Exam
Ph.D. Preliminary Examination in Ordinary Differential Equations
Spring 2025 · 6 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Consider the ODE system
\[ \begin {cases} \dot {x}=p,\\ \dot {p}=-V'(x), \end {cases} \qquad \text {where } V(x)=\frac 14(1-x^2)^2. \]- Define Lyapunov stability. Are the stationary points at \((\pm 1,0)\) Lyapunov stable? Justify.
- Find all fixed points, classify their stability (with justification), and make an accurate drawing of the phase plane including any heteroclinic / homoclinic orbits.
- Define the notion of asymptotically stable stationary point. Are any of the stationary points asymptotically stable?
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Question 2
Consider the family of ODE systems
\[ \dot {x}_\epsilon =f(t,x_\epsilon )+\epsilon g(t,x_\epsilon ) \]with \(\|Df\|\le L\) and \(|g|\le M\). Show that
\[ |x_\epsilon (t)-x_0(t)| \le e^{Lt}|x_\epsilon (0)-x_0(0)|+\frac {M\epsilon }{L}e^{Lt}. \]Hint: Look at the integral equation for \(z_\epsilon (t):=x_\epsilon (t)-x_0(t)\), estimate using Grönwall-type ideas.
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Question 3
Let \(f(t,x):\mathbb {R}\times \mathbb {R}^n\to \mathbb {R}^n\) be continuous such that for some
\[ L\in \mathbb {R} \]\[ |f(t,x)-f(t,y)|\le L|x-y| \]for all \(t\in \mathbb {R}\) and \(x,y\in \mathbb {R}^n\). Show that the initial value problem for \(x_0\in \mathbb {R}^n\) \[ \dot {x}=f(t,x)\quad \text {with }x(0)=x_0 \] has a unique solution that exists on \(t\in [0,\infty )\).
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Question 4
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 \(\Phi (0)=I\).
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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 \(\Phi (T)\).
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Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are \(n\) 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 \(\mu _i\)?
- Consider the equation \(\dot {x}=f(t)A_0x\), \(x\in \mathbb {R}^2\), with \(f(t)\) a scalar \(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigen- values. Determine the corresponding Floquet multipliers.
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Question 5
Consider the system
\[ \begin {cases} \dot {x}=x-y-x^3,\\ \dot {y}=x+y-y^3. \end {cases} \]- Plot the sets \(\{\dot {x}=0\}\) and \(\{\dot {y}=0\}\) in the \((x,y)\)-plane, find the fixed points of the system and determine their stability.
- Show that the annular region \(A=\{1<|(x,y)|<\sqrt {2}\}\) is forward invariant.
- State the Poincaré-Bendixson theorem.
- Show that the system has at least one non-constant periodic orbit.
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Question 6
Let \(A\) be a \(d\times d\) matrix with all eigenvalues having negative real part and consider the system
\begin{equation} \tag {1} \dot {x}=Ax+h(t) \end{equation}where \(h(t)\in \mathbb {R}^d\) is a \(T\)-periodic continuous function.
- Define the Poincaré map in this context and give an explicit formula in terms of \(h\) and \(A\).
- Show that there is a unique \(T\)-periodic solution \(x(t)\) of equation (1). Hint: there is again an explicit formula in terms of \(A\) and \(h\)
- What stability property does this periodic orbit have? (You do need to justify your answer with explicit calculation/proof)
- How could the requirements stipulated on the eigenvalues of \(A\) be changed and still maintain the existence and uniqueness result for the periodic orbit from (b).