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\noindent
Math 325

\noindent
Spring, 2001

\bigskip

\begin{center}
GROUP PROJECT 1, due Wednesday, April 4
\end{center}

\bigskip

Let 
\begin{eqnarray}
x'&=&f(x,y,\mu)
\nonumber
\\
y'&=&g(x,y,\mu)
\label{autonomous}
\end{eqnarray}
be
a planar autonomous
system of ordinary differential equations depending on a parameter
$\mu\/$.  
The purpose of this project is to study the dependence
of the behavior of the system
on the parameter  $\mu\/$.  This is a group project.  
Please follow the guidelines for group projects.

Assume that $f$ and $g$ vanish at the origin.
We can write the system (\ref{autonomous})
in the form 
$$
{\bf x}' = A(\mu) {\bf x} + Q({\bf x},\mu) 
$$
where  
$$
{\bf x}=\left[\begin{array}{c} x\\y\end{array}\right],
$$
$A(\mu) \/$
is a matrix depending on $\mu\/$ but not on  ${\bf x}\/$, 
and  $Q\/$ contains the higher order terms in (\ref{autonomous}).
The matrix  $A(\mu)\/$
is called the {\it linearization} of (\ref{autonomous}) at the origin.

\medskip
I.  Consider the system 
\begin{eqnarray}
x'&=&\mu x+y-x(x^2+y^2)
\nonumber
\\
y'&=&-x+\mu y-y(x^2+y^2).
\label{hopf}
\end{eqnarray}

\smallskip
1.  Show that the origin is the only equilibrium point of the system
(\ref{hopf}).

\smallskip
2.  Find the eigenvalues of the linearization of (\ref{hopf}).
Show that they are of the form 
 $\alpha(\mu)+i\beta(\mu)\/$
where  $\alpha\/$
and $\beta\/$
are
differentiable functions of  $\mu\/$,  $\alpha(0)=0\/$,  $\alpha'(0)>0\/$
and  $\beta (0)\ne 0\/$.

\smallskip
3.  Identify the type of critical point and
stability of the linearization. 
(The answer will depend on  $\mu\/$.)

\smallskip
4.  For 
 $\mu=-1,-0.5,0,0.5,1\/$   plot trajectories of the linearization
and
describe the behavior of the trajectories as  $t\to \infty\/$.

\smallskip
5.  For the same values of  $\mu\/$, plot trajectories of the
system (\ref{hopf}).  What do you notice about the behavior of
the trajectories as  $t\to \infty\/$?  
How does it depend on  $\mu\/$?

\smallskip
6.  Rewrite the system (\ref{hopf}) in polar coordinates.  You should
get the system 
\begin{eqnarray}
r'&=& r(\mu-r^2)
\nonumber
\\
\theta'&=&-1.
\label{polarhopf}
\end{eqnarray}

\smallskip
7. Use the equations (\ref{polarhopf}) to analyze the behavior 
of trajectories of (\ref{hopf}) as  $t\to \infty\/$.

\smallskip
a.  Show that  if $\mu\le 0\/$, the trajectories spiral clockwise
toward the origin as  $t\to \infty\/$.

\smallskip
b.  Show that if  $\mu>0\/$
the circle  $r=\sqrt \mu\/$
is a trajectory.

\smallskip
c.  Show that if  $\mu>0\/$
then a trajectory through a point with  $0<r<\sqrt \mu\/$
spirals clockwise outward to   $r=\sqrt \mu\/$
as  $t\to \infty\/$
and   a trajectory through a point with $\sqrt \mu<r\/$
spirals clockwise inward to  $r=\sqrt \mu\/$ as  $t\to \infty\/$.  
Such a limiting trajectory is called a {\it limit cycle}.


\smallskip
{\bf Hints}:  You should have seen the behavior described in  a-c
in your plots.  For  a and c, use the information you obtain from the sign of 
the derivatives in (\ref{polarhopf}). 

