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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Bifurcations}




\markboth{Bifurcations}{Bifurcations}

\author{Edward W. Swim\\
Department of Mathematics and Statistics\\
Sam Houston State University\\
Huntsville TX 77341-2206 USA\\
edward.swim@shsu.edu
}

\makeStitlePDFLaTex


\begin{abstract}
We lead students to investigate first-order differential equations that contain unknown parameters.  Students discover what happens to the qualitative behavior of solutions to these equations as these parameters vary.
\end{abstract}


\title{TEACHER VERSION\\BIFURCATIONS}



 
\section*{INTRODUCTION}

Bifurcation analysis is a useful mathematical tool applied in a wide variety of application areas such as neural network models for human sensory systems \cite{Freeman}, quantum motion in molecular systems \cite{Yang}, density dependent models of logistic growth in ecology \cite{Gop}, etc.  In these situations, solutions to initial value problems are often sensitive to changes in the value of one or more parameter within the model.  In particular, it is possible that an equilibrium solution may suddenly change from being asymptotically stable to unstable as parameter values are varied.  For example, in quantum mechanics a system of particles may change phase at absolute zero temperature based upon changes in quantum momentum or some other parameter.  Whenever quantum momentum is high enough, the system is typically disordered, while at lower momentum values a topologically ordered state of matter may be observed \cite{Yang}.  Unlike thermal phase transitions (\emph{e.g.}, between solid and liquid states) where changes in pressure at a fixed positive temperature may produce mesophases such as liquid crystal, a quantum change of phase occurs suddenly for a single value of one or more parameters.  Frequently, these changes are modeled using differential equations for local quantities such as magnetization where a singularity in a derivative indicates the instantaneous change in topological order \cite{Chamon}.

In the setting of ordinary differential equations in one real variable, models for various biological systems include a similar behavior.  Species of animals where reproductive rates depend highly on food availability at a time prior to when eggs hatch have population dynamics modeled by a delay modification to the logistic equation:
\[\frac{dx}{dt}=rx\left(1-\frac{x(t-T)}{K}\right),\]
commonly referred to as the Hutchinson-Wright equation \cite{Allen}.
Here, $r$ is the usual intrinsic growth rate, $T$ is a delay parameter, and $K$ is the carrying capacity.  It is worth noting that whenever $r>2$, solutions become chaotic \cite{Allen}.  Moreover, there is a critical value, $T=\pi/(2r)$, at which the equilibrium solution $x(t)=K$ shifts between being unstable and asymptotically stable.  Hence, whenever the delay parameter $T$ becomes very large, the population will not tend towards carrying capacity in the long run but rather will decay exponentially.


We investigate first-order differential equations that contain an unknown parameter such as this and identify a methodology for characterizing such a shift in the behavior of equilibrium solutions.  Our goal is to understand what happens to the qualitative behavior of solutions to differential equations as we vary the value of this parameter.  We will construct an array of phase portraits for different values of the parameter and then summarize the qualitative behavior of solutions in a new graph called a \emph{bifurcation diagram}.  The name for this diagram comes from the Latin word \emph{bifurcatus}, meaning forked in two.  In this first exercise in bifurcation it will become obvious why we use this terminology.

We begin by examining the differential equation

\begin{equation}\label{eq:first}
\frac{dx}{dt} =r+x^2\, .
\end{equation}

\begin{itemize}
  \item [1)]
 Consider equation (\ref{eq:first}) for four distinct values of the parameter $r$:  $r = -4$,  $r = -1$, $r = 0$, and $r = 1$.  In each of these cases, find all equilibria, determine their stability properties, and create a plot of several solutions to the equation where each one begins with a different initial condition. Summarize and report your results in a structured presentation with diagrams and annotation as well as a good narrative.
\end{itemize}

\noindent
We make a slight change in (\ref{eq:first}) and study the results as we consider the differential equation

\begin{equation}\label{eq:second}
\frac{dx}{dt} =r-x^2\, .
\end{equation}

\begin{itemize}
  \item [2)]
 Consider equation (\ref{eq:second}) for four distinct values of the parameter $r$:  $r = -4$,  $r = -1$, $r = 0$, and $r = 1$.  In each of these cases, find all equilibria, determine their stability properties, and create a plot of several solutions to the equation where each one begins with a different initial condition.    Summarize and report your results in a structured presentation with diagrams and annotation as well as a good narrative.
\end{itemize}




\noindent
We now increase our power of $x$, thus making  a slight change in (\ref{eq:second}), and study the results as we consider the differential equation

\begin{equation}\label{eq:third}
\frac{dx}{dt} =r x-x^3\, .
\end{equation}

\begin{itemize}
  \item [3)]
 Consider equation (\ref{eq:third}) for four distinct values of the parameter $r$:  $r = 1$,  $r = -1$, $r = 0$, and $r = 4$.  In each of these cases, find all equilibria, determine their stability properties, and create a plot of several solutions to the equation where each one begins with a different initial condition.    Summarize and report your results in a structured presentation with diagrams and annotation as well as a good narrative.
\end{itemize}

