Project 3.10.1. Project—Parameter Space Analysis.
This project is about classifying all of the possible behaviors for a system of linear differential equations. Consider the family of systems of linear differential equations, \(d \mathbf x/dt = A \mathbf x\text{,}\) where
\begin{equation*}
A = \begin{pmatrix} a \amp c \\ b \amp a \end{pmatrix}.
\end{equation*}
The goal of this project is to produce a picture of how the values of \(a\text{,}\) \(b\text{,}\) and \(c\) affect the behavior of solutions to your system of differential equations. This means that you will need to precisely describe each possible region of \(abc\)-space and the corresponding types of behaviors in that region (spiral, sink, repeated eigenvalue, etc.). Representing these regions in three dimensional space can be difficult, so start early on this project and be creative in your creations. Your report should address the following questions at a minimum (but you will likely need to explore more than just these questions to fully understand your parameter space).
You might want to consider a strategy similar to the following.
- First examine the case when \(a = 0\text{.}\) You should compute the eigenvalues for this case and determine how the behavior of solutions to your system depend on \(b\) and \(c\text{.}\) Be sure to think about what is happening on the boundary between different regions and specify what is happening to solutions for each case. You may want to draw example phase planes for each of the regions. You should have a clear, accurate, and complete picture of the \(bc\)-space in terms of these behaviors.
- You should now do the same analysis for the case \(a = 1\) as you did for \(a = 0\text{.}\)
- Describe the behavior of your system when \(0 \lt a \lt 1\) and be specific about what is changing.
- Completely describe what behaviors occur when \(a = -1\text{.}\)
- You should now use the cases from your previous parts to help make a general picture. You need to draw the three dimensional parameter space showing all of the possible behaviors in the system and identify which regions (in terms of \(a\text{,}\) \(b\text{,}\) and \(c\)) exhibit which behaviors. Be creative in your display of this information. You may even want ot build a physical model to display your findings.
(a)
Describe the parameter space for \(d \mathbf x/dt = A \mathbf x\text{,}\) where
\begin{equation*}
A = \begin{pmatrix} a \amp c \\ b \amp a \end{pmatrix}.
\end{equation*}
(b)
Describe the parameter space for \(d \mathbf x/dt = A \mathbf x\text{,}\) where
\begin{equation*}
A = \begin{pmatrix} a \amp b \\ c \amp 0 \end{pmatrix}.
\end{equation*}
(c)
Describe the parameter space for \(d \mathbf x/dt = A \mathbf x\text{,}\) where
\begin{equation*}
A = \begin{pmatrix} a \amp c \\ 1 \amp b \end{pmatrix}.
\end{equation*}