\medskip
This is an example of a {\it bifurcation} of an asymptotically
stable equilibrium point into a limit cycle.  This kind of behavior
 is called a {\it Hopf bifurcation}.

\medskip
In the next three groups of problems, you will explore a linear system
and two nonlinear systems with the same linearization.
In the final problem you will compare the three systems.  

\medskip

II.  Consider the system
\begin{eqnarray}
x'&=&\mu x-y
\nonumber
\\
y'&=&x.
\label{linearvan}
\end{eqnarray}

\smallskip
8.  Show that the origin is the only equilibrium point of 
(\ref{linearvan}).  

\smallskip
9.  Find the eigenvalues of the linearization of (\ref{linearvan}) 
and show that they
have the same properties as the eigenvalues
in problem  2 if  $|\mu|\/$ is small.  

\smallskip
10.  Identify the type of critical point and stability of
(\ref{linearvan}).

\smallskip
11.  For  $\mu=-3,-2,-1,-0.5, 0 , 0.5,1,2,3\/$
plot trajectories of (\ref{linearvan}) and describe the behavior 
of the trajectories as  $t\to \infty\/$.

\smallskip
12.  Show that for  $\mu =0\/$, the trajectories, other than
the equilibrium point, are circles
centered at the origin.  


\medskip

III.  Consider the system
\begin{eqnarray}
x'&=&\mu x-y -x^3
\nonumber
\\
y'&=&x.
\label{van}
\end{eqnarray}

\smallskip
13. Show that the origin is the only equilibrium point of
(\ref{van}).

\smallskip

14.  For the values of  $\mu \/$
in  11, plot trajectories of the system (\ref{van}).  What do you 
notice about the behavior of the trajectories as  $t\to\infty\/$?  How does it 
depend on  $\mu\/$?

\smallskip 
15.  For  $0<\mu<2\/$, (\ref{van}) has a limit cycle.  Label its 
approximate location in your plots in 14.  Estimate the
  $x\/$ coordinate of the intersection of the limit
cycle with the positive  $x\/$ axis as accurately
as you can.  
Explain how you got your estimate.

\smallskip
16.   Write  (\ref{van}) in polar coordinates
and  show that for  $\mu=0\/$,   $r\/$
is a decreasing function of  $t\/$
on trajectories of the system.   From this, how do you expect
trajectories to behave as  $t\to \infty\/$?  Is this expectation
consistent with the behavior you found in your plot for  $\mu=0\/$?

\medskip  The system (5), known as the {\it van der Pol}
system, also has a Hopf bifurcation at $\mu =0\/$.   

\medskip

IV.  Consider the system
\begin{eqnarray} 
x'&=&\mu x-y -y^3 
\nonumber 
\\ 
y'&=&x. 
\label{typo}
\end{eqnarray} 

\smallskip
17. Show that the origin is the only equilibrium point of
(\ref{typo}).

\smallskip 
18.  For the values of  $\mu \/$ 
in  11, plot trajectories of the system (\ref{typo}).  What do you 
notice about the behavior of the trajectories as  $t\to\infty\/$? 
 How does it  
depend on  $\mu\/$? 
Did you find limit cycles for  $0<\mu<2\/$?

\smallskip

19.
Show that the energy function 
$$
H(x,y)=\frac {x^2}2 + \frac {y^2}2+\frac {y^4}4
$$
is constant on trajectories of (\ref{typo}) for  $\mu=0\/$.  
Here  $\frac {x^2}2\/$
is the potential energy and  $\frac {y^2}2+\frac {y^4}4 \/$
is the kinetic energy of the system.
Conclude that the trajectories are closed curves.  
You should have observed these in your plot.

\medskip

The system (\ref{typo}) exhibits a different type of bifurcation
at  $\mu=0\/$. 

\medskip

V. 20. Compare the behavior of the trajectories of (\ref{linearvan}),
(\ref{van}) and (\ref{typo}) as  $t\to \infty\/$
for the values of  $\mu\/$
in problem 11.


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