MATLAB Contour Plots for ODE Solutions

In summary, MATLAB has a contour plot application that can be used to generate level curves for a two variable function, such as the implicit solution mentioned in the conversation.
  • #1
mathema
1
0
Salut –

I will be starting the control systems track at my university in a couple of weeks, so naturally I have been brushing up on my skills concerning the solution of DEs and difference equations. I was just wondering if there is an easy way to generate some of the graphs of a 1-parameter family of solutions to a first order ODE using MATLAB? (Particularly implicit solutions.) For example, an implicit solution to the simple ODE

(cosxsinx – xy^2)dx + y(1 – x^2)dy = 0

is given by

y^2(1 – x^2) – (cosx)^2 = c.

Because the equation is exact its solution is routine, but since my freshmen year I have always thought the family of curves it generates (especially near x=0) to be fairly interesting, so one of the first things I tried to do with my new copy of MATLAB was to generate this family for a few values of c… so far without success.

Anyway, my question is:

Does MATLAB have some sort of contour plot application for producing the level curves of a two variable function of the form G(x,y) = c ?
 
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  • #2




Hello! Congratulations on starting your control systems track at university. It sounds like you have been diligently preparing for your studies. To answer your question, yes, MATLAB does have a contour plot application for generating level curves of a two variable function. You can use the contour function in MATLAB to plot the implicit solution you mentioned for different values of c. Here's an example code:

% Define the implicit solution function
f = @(x,y,c) y.^2.*(1-x.^2) - (cos(x)).^2 - c;

% Define the range of x and y values
x = linspace(-5,5,100);
y = linspace(-5,5,100);

% Create a meshgrid of x and y values
[X,Y] = meshgrid(x,y);

% Choose a few values of c
c = [-2 -1 0 1 2];

% Plot the contour lines for each value of c
figure
contour(X,Y,f(X,Y,0),[0 0],'k'); % plot the solution curve for c=0
hold on
for i = 1:length(c)
contour(X,Y,f(X,Y,c(i)),[c(i) c(i)]);
end
title('Family of curves for the implicit solution');
xlabel('x');
ylabel('y');
legend('c=0','-2','-1','0','1','2');
hold off

This will generate a plot with the solution curve for c=0 and the family of curves for c=-2,-1,0,1,2. You can adjust the range of x and y values and the values of c to explore different parts of the solution curve. I hope this helps and good luck with your studies!
 
  • #3


I can say that MATLAB does indeed have the capability to generate contour plots for ODE solutions. Contour plots are a useful tool for visualizing solutions to ODEs because they show how the solution varies with respect to different parameters. To generate a contour plot in MATLAB, you can use the "contour" or "contourf" functions, which take in a grid of points and a corresponding function value at each point. For your specific example, you would first need to rearrange the implicit solution into the form y = f(x,c) and then create a grid of x and c values to plot. This can be done using the "meshgrid" function. Once you have your grid and function values, you can use the "contour" function to plot the level curves. You can also add labels and a colorbar to make the plot more informative. Overall, using MATLAB to generate contour plots for ODE solutions is a simple and effective way to visualize the behavior of a 1-parameter family of solutions.
 

1. What is a MATLAB contour plot?

A MATLAB contour plot is a graphical representation of a function of two variables, showing the level curves of the function on a 2D plane. The contours are lines that connect points of equal value, allowing for visualizing the behavior of the function across the entire plane.

2. How can MATLAB contour plots be used to analyze ODE solutions?

Contour plots can be used to visualize the behavior of a solution to a system of ordinary differential equations (ODEs). The contour lines represent the trajectories of the solution over time, allowing for easy identification of equilibrium points and understanding of the overall behavior of the system.

3. What are the inputs required for creating a MATLAB contour plot for ODE solutions?

The inputs required for creating a MATLAB contour plot for ODE solutions are the ODE system itself, initial conditions, and the range of values for the independent variables. Additionally, the user may choose to specify the number of contour lines and their levels to be plotted.

4. How can I customize the appearance of a MATLAB contour plot?

There are several ways to customize the appearance of a MATLAB contour plot. This includes changing the color, line style, and thickness of the contour lines, as well as adding labels, titles, and legends. The user can also adjust the axes and grid settings to enhance the visualization of the plot.

5. Are there any limitations to using MATLAB contour plots for ODE solutions?

While MATLAB contour plots are a useful tool for visualizing ODE solutions, they have some limitations. They can only be used for systems with two independent variables, and the accuracy of the plot is dependent on the resolution of the grid used. Additionally, contour plots may not be suitable for very complex systems with many equilibrium points and intricate behavior.

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