Python Tutorial for Phase Planes

This notebook demonstrates the use of functions related to phase plane portraits from the ode_tools Python module.

The actual Python code for each function can be found in the file named ode_tools.py located in the directory ../utils (or wherevever you have previously saved this file).
Good news! The functions defined in ode_plot_tools.py are coded and ready for use with no mofications needed to the source file!
- You do not even have to view the source file to understand how to use and adjust the functions to fit your needs.
See the documentation below for a “Quick Reference Guide” to working with phase plane portrait functions in the ode_tools module.

Section 1: Loading `ode_tools` from GitHub

Run the code cell below to load the most up to date modules stored in GitHub.
You will only need to run this code cell one time during an active session.

!pip install git+https://github.com/CU-Denver-MathStats-OER/ODEs
from IPython.display import clear_output
clear_output()

Section 2: Plotting a Phase Plane Portrait

A function named phase_portrait() plots a phase plane portrait (vector field) for a system of first order differential equations over a specified range of values for the dependent variables \(x\) and \(y\).

We use x for dependent variable 1.
We use y for dependent variable 2.
We use the function f(Y, t) for the system of differential equations.
- The output of this function is a vector.
- The first entry is a formula for the first differential equation \(\dfrac{dx}{dt}\).
- The second entry is a formula for the second differential equation \(\dfrac{dy}{dt}\).

How to Plot with `phase_portrait(x, y, f)`

Consider plotting the system the of differential equations below in the phase plane with a window of \(0 \leq x \leq 5\) and \(0 \leq y \leq 5\).

\[\begin{array}{l} \dfrac{dx}{dt} = 3x-1.4xy \\ \dfrac{dy}{dt} = -y + 0.8xy\\ \end{array}\]

Input vectors of values for the dependent variables x and y.
- In the code cell below, we set up a grid of equally spaced values along the intervals \(0 \leq x \leq 5\) and \(0 \leq y \leq 5\).
Define the system of two differential equations f(Y,t).
- In the code cell below, we enter the system of differential equations above.
- The formula for \(\frac{dx}{dt}=3x-1.4xy\) is entered as 3*x - 1.4*x*y.
- The formula for \(\frac{dy}{dt}=-y+0.8xy\) is entered as -y + 0.8*x*y.

import numpy as np

# Set plot range
x = np.linspace(0.0, 5.0, 20)  # range of values for x
y = np.linspace(0.0, 5.0, 20)  # range of values for y

# Enter differential equation
def f(Y, t):
    x, y = Y
    return [3*x - 1.4*x*y,  # formula for dx/dt
            -y + 0.8*x*y]  # formula for dy/dt

Import the phase_portrait function.

Like packages, you only need to import a function one time after opening a notebook.
Be sure you have first loaded ode_tools.py from GitHub..
If you get an error message, it is likely you forgot to first load the ode_tools module from GitHub. See Section 1: Loading ode_tools from GitHub

from utils.ode_tools import phase_portrait  # Only need to import one time.

We generate the slope field by running the function with the command phase_portrait(x, y, f).

# Plots a phase portrait

phase_portrait(x, y, f)

Each time you want to plot a new solution:

Redefine x, y, and f(Y, t) as needed.
Then run the command phase_portrait(x, y, f).

Section 3: Adding a Solution to a Phase Plane Portrait

A function named plot_phase_sol plots particular solutions in a phase plane portrait (vector field) for a system of first order differential equations over a specified range of values for the indpendent and dependent variables.

We use x for dependent variable 1.
We use y for dependent variable 2.
We use the function f(Y, t) for the system of differential equations.
- The output of this function is a vector.
- The first entry is a formula for the first differential equation \(\dfrac{dx}{dt}\).
- The second entry is a formula for the second differential equation \(\dfrac{dy}{dt}\).
We use tspan for the range of time to visualize the solution.
We use x0 for the initial value of the first dependent variable \(x\).
We use y0 for the initial value of the first dependent variable \(y\).

How to Plot with `plot_phase_sol(x, y, f, tspan, x0, y0)`

Consider plotting the solution to the system the of differential equations below with initial condition \((x(0),y(0)) = (2,3)\) in the phase plane with over a window of \(0 \leq x \leq 5\) and \(0 \leq y \leq 5\).

\[\begin{array}{l} \dfrac{dx}{dt} = 3x-1.4xy \\ \dfrac{dy}{dt} = -y + 0.8xy\\ \end{array}\]

Input vectors of values for the dependent variables x and y.
- In the code cell below, we set up a grid of equally spaced values along the intervals \(0 \leq x \leq 5\) and \(0 \leq y \leq 5\).
Define the system of two differential equations f(Y,t).
- In the code cell below, we enter the system of differential equations above.
- The formula for \(\frac{dx}{dt}=3x-1.4xy\) is entered as 3*x - 1.4*x*y.
- The formula for \(\frac{dy}{dt}=-y+0.8xy\) is entered as -y + 0.8*x*y.
Enter the range of time over which the solution will be plotted using tspan.
Enter the initial value for \(x\) at \(t=0\) as x0.
Enter the initial value for \(y\) at \(t=0\) as y0.

import numpy as np

# Set plot range
x = np.linspace(0.0, 5.0, 20)  # range of values for x
y = np.linspace(0.0, 5.0, 20)  # range of values for y

# Enter differential equation
def f(Y, t):
    x, y = Y
    return [3*x - 1.4*x*y,  # formula for dx/dt
            -y + 0.8*x*y]  # formula for dy/dt

# Enter range of time
tspan = np.linspace(0, 50, 200) # range of time to visualize solution

# Enter initial values
x0 = 2  # initial value of x
y0 = 3  # initial value of y

Import the plot_phase_sol function.

Like packages, you only need to import a function one time after opening a notebook.
Be sure you have first loaded ode_tools.py from GitHub..
If you get an error message, it is likely you forgot to first load the ode_tools module from GitHub. See Section 1: Loading ode_tools from GitHub

from utils.ode_tools import plot_phase_sol  # Only need to import one time.

We generate the slope field by running the function with the command plot_phase_sol(x, y, f, tspan, x0, y0).

# Plots a solution in a phase plane portrait

plot_phase_sol(x, y, f, tspan, x0, y0)

Each Time You Want To Plot A New Solution:

Redefine x, y, f(Y, t), tspan, x0, and y0 as needed.
Then run the command plot_phase_sol(x, y, f, tspan, x0, y0).

Creative Commons License Information

Exploring Differential Equations by Adam Spiegler is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Python scripts created by Jonathon Hirschi, Troy Butler, and Adam Spiegler.

Section 1: Loading ode_tools from GitHub