DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN DIFFERENTIAL EQUATIONS January 2007

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. Suppose that the autonomous nonlinear equation \(\dot {x}=f(x)\) exhibits a limit cycle. By linearizing about this solution, explain how Floquet theory can be used to determine the linear stability of the limit cycle.

Consider the nonlinear equation \[ \ddot {x}+\varepsilon \left (\frac {1}{3}\bigl (\dot {x}\bigr )^3-\dot {x}\right )+x=0, \quad 0<\varepsilon \ll 1 \] and choose initial conditions \(x(0)=a\), \(\dot {x}(0)=0\).

1. Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim 2R(\varepsilon t)\cos (t+\theta (\varepsilon t))+O(\varepsilon ) \] with \[ \theta _\tau =0,\quad R_\tau =\frac {1}{2}R(1-R^2) \] where \(\tau =\varepsilon t\).
2. Derive the solution \[ R(\tau )=\left (1+a_0e^{-\varepsilon t}\right )^{-1/2} \] and determine \(a_0\) from the initial conditions. Hence, establish that there exits a stable periodic orbit.

Consider a second order, linear autonomous system \(\dot {x}=Ax\), \(x\in \mathbb {R}^2\).

1. Suppose that \(A\) has a pair of complex conjugate eigenvalues. By performing an appropriate change of coordinates, convert the equation into Jordan normal form and thus obtain the general solution.
2. Under what conditions is the fixed point at the origin hyperbolic? Explain the significance of hyperbolic fixed points in the theory of nonlinear ODEs.

1. Use the Poincaré–Bendixson theorem to establish the existence of a limit cycle for the system \[ \begin {aligned} \dot {x} &= y+\frac {x}{4}\left (1-2(x^2+y^2)\right ),\\ \dot {y} &= -x+\frac {y}{2}\left (1-(x^2+y^2)\right ). \end {aligned} \]
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 explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-xy,\quad \dot {y}=-y-x^2. \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions.

For each \(n\in \N \), consider the Cauchy problem \[ \begin {aligned} -\Delta u_n &= 0, && \text {in } U,\\ u_n &= \frac {1}{n^2}\sin nx, && \text {on } \{(x,y):y=0\},\\ \frac {\partial u_n}{\partial y} &= \frac {1}{n}\sin nx, && \text {on } \{(x,y):y=0\}, \end {aligned} \] where \(U=\{(x,y):0<y<1\}\). Find a sequence \(\{u_n\}\) of solutions to these problems, prove that \(\{u_n\}\) does not converge to zero, and explain why this implies that the Cauchy problem above is not “well posed”.

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 \(\mathbb {R}^n\times \mathbb {R}\)). Prove that functions \(u(x,t)\) satisfying \(u\in C^{2,1}(U_T)\cap C(\overline {U_T})\) and \[ \begin {aligned} \Delta u-u &\ge u_t, && \text {in } U_T,\\ u &\ge 0, && \text {in } \overline {U_T}, \end {aligned} \] also satisfy the maximum principle \[ \max _{\overline {U_T}} u=\max _{\Gamma _T} u. \]

Consider the wave equation in three spatial dimensions \[ u_{tt}-\Delta u=0 \quad \text {in } \mathbb {R}^3\times (0,\infty ). \]

1. Explain the concept of domain of dependence for solutions of the wave equation, and describe what this has to do with finite propagation speed.
2. Sketch a proof, either using a representation formula or energy methods, that solutions \(u\in C^2\) of the wave equation have an explicitly defined domain of dependence.

Denote \(U=\{x\in \mathbb {R}^2:x_1>0\}\), and \(\Gamma =\{x\in \mathbb {R}^2:x_1=1\}\). Use the method of characteristics to solve the first-order problem \[ \begin {aligned} x_1u_{x_1}+uu_{x_2}-x_2 &= 0, && \text {in } U,\\ u(1,x_2) &= 2x_2, && \text {on } \Gamma . \end {aligned} \] (Hint: to solve the characteristic equations, note that the linear combinations \(x_2+z\) and \(x_2-z\) decouple.) Determine the region around \(\Gamma \) upon which the solution is well-defined.

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

Prove a Poincaré-type inequality for this problem, and use it to show that there exists a unique weak solution \(u\) in an appropriate Hilbert space.