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\noindent
\large
\centerline{{\bf Homework 7 (Math 455) \\
Due: February 15-16, 2019.}}
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\large
\noindent
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\vspace{0.2in}
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\noindent
{\bf 1.} [10] Use the Poincare-Bendixson Theorem
to prove that the planar system defined by
\begin{eqnarray*}
\dot{x} & = & f_{1}(x,y) =x-y-x^3 \\
\dot{y} & = & f_{2}(x,y) =x+y-y^3
\end{eqnarray*}
has at least one periodic orbit. Choose an annular
trapping region $M$. The outer circle is $\partial M_o$
of radius $r_o$. The inner circle is $\partial M_i$ with
radius $r_i$.
The sole fixed point is the origin.
\begin{itemize}
\item[a)] Show the fixed point is an unstable spiral
so that flow is into $M$ if $r_i$ sufficiently small.
\item[b)] On $\partial M_o$ an inward normal vector
is $\vec{N}=(-x,-y)$. Show that on $\partial M_o$
\[
\vec{N} \cdot (f_1,f_2) = -r^2 +r^4 F(\theta)
\]
for some function $F(\theta)$ in polar coordinates.
\item[c)] Show $\frac{1}{2} \le F(\theta) \le 1$ and
use this to find an $r_o$ value such than $r>r_o \Rightarrow$
$\vec{N} \cdot f >0$.
\end{itemize}
\vspace{0.4in}
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\noindent
{\bf 2.} [10] Use the Poincare-Bendixson Theorem
to prove that the \underline{perturbed} planar system defined by
\begin{eqnarray*}
\dot{x} & = & f_{1}(x,y) =x-y-x(x^2+y^2) + \epsilon x \\
\dot{y} & = & f_{2}(x,y) =x+y-y(x^2+y^2)
\end{eqnarray*}
has at least one periodic orbit for sufficiently small $\epsilon$. Choose an annular
trapping region $M$. The outer circle is $\partial M_o$
of radius $r_o$. The inner circle is $\partial M_i$ with
radius $r_i$.
The sole fixed point is the origin.
\begin{itemize}
\item[a)] Show the origin is unstable (non-saddle) for all $\epsilon>0$
so that (certainly) for sufficiently small $\epsilon$ flow across $\partial M_i$
is into $M$.
\item[b)] Convert the system into polar coordinates (posted Notes17)
and show that when $\epsilon=0$ the system is
\[ \dot{r} = r (1-r^2) \quad \quad \dot{\theta}=1 \]
when $\epsilon \ne 0$ you should get
\[ \dot{r} = r \left( \epsilon \cos^2\theta + 1 - r^2\right)
\quad \quad \dot{\theta} = 1 -\frac{1}{2} \epsilon \sin(2\theta) \]
This implies the unperturbed ($\epsilon =0$) system has a stable limit cycle $r=1$.
\item[c)] Using your results in a)-b), give a carefully stated argument why $M$ is a trapping region
for $\epsilon>0$ small enough.
\end{itemize}
\newpage
\noindent
{\bf 3.} [10] For each of the following systems
draw phase portraits for the cases $\mu <0, \mu = 0, \mu>0$
and then draw a bifurcation diagram of the
$x$ coordinate of the fixed point branches versus
$\mu$.
\begin{eqnarray*}
\dot{x} & = & \mu x - x^{2}
\quad \dot{y} = - y \\
\dot{x} & = & \mu x + x^{3}
\quad \dot{y} = - y \\
\end{eqnarray*}
What types of bifurcations are these?
\vspace{0.1in}
\noindent
{\bf 4.} [10] (Bistability) Consider the following planar system
where $\mu$ is a parameter.
\begin{eqnarray*}
\dot{x} & = & f_1(x,y) = \phi(x)-\mu \\
\dot{y} & = & f_2(x,y) = -y
\end{eqnarray*}
where $\phi(x)$ is a smooth function of $x$.
\begin{itemize}
\item[a)] Fixed points are
\[ X(\mu) = \left(
\begin{array}{c}
\bar{x}(\mu) \\
0 \end{array}
\right) \]
where $\bar{x}$ are roots of $\phi(x)-\mu=0$.
Use linear stability analysis to show the
stability of such fixed points depends only on
the sign of the derivative $\phi'(x)$
\item[b)] Let $\phi(x)=x-x^3$ and plot $\mu$ as a function of $x$
This is the locus of all fixed points. Rotate the figure and
use the results in a) to draw a bifurcation diagram
of $x$ versus $\mu$ labeling branch stability.
\item[c)] For what $\mu$ is the system "bistable" - has two stable
fixed points. Here I want an interval, i.e., $\mu \in (\mu_-,\mu_+)$.
\end{itemize}
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