DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN DIFFERENTIAL EQUATIONS August 2006

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\quad x\in \mathbb {R}^n,\quad A(t)=A(t+T). \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(x=\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Explain how Floquet theory can be used to determine the linear stability of a limit cycle.

1. Consider the van der Pol oscillator \[ \ddot {x}+\varepsilon \dot {x}(x^2-1)+x=0,\quad 0<\varepsilon \ll 1 \] and choose initial conditions \(x(0)=1\), \(\dot {x}=0\). Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim R(\varepsilon t)\cos (t+\theta (\varepsilon t))+O(\varepsilon ) \] with \[ \theta _\tau =0,\quad R_\tau =\frac {1}{8}R(4-R^2) \] where \(\tau =\varepsilon t\), \(\theta (0)=0\) and \(R(0)=1\). Hence, establish that there exits a stable periodic orbit.
2. Consider the Duffing equation \[ \ddot {x}+x=-\varepsilon x^3,\quad 0<\varepsilon \ll 1. \] Using the method of multiple scales show that the leading order solution is of the form \[ x(t)\sim R_0\cos \left (t+t_0+\frac {3}{8}R_0^2\varepsilon t\right )+O(\varepsilon ). \]

The response of a certain biological oscillator, \((x,y)\), \(x\ge 0\), \(y\ge 0\), to a stimulus of size \(b\) satisfies the differential equation \[ \begin {aligned} \dot {x}&=x-ay+b,\quad &\dot {y}&=x-cy && \text {for } x\ge 0,\ y\ge 0;\\ &&\dot {y}&=-cy && \text {for } x=0, \end {aligned} \] with \(a,b,c>0\).

1. Using phase-plane analysis show that there exists a limit cycle, part of which lies on the \(y\)-axis, when \(c<1\) and \(4a>(1+c)^2\). [Hint: existence of limit cycle depends on existence of an unstable spiral].
2. Show that the period of the orbit is independent of \(b\).

1. Give the definitions of Poincare and Liapunov stability. Show that solutions of the system \(\dot {x}=y\), \(\dot {y}=0\) are Poincare but not Liapunov stable.
2. Consider the system \[ \begin {aligned} \dot {x} &= x-y-x(x^2+y^2)+\frac {xy}{\sqrt {x^2+y^2}},\\ \dot {y} &= x+y-y(x^2+y^2)-\frac {x^2}{\sqrt {x^2+y^2}}. \end {aligned} \] Show that the above pair of equations can be rewritten in polar coordinates as \[ \dot {r}=r(1-r^2),\quad \dot {\theta }=2\sin ^2(\theta /2) \] and sketch the phase portrait. Determine whether or not the fixed point \((1,0)\) is Liapunov stable.

State the center manifold theorem and briefly explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-2x+y-x^2,\quad \dot {y}=x(y-x). \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions. [Hint: first perform a change of variables in order to diagonalize the linear part of the system].

Let \(U\) be an open subset of \(\mathbb {R}^2\). Prove that if \(u\in C^2(U)\) satisfies \[ \Delta u-u=0 \quad \text {in } U, \] then \[ u(x)=\frac {1}{I(r)}\int _{\partial B(x,r)} u\,dS_y \] for every ball \(B(x,r)\subset U\), where \[ I(r)=\frac {1}{\pi }\int _0^\pi e^{r\cos t}\,dt. \] Assume without proof that \(I(r)\) is the unique solution to the modified Bessel equation \[ r\frac {d^2\varphi }{dr^2}+\frac {d\varphi }{dr}-r\varphi =0,\quad r>0, \] which satisfies \(\varphi (0)=1\).

Let \(U\subset \mathbb {R}^n\) be open and bounded, with smooth boundary. For \(T>0\), denote \(U_T=U\times (0,T]\). Let \(\Gamma _T=\overline {U_T}-U_T\) (closure taken in \(\mathbb {R}^n\times \mathbb {R}\)). Recall that for solutions \(u(x,t)\) of \[ \begin {aligned} u_t-\Delta u &= f, && \text {in } U_T,\\ u &= g, && \text {on } \Gamma _T, \end {aligned} \] we define the energy \[ e(t)=\int _U u^2(x,t)\,dx. \]

1. Use energy methods to show that the problem above has at most one solution \(u\in C^{2,1}(U_T)\) (\(C^2\) in the \(x\) variables, and \(C^1\) in \(t\)).
2. Assuming \(g=0\), and \(u\in C^{2,1}(U_T)\), prove that for \(0<t<T\), \[ e(t)\le \left (\int _0^t\left (\int _U f^2(x,s)\,dx\right )^{1/2}\,ds\right )^2. \]

Let \(\mathbb {R}_+=(0,\infty )\). Consider the problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0, && \text {in } \mathbb {R}_+\times \{t>0\},\\ u(x,0) &= g(x),\quad u_t(x,0)=h(x), && x\in \mathbb {R}_+ && \text {(initial conditions)},\\ u_x(0,t) &= 0, && t>0 && \text {(boundary condition)}. \end {aligned} \] Assume that \(g,h\in C^1(\overline {\mathbb {R}_+})\), and that \(g_x(0)=h_x(0)=0\).

1. Find an explicit formula for the solution \(u(x,t)\) (d’Alembert’s formula is a good place to start).
2. Suppose that for \(0\le t<\frac {1}{2}\) and \(x>0\), we know \(u(x,t)=g(x+t)\), and assume \(\operatorname {supp}g\subset (1,2)\). Thus, initially a wave is travelling toward the origin \(x=0\). What happens to the wave after it hits the origin, say for \(t>2\)? Is it absorbed, reflected, damped, inverted? Calculate \(u(x,3)\).

Consider the scalar problem \[ \begin {aligned} u_t+H(Du) &= 0, && \text {in } \mathbb {R}^n\times (0,\infty ),\\ u &= x_1, && \text {on } \mathbb {R}^n\times \{t=0\}. \end {aligned} \] Assume the Hamiltonian \[ H(p)=\frac {1}{4}\sum _{j=1}^n\frac {1}{a_j}(p_j-b_j)^2, \] where the coefficients \(a_j,b_j\) are fixed real numbers, with \(a_j>0\).

1. Use the Legendre transform \(H^*(q)=\sup _p\{p\cdot q-H(p)\}\) to find the Lagrangian.
2. Setting the coefficients \(a_j=1\), \(b_j=0\) for all \(j\), use the Hopf-Lax formula \[ u(x,t)=\min _y\left \{tL\left (\frac {x-y}{t}\right )+g(y)\right \} \] to find an explicit weak solution to the problem above.

Let \(U=\{x\in \mathbb {R}^2:\frac {1}{2}<|x|<1\}\), and denote \(\partial U=\Gamma _1\cup \Gamma _2\) where \[ \Gamma _1=\{|x|=\tfrac {1}{2}\},\quad \Gamma _2=\{|x|=1\}. \] Consider \[ \begin {aligned} -\Delta u &= f, && \text {in } U,\\ \alpha u+\frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _1,\\ \frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _2, \end {aligned} \] where \(\alpha >0\), \(\eta \) is the unit outward normal on \(\partial U\), and \(f\in L^2(U)\).

Prove that there exists a constant \(C>0\) such that \[ C\int _U u^2\,dx \le \int _U |Du|^2\,dx+\int _{\Gamma _1} u^2\,dS \] for all \(u\in C^1(\overline {U})\). Polar coordinates may be helpful. Explain how this estimate could be used in a proof of existence and uniqueness for solutions of the problem above.