Preface

This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The Mathematica commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts.

Finally, you can copy and paste all commands into your Mathematica notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License (GPL). You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately. The tutorial accompanies the textbook Applied Differential Equations. The Primary Course by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

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Introduction to Linear Algebra

Glossary

Hopf Bifurcation

A very important bifurcation for two-dimensional systems is the Hopf bifurcation. It is characterized by the emergence of periodic solutions. The generic system that produces a (supercritical) Hopf bifurcation is given by

\[ \begin{cases} \dot{x} &= -y + x \left[ r- \left(x^2 + y^2 \right) \right] , \\ \dot{y} &= x + y \left[ r- \left(x^2 + y^2 \right) \right] . \end{cases} \]
We see immediately that (0,0) is a point equilibrium. The following analysis shows that for r<0, the point equilibrium (0,0) is locally stable and for r>0, it is unstable. Define
\[ f(x,y) = -y + x \left[ r- \left( x^2 + y^2 \right) \right] , \qquad g(x,y) = x + y \left[ r- \left( x^2 + y^2 \right) \right] . \]
Then
\[ f(x,y) = -y + rx -x^3 + x\,y^2 , \qquad g(x,y) = x + ry -x^2 y + y^3 . \]
The Jacobian matrix is given by
\[ {\bf J}(x,y) = \begin{bmatrix} r -3x^2 - y^2 & -1 -2xy \\ 1-2xy & r - x^2 -3 y^2 \end{bmatrix} . \]
Evaluating this matrix at the equilibrium (0,0) yields
\[ {\bf J}(0,0) = \begin{bmatrix} r & -1 \\ 1 & r \end{bmatrix} . \]
The eigenvalues of this matrix are
\[ \lambda = r \pm {\bf j} . \]
This shows that the equilibrium (0,0) is locally stable for r<0 and unstable for r>0.

As r crosses 0 from the left to the right, a periodic solution appears. That is, produce two graphs, one for r = -0.2, the other for r = 0.2. You will see the emergence of a stable limit cycle. This is called a Hopf bifurcation. The analytical analysis only shows that the equilibrium (0,0) becomes unstable as r crosses 0 from the left to the right. This does not imply that there will be a locally stable periodic solution.

 

 

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