gibson:teaching:fall-2012:math445:lab11
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gibson:teaching:fall-2012:math445:lab11 [2012/11/26 19:35] gibson |
gibson:teaching:fall-2012:math445:lab11 [2012/12/04 06:03] (current) gibson [Problem 3: projectile motion] |
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| - | ===== Problem 1 ===== | + | ===== Problem 1: dy/dt = sin(y+t) ===== |
| Show a quiver plot in the //(t,y)// plane of the differential equation //dy/dt = sin(y+t)// over the region //0 ≤ t ≤ 2pi// and //-pi ≤ y ≤ pi//. | Show a quiver plot in the //(t,y)// plane of the differential equation //dy/dt = sin(y+t)// over the region //0 ≤ t ≤ 2pi// and //-pi ≤ y ≤ pi//. | ||
| Use Matlab's //ode45// function to compute solution curves //y(t)// for each of the following initial conditions, and superimposed the solution curves on the quiver plot in the indicated colors //y(0) = -1// in red, //y(0) = 0// in blue , //y(0) = 1// in black. Be sure to set the axes tight around the quiver plot, set the unit length of the axes to be equal, label the axes, and title the graph. | Use Matlab's //ode45// function to compute solution curves //y(t)// for each of the following initial conditions, and superimposed the solution curves on the quiver plot in the indicated colors //y(0) = -1// in red, //y(0) = 0// in blue , //y(0) = 1// in black. Be sure to set the axes tight around the quiver plot, set the unit length of the axes to be equal, label the axes, and title the graph. | ||
| - | ===== Problem 2 ===== | + | ===== Problem 2: van der Pol oscillator ===== |
| - | The van der Pol oscillator is defined by the second-order differential equation | + | (a) The van der Pol oscillator is defined by the second-order differential equation |
| <latex> | <latex> | ||
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| This second-order equation can be converted to a system of two first-order equations by letting | This second-order equation can be converted to a system of two first-order equations by letting | ||
| - | //y// be a 2-d vector $y = [y_1, y_2] = [x, ~dx/dt]$. Then $dy/dt = [y_2, ~\mu (1-y_1^2) y_2 + y_1]$. | + | //y// be a 2-d vector |
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| + | $y = [y_1, y_2] = [x, ~dx/dt]$. | ||
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| + | Then | ||
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| + | $dy/dt = [dy_1/dt, ~dy_2/dt ] = [y_2, ~\mu (1-y_1^2) y_2 - y_1]$. | ||
| Use this definition of the vector $dy/dt$ to produce a quiver plot of the van der Pol oscillator | Use this definition of the vector $dy/dt$ to produce a quiver plot of the van der Pol oscillator | ||
| in the $y_1, y_2$ plane for $\mu = 1$, over the range $-3 \leq y_1 \leq 3$ and | in the $y_1, y_2$ plane for $\mu = 1$, over the range $-3 \leq y_1 \leq 3$ and | ||
| - | $-3 \leq y_2 \leq 3$. Using Matlab's //ode45//, compute solution curves $y(t)$ for the initial values | + | $-3 \leq y_2 \leq 3$. Using Matlab's //ode45//, compute solution curves $y(t)$ for the five different initial values |
| - | $y(0) = [0.01, 0]$ and $y(0) = [3, 0]$, and superimpose them on the quiver plot in red and black. | + | $y(0) = [0.1~ 0], [0.2~0], [1~0], [2~0],$ and $[3~0]$, and superimpose them on the quiver plot in red. |
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| + | Hint: put the call to //ode45// inside a //for// loop that sets the value of $y_1$ to the five different initial values. | ||
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| + | (b) Put your work for part (a) in a script file, and run the script for a range of values of $\mu$ between -1 and 1. How does the character of the solutions of the equation change as $\mu$ passes through 0? Turn in plots for a negative, zero, and a positive value of $\mu$, whichever values you think best illustrate the change as $\mu$ passes through zero. | ||
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| + | Hint: To avoid the need to change the value of $\mu$ in both the script and the function file that defines $dy/dt = f(t,y)$, define $dy/dt = f(t,y)$ as an anonymous function within your script file, so that it inherits the $\mu$ of | ||
| + | the script file. | ||
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gibson/teaching/fall-2012/math445/lab11.1353987327.txt.gz · Last modified: 2012/11/26 19:35 by gibson
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