# UTPA STEM/CBI Courses/Calculus/Ordinary Differential Equations (ODE's)/Nonlinear Systems of Differential Equations

Course Title: Ordinary Diﬀerential Equations
Lecture Topic: Nonlinear Systems of Diﬀerential Equations
Instructor: Josef Sifuentes
Institution: UTRGV

## Backwards Design

Course Objectives
* Primary Objectives- By the next class period students will be able to:
** Find equilibrium solutions to nonlinear systems of diﬀerential equations.
** Determine whether each equilibrium solution is stable (attractors) or unstable (repellents).
** Analyze the behavior of the system and its overall stability.
* Sub Objectives- The objectives will require that students be able to:
** Compute Jacobian matrices. – Compute eigenvalues. – Use Computer systems to approximate and illustrate solutions
* Difficulties- Students may have difficulty:
** Understanding the derivation of the mathematical model given a physical system.
** Understanding the Taylor expansion of a nonlinear, multidimensional function.
** Understanding how a linear system can locally approximate a nonlinear system.
** Computing Jacobian Matrices.
** Reconciling stable systems with physical attributes.
** Understanding the relationship between eigenvalues and stability.
* Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
** Modeling an outbreak of infectious disease
** Modeling populations
** Modeling predator / prey relationships.
** Modeling pendulum behavior.
** Modeling the dynamics of free falling on parameterized curves (e.g. the Brachistochrone and Tautochrone problem)
** Modeling weather dynamics (e.g. The Lorenz System)
Model of Knowledge
* Concept Map
** Nonlinear Systems
** Jacobians
** Eigenvalues
** Equilibrium Solutions
** Stability
* Content Priorities
** Enduring Understanding
*** Understanding the link between eigenvalues and stability
*** Understanding the diﬀerence between stable and unstable equilibria *** Explaining the long term behavior of a nonlinear, dynamical system
** Important to Do and Know
*** Compute Jacobian.
*** Compute eigenvalues.
*** Use computer systems to approximate and illustrate solutions.
** Worth Being Familiar with
*** Local Linearization on nonlinear functions.
*** Taylor expansion.
*** The relationship between the imaginary values of eigenvalues and natural frequencies.
*** Criteria for local linearity.
Assessment of Learning
* Formative Assessment
** In Groups (Class or Computer Lab setting)
*** Analyze how changing parameters can lead to stability or instability.
*** Use Computer systems to approximate and illustrate solutions.
*** Compute Jacobian at equilibria and the eigenvalues of Jacobian
** Individually (Homework)
*** Compute Jacobian of various nonlinear, multivariable functions.
*** Compute equilibria and stability of each equilibrium solution.
* Summative Assessment
** In class exams.
** Class presentation on Real World Example.

## Legacy Cycle

OBJECTIVE
* Analyze a nonlinear diﬀerential equation and the implications this analysis has on a real world example. This objective will require students to
** Find equilibrium solutions to nonlinear systems of diﬀerential equations.
** Determine whether each equilibrium solution is stable (attractors) or unstable (repellents).
** Analyze the behavior of the system and its overall stability.
** Compute Jacobian matrices.
** Compute eigenvalues.
** Use Computer systems to approximate and illustrate solutions.
THE CHALLENGE
An Ebola outbreak has begun in the United States. The students will model the disease under current parameters as well as under various parameter changes as a result of public health campaigns. The students must decide which set of parameters are best.
GENERATE IDEAS
Which public health campaign do the students think will be most eﬀective? Why? (After the research and revise, the students can match up their intuition with more formal mathematical reasoning)
MULTIPLE PERSPECTIVES
The students will share their ideas on best strategies with each other and their reasoning. Then they will come to a revised opinion as a group.
RESEARCH & REVISE
This is where I lecture on the on the primary and sub objectives listed in the Backwards Design component of this template.
TEST YOUR METTLE
* Compute Jacobian of various nonlinear, multivariable functions.
* Compute equilibria and stability of each equilibrium solution.
GO PUBLIC
Students will present their ﬁndings on which public health campaign is best (each public health campaign corresponds to a diﬀerent set of parameters). They will show how the eigenvalues of the Jacobian, evaluated at the equilibrium solutions change under the diﬀerent set of parameters. They will explain how these eigenvalues predict the behavior demonstrated in the approximate solutions illustrated on via a computer modeling program.

## Pre-Lesson Quiz

Various material can be tested here, depending on time constraints.

## Test Your Mettle Quiz

* Consider the function
** Demonstrate that x$_e$ = [1 0]$^T$ satisﬁes f(x$_e$) = 0.
** Compute the Jacobian of f(x) evaluated at x$_e$ and compute the eigenvalues of that matrix.
** Is the system y'(t) = f(y) stable?
** If y is a solution to y'(t) = f(y) for a given initial condition, what is ?
* Consider the function
** Demonstrate that x$_u$ = [π 0]$^T$ satisﬁes f(x$_e$) = 0. Demonstrate that x$_s$ = [0 0]$^T$ satisﬁes f(x$_s$) = 0.
** Compute the Jacobian of f(x) evaluated at both equilibrium solutions and compute the eigenvalues of each matrix.
** Which of the equilibrium solutions are stable?
** If y is a solution to y'(t) = f(y) for an initial condition close to the zero vector, Can you bound ||y(t)|| for all values of t?
* Consider the function
** Demonstrate that x$_u$ = [π 0]$^T$ satisﬁes f(x$_e$) = 0. Demonstrate that x$_s$ = [0 0]$^T$ satisﬁes f(x$_s$) = 0.
** Compute the Jacobian of f(x) evaluated at both equilibrium solutions and compute the eigenvalues of each matrix.
** Which of the equilibrium solutions are stable?
** If y is a solution to y'(t) = f(y) for an initial condition, what is ?