\noindent
We make a slight change in (\ref{eq:third}) and study the results as we consider the differential equation


\begin{equation}\label{eq:fourth}
\frac{dx}{dt} =r x+x^3\, .
\end{equation}

\begin{itemize}
  \item [4)]
Consider equation (\ref{eq:fourth}) for four distinct values of the parameter $r$:  $r = 1$,  $r = -1$, $r = 0$, and $r = 4$.  In each of these cases, find all equilibria, determine their stability properties, and create a plot of several solutions to the equation where each one begins with a different initial condition.   Summarize and report your results in a structured presentation with diagrams and annotation as well as a good narrative.
\end{itemize}



\noindent
Finally, we make a change in (\ref{eq:third}) and study the results as we consider the differential equation
formed by adding a constant to our right hand side of (\ref{eq:third}):

\begin{equation}\label{eq:fifth}
\frac{dx}{dt} =d+cx-x^3\, .
\end{equation}

\noindent
First, note that if $d = 0$ we have the same system as in equation (\ref{eq:third}) which we have studied already.  Thus now consider what happens for different values of $d$.

\begin{itemize}
  \item [5)]
 Considering equation (\ref{eq:fifth}) for each of the following values of the parameter $d$,  allow the value of the parameter $c$ to vary as in our previous bifurcation studies and find all equilibria, determine their stability properties, and create a plot of several solutions to the equation where each one begins with a different initial condition.

First, for $d = 1$, find and classify equilibria for varying values of $c$. Discuss how the fixed point moves as $c$ moves from $c = -4$ to $c=-1$. What happens when $c=0$?  When $c>0$? What happens if we go up to and beyond $c=1.8$ or 1.9?  Construct a bifurcation diagram for this value of the parameter ``d.''

Now repeat this process for $d = 4$, $d = -1$, and $d = -4$.  Summarize and report your results in a structured presentation with diagrams and annotation as well as a good narrative.

\end{itemize}


\teacher{
\section*{NOTES FOR TEACHERS}

We include a Mathematica notebook both in Mathematica notebook format (1-009-T-Bifurcation-Mathematica.nb) and universally readable pdf (1-009-T-Bifurcation-Mathematica.pdf) format in which we offer a discussion of what students might see either through guidance from the teacher or on their own after some initial discussion in class.  We use Mathematica, but any computer algebra system will suffice here for this study.

In the INTRODUCTION  for students, we have intentionally been somewhat vague about the behavior that the student should expect to observe so that they may discover on their own why we call this bifurcation analysis.  In the first two activities, they will analyze the classical saddle-node (or fold) bifurcations while in the subsequent activities they will encounter pitchfork bifurcations.

When implementing these exercises in a first course in ordinary differential equations, we have typically spent either two 50 minute classes or one 80 minute class completing the first two activities, followed by a lecture on bifurcations that explores other autonomous nonlinear systems which exhibit a saddle-node bifurcation and introduces transcritical bifurcations.  This follows the pattern established in the book, \emph{Nonlinear Dynamics and Chaos} by Strogatz \cite{Strogatz}.  It is essential that prior to these activities, students have become familiar with methods for solving first order differential equations and initial value problems, specifically the method of separating variables.  Another important prerequisite is the usual development of stability analysis for autonomous differential equations.  Students frequently need some guidance in making the transition from a sequence of phase portraits to a bifurcation diagram in the first activity.  One successful technique has been to set up the required axes on a whiteboard in front of the class, add the usual circles for an unstable and stable equilibrium when $r = -4$, and then have different students add more and more circles for varying values of $r$.  We try to get the class to come to a consensus about what will happen for all values of $r$ and why this must be true before they proceed to the second activity.

The remaining activities provide a more robust view of the classically studied bifurcations, including pitchfork and imperfect forms.  Along the way it is helpful to have a discussion about the fact that one can create the bifurcation diagrams without ever actually solving an initial value problem since phase portraits provide sufficient information.
}

\begin{thebibliography}{98}

\bibitem{Allen}  Allen, L. S. J.  2007.  {\it An Introduction to Mathematical Biology\/}.  Upper Saddle River NJ:  Pearson Education.

\bibitem{Chamon}  Castelnovo, C. and C. Chamon.  2008.  Quantum topological phase transition at the microscopic level.  {\it Phys. Rev. B\/}.  77 (5): 054433.

\bibitem{Freeman}  Freeman, W. J. and S. Jakubith.  1993.  Bifurcation analysis of continuous time dynamics OS oscillatory neural networks.  In {\it Brain Theory.  Spatio-Temporal Aspects of Brain Function\/}.  Amsterdam:  Elsevier.

\bibitem{Gop}  Gopalsamy, K.  1992.  {\it Stability and Oscillations in Delay Differential Equations of Population Dynamics\/}.  Dordrecht:  Kluwer Academic Publishers.

\bibitem{Strogatz} Strogatz, S. H.  1994. {\it Nonlinear Dynamics and Chaos\/}. Cambridge MA:  Perseus Books Publishing.

\bibitem{Yang} Yang, C. D. and H. J. Wend.  2011.  Quantum bifurcation in a coulombic-like potential.  {\it Int. J. Quantum Chem.\/}.  111 (15):  4330-4351.


\end{thebibliography}

\end{document}
