UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations August 15th, 2016.

Let \(f\in C^1(U,\mathbb {R}^n)\) for \(U\subset \mathbb {R}^n\) and \(x_0\in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\norm {x}=\max _{0\le t\le T}\abs {x(t)}\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\dd s \] for \(x\in X\). Define \(V=\{x\in X\mid \norm {x-x_0}\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\).

1. Give the definition of a locally Lipschitz function in an open set \(U\) of a normed vector space.
2. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that \[ \abs {K(x(t))-K(y(t))}\le L_0t\norm {x-y}. \] Hence, show that \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y}\qquad x,y\in V. \]
3. State the contraction mapping principle on a Banach space.
4. Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the integral equation has a unique continuous solution \(x(t)\) for all \(t\in [0,T]\) and sufficiently small \(T\). Hence establish existence and uniqueness of the initial value problem \[ \frac {dx}{dt}=f(x),\qquad x(0)=x_0. \]

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

1. Show that there exists at least one nontrivial solution \(\chi (t)\) 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. By determining the time derivative of the Wronskian \(W(t)=\det [\Phi (t)]\), prove that the product of the Floquet multipliers \(\mu _i\) satisfies \[ \prod _{i=1}^n\mu _i =\exp \left (\int _0^T\operatorname {tr}[A(s)]\dd s\right ) \]
4. Using part (c) show that \(\mu _1\mu _2=1\) in the case of Mathieu’s equation for a parametric oscillator: \[ \ddot {x}+(\alpha +\beta \cos t)x=0. \]

Consider the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=\Gamma \cos (\omega t), \qquad 0<\epsilon \ll 1 \] with \(\Gamma =O(1)\) and \(\omega \ne 1/3,1,3\). Using the method of multiple scales show that the solution is attracted to \[ x(t)=\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2\ge 2(1-\omega ^2)^2\) and \[ x(t)=2\left [1-\frac {\Gamma ^2}{2(1-\omega ^2)^2}\right ]^{1/2}\cos t +\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2<2(1-\omega ^2)^2\). Explain why this result breaks down when \(\omega =1/3,1,3\).

1. Give definitions for the following: invariant set, attracting set, \(\omega \)-limit set.
2. Use the Poincare’-Bendixson (PB) Theorem and the fact that the planar system \[ \dot {x}=x-y-x^3,\qquad \dot {y}=x+y-y^3 \] has only the one critical point at the origin to show that this system has a periodic orbit in the annular region \(A=\{x\in \mathbb {R}^2\mid 1<\abs {x}<\sqrt {2}\}\).
3. Show that the system \[ \dot {x}=x-rx-ry+xy,\qquad \dot {y}=y-ry+rx-x^2 \] can be written in polar coordinates as \(\dot {r}=r(1-r)\), \(\dot {\theta }=r(1-\cos \theta )\). Show that it has an unstable node at the origin and a saddle node at \((1,0)\). Use this information and the PB Theorem to sketch the phase portrait and then deduce that for all \((x,y)\ne (0,0)\), the flow \(\phi _t(x,y)\to (1,0)\) as \(t\to \infty \) but that \((1,0)\) is not linearly stable.

The displacement \(x\) of a spring-mounted mass under the action of dry friction is assumed to satisfy \[ m\ddot {x}+kx=-F_0\sgn (\dot {x}) \] An example would be a mass \(m\) connected to a fixed support by a spring of stiffness \(k\) and resting on a surface with frictional force \(F_0\), \(F_0>0\). Set \(m=k=1\) for convenience and let \(y=\dot {x}\).

1. Consider the initial conditions \(x=x_0>0\), \(\dot {x}=0\) at \(t=0\). Show that the phase path will spiral exactly \(n\) times before reaching equilibrium if \[ (4n-1)F_0<x_0<(4n+1)F_0. \]
2. Suppose the initial conditions at \(t=0\) are \(x=x_0>3F_0\) and \(\dot {x}=0\). Subsequently, whenever \(x=-\alpha \), where \(2F_0-x_0<-\alpha <0\) and \(\dot {x}>0\), a trigger operates to increase suddenly the forward velocity so that the kinetic energy increases by a constant amount \(E\). Show that if \(E>8F_0^2\) then a periodic motion is approached, and show that the largest value of \(x\) in the periodic motion is equal to \(F_0+E/(4F_0)\).

Green’s integral identity is \[ \int _\Omega (v\nabla ^2u-u\nabla ^2v)d x =\int _{\partial \Omega }(vu_\nu -uv_\nu )d\sigma \] where \(u_\nu \) refers to the outward normal derivative.

1. Does a solution of the equation \(u_x+xu_y=u\) with \(u(x,x^2)=x\) exist for \(-\infty <x<\infty \), \(y>0\)? Explain why or why not.
2. Does a solution of the equation \(xu_x-yu_y=u\) with \(u(x,x^2)=x\) exist for \(-\infty <x<\infty \), \(y>0\)? Explain why or why not.

The fundamental solution for the Laplacian in two dimensional space is \(G(\mathbf {x},\mathbf {y})=\frac {1}{2\pi }\ln (\abs {\mathbf {x}-\mathbf {y}})\), where \(\mathbf {x}=(x_1,x_2)\). What is the Green’s function for the Laplacian operator on the first quadrant \(0<x_1,x_2<\infty \), subject to boundary conditions \(G(x_1,0)=0\), \(G_{x_1}(0,x_2)=0\)? Be sure to verify the validity of your solution.