Advanced Differential Equations 001

From Prof. J. Nathan Kutz’ Lecture 001

Phase-Plane Analysis For Nonlinear Dynamics

We begin by reviewing linear system

\[\mathbf{x}'=\mathbf{Ax}\]

where

\[\mathbf{x} \in \mathbb{R}^{2}, \mathbf{A} \in \mathbb{R}^{2 \times 2}\]

The equilibrium points of this system are determined by setting \(\mathbb{x}'=0\).

\[\mathbf{x}'=\mathbf{Ax}=0 \implies \mathbf{x}=0\]

Here we assume \(A\) is not singular. The differential equation can be solved by

\[\mathbf{x}=\mathbf{v}e^{\lambda t} \implies (\mathbf{A}-\lambda \mathbf{I})\mathbf{v}=0\]

There are five cases

• Cases 1: eigenvalues are real, unequal, same sign

\[\mathbf{x} = c_1 \mathbf{v}^{(1)}e^{\lambda_1 t} + c_2 \mathbf{v}^{(2)} e^{\lambda_2 t}\]

• Cases 2: eigenvalues are real, opposite sign

\[\mathbf{x} = c_1 \mathbf{v}^{(1)}e^{\lambda_1 t} + c_2 \mathbf{v}^{(2)} e^{-\lambda_2 t}\]

• Cases 3: eigenvalues are real and equal

For the case of a double root, two possibilities exist: either we can find two linearly independent eigenvectors so that our solution is

\[\mathbf{x} = c_1 \mathbf{v}^{(1)}e^{\lambda_1 t} + c_2 \mathbf{v}^{(2)} e^{\lambda_1 t}\]

or, there is only one eigenvector, and we must generate a generalized eigenvector via the methods

\[\mathbf{x} = c_1 \mathbf{v}^{(1)}e^{\lambda_1 t} + c_2 [\mathbf{v}^{(1)}t e^{\lambda_1 t}+\boldsymbol{\eta}e^{\lambda_1 t}]\]

• Cases 4: eigenvalues are complex eigenvalues

\[\lambda_{\pm} = \beta \pm i\mu\]

• Cases 5: eigenvalues are purely imaginary

\[\lambda_{\pm} = \pm i\mu\]