damped, spring-mass system. In Exercises \(23-26\), values of the parameters \(m, d\), and \(k\) have been chosen to create a specific example of a damped, spring-mass system. We are looking for the solution \(y\) with the given initial position and velocity. Let $$ x_{1}=y, \quad \text { and } \quad x_{2}=y^{\prime}, $$ and use this change of variables to write each spring-mass system as a planar, autonomous system. Use pplane6 to obtain a printout of the graph of \(y\) versus \(t\). $$ \begin{aligned} &y^{\prime \prime}+4 y=0 \\ &y(0)=-4, y^{\prime}(0)=-2 \end{aligned} $$

The planar, autonomous system having the form $$ \begin{aligned} &x^{\prime}=a x+b y \\ &y^{\prime}=c x+d y \end{aligned} $$ where \(a, b, c\), and \(d\) are arbitrary real numbers, is called a linear system of first order differential equations. Exercises 1 \(-6\) each contain a solution of some linear system. Use MATLAB to create a plot of \(x\) versus \(t, y\) versus \(t\), and a plot of \(y\) versus \(x\) in the phase plane. Use the subplot command, as in subplot (221), plot (t, \(x)\), axis tight subplot(222), plot(t,y), axis tight subplot (221), plot \((t, x)\), axis tight subplot (222), plot (t,y), axis tight subplot (212), plot( \(x, y)\), axis equal subplot (212), plot \((x, y)\), axis equal to produce a plot containing all three plots. Use the suggested time interval. $$ \begin{aligned} &x=-2 e^{-2 t}+3 e^{-3 t} \\ &y=4 e^{-2 t}-3 e^{-3 t} \\ &{[-0.5,2]} \end{aligned} $$

The planar, autonomous system having the form $$ \begin{aligned} &x^{\prime}=a x+b y \\ &y^{\prime}=c x+d y \end{aligned} $$ where \(a, b, c\), and \(d\) are arbitrary real numbers, is called a linear system of first order differential equations. Exercises 1 \(-6\) each contain a solution of some linear system. Use MATLAB to create a plot of \(x\) versus \(t, y\) versus \(t\), and a plot of \(y\) versus \(x\) in the phase plane. Use the subplot command, as in subplot (221), plot (t, \(x)\), axis tight subplot(222), plot(t,y), axis tight subplot (221), plot \((t, x)\), axis tight subplot (222), plot (t,y), axis tight subplot (212), plot( \(x, y)\), axis equal subplot (212), plot \((x, y)\), axis equal to produce a plot containing all three plots. Use the suggested time interval. $$ \begin{aligned} &x=e^{-t}(\cos t+\sin t) \\ &y=e^{-t}(\cos t-\sin t) \\ &{[-\pi, \pi]} \end{aligned} $$

The equation \(m y^{\prime \prime}+d y^{\prime}+k y=0\) represents a damped, spring-mass system. In Exercises \(23-26\), values of the parameters \(m, d\), and \(k\) have been chosen to create a specific example of a damped, spring-mass system. We are looking for the solution \(y\) with the given initial position and velocity. Let $$ x_{1}=y, \quad \text { and } \quad x_{2}=y^{\prime}, $$ and use this change of variables to write each spring-mass system as a planar, autonomous system. Use pplane6 to obtain a printout of the graph of \(y\) versus \(t\). $$ \begin{aligned} &y^{\prime \prime}+3 y^{\prime}+2 y=0 \\ &y(0)=3, y^{\prime}(0)=2 \end{aligned} $$
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