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[edit  edit source]
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.
 RealWorld Contexts There are many ways that students can use this material in the realworld, 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.
 Enduring Understanding
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.
 In Groups (Class or Computer Lab setting)
 Summative Assessment
 In class exams.
 Class presentation on Real World Example.
Legacy Cycle[edit  edit source]
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.
PreLesson Quiz[edit  edit source]
Various material can be tested here, depending on time constraints.
Test Your Mettle Quiz[edit  edit source]
 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 ?
