UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS

Show that the ODE

cannot have any non-constant time periodic solution if \(\mu > 0\).

Consider the gradient flow defined by the ODE

where \(f(x)=x^{\top }Qx\) with \(Q\in \mathbb {R}^{n\times n}\), \(n\ge 2\) being a symmetric matrix.

1. Assume \(Q\) is positive semi-definite, show that \[ f(x(t))\le \frac {C\|x_0\|_2^2}{t} \] for some constant \(C\).
2. Assume \(Q\) is positive definite, show that \[ f(x(t))\le \lambda _{\max }e^{-4\lambda _{\min }\cdot t}\|x_0\|_2^2, \] where \(\lambda _{\min }\) and \(\lambda _{\max }\) are the smallest and largest eigenvalues of \(Q\), respectively.

Consider the nonlinear system in \(\mathbb {R}^3\): \[ \begin {cases} \dot {x}=y-x^2,\\ \dot {y}=-z+xy,\\ \dot {z}=-\lambda z+x^2, \end {cases} \]

where \(\lambda > 0\) is a parameter.

1. Find all critical points of the system and classify them when \(\lambda > 0\).
2. Analyze the stability of the origin \((0,0,0)\) using the center manifold theorem. Compute the reduced system on the center manifold up to quadratic terms.
3. Discuss how the stability of the origin changes as \(\lambda \to 0^+\). What happens if \(\lambda =0\)?

Let \(\phi _t\) and \(\psi _t\) be two flows on \(\mathbb {R}^2\setminus \{0\}\) induced by the autonomous linear ODEs \[ \frac {d}{dt}(x,y)=(x,y),\qquad \frac {d}{dt}(u,v)=(\alpha u,\beta v), \]

where \(\alpha , \beta > 0\) are constants.

1. Show that for any \(\alpha ,\beta >0\), the flows \(\phi _t\) and \(\psi _t\) are topologically conjugate on \(\mathbb {R}^2\setminus \{0\}\); i.e., there exists a homeomorphism \(h:\mathbb {R}^2\setminus \{0\}\to \mathbb {R}^2\setminus \{0\}\) such that \[ h(\phi _t(x,y))=\psi _t(h(x,y)) \] for all \((x,y)\ne (0,0)\) and all \(t\in \mathbb {R}\).
2. Find an explicit homeomorphism \(h\) realizing the conjugacy and prove it is a bijective continuous map with continuous inverse.
3. Discuss whether the conjugacy can be chosen differentiable (i.e., a \(C^1\) diffeomorphism). Under what conditions on \(\alpha ,\beta \) does a differentiable conjugacy exist?

Consider the nonlinear system on \(\mathbb {R}^2\): \[ \begin {cases} \dot {x}=-y+x(1-x^2-y^2),\\ \dot {y}=x+y(1-x^2-y^2). \end {cases} \]

1. Show that the unit circle \(S^1=\{(x,y):x^2+y^2=1\}\) is an invariant set for the flow.
2. Prove that every trajectory starting near \(S^1\) spirals toward it as \(t\to \infty \).
3. Define a Poincaré section \(\Sigma =\{(x,y):y=0,\ x>0\}\) and compute the Poincaré map \(P:\Sigma \to \Sigma \) for points near the unit circle.
4. Linearize the Poincaré map near the fixed point corresponding to the periodic orbit on \(S^1\), and determine its stability.

Consider the following two second-order differential equations:

where \(f(t)\) is continuous and \(2\pi \)-periodic, i.e., \(f(t)=f(t+2\pi )\) for all \(t\in \mathbb {R}\). For both equations, we have the initial conditions \(x(0)=1\) and \(x'(0)=1\).

1. For equation (I), show the solution \(x(t)\) is unbounded.
2. For equation (II), show that \(x(t)\) is bounded.