6410-2022-08

Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\) and \(L\in \mathbb {R}\) be such that

Show that there is \(T>0\) such that the Picard iterates converge to a continuously differentiable solution of the initial value problem

on \([0,T]\). [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

1. Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuously differentiable function and \(x_0\in \mathbb {R}^d\) be such that \(f(t,x_0)=0\) for all \(t\ge 0\). Define: the constant solution \(x(t)=x_0\) on \(t\ge 0\) is a stable (Liapunov stable) solution of

   \[ \dot {x}=f(t,x). \]

2. Determine whether the zero solution \(z(t)=0\) is stable for \[ \begin {bmatrix}\dot {x}\\ \dot {y}\end {bmatrix} = \begin {pmatrix}1 & 1\\ 0 & h(t)\end {pmatrix} \begin {bmatrix}x\\ y\end {bmatrix} \]

   where

   \[ h(t)=\frac {\cos t+\sin t}{2+\sin t-\cos t}. \]

   [Hint: the general solution is \[ \begin {bmatrix}x\\ y\end {bmatrix} = \begin {bmatrix} ae^t-b(2+\sin t)\\ b(2+\sin t-\cos t) \end {bmatrix} \]

   where \(a\) and \(b\) are constants.]

Let \(A\) be a real matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\).

1. Show that there is \(\delta >0\) such that if \(|x_0|\le \delta \), then the initial value problem

   \[ \dot {x}=Ax+e^{-t}|x|x \]

   (1)

   \[ x(0)=x_0 \]

   has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem.]

2. Is the zero solution of (1) asymptotically stable? Explain. [Hint: let \(f(t)\), \(\phi (t)\) be nonnegative continuous functions on the interval \(J=(\alpha ,\beta )\) containing \(t_0\). Let \(c_0\ge 0\). Gronwall’s Lemma says that if \[ f(t)\le c_0+\left |\int _{t_0}^t \phi (s)f(s)\,ds\right | \] for all \(t\in J\), then \[ f(t)\le c_0\exp \left |\int _{t_0}^t \phi (s)\,ds\right | \] for all \(t\in J\).]

Describe the bifurcations that occur in the equation as the parameter \(a\in \mathbb {R}\) increases.

Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) have a non-constant, \(T>0\) periodic trajectory \(\gamma (t)\) satisfying

1. Define: the periodic solution \(\gamma (t)\) is orbitally stable.
2. Define: the Poincare Map for the orbit \(\gamma \).
3. \([17]\) \(\gamma (t)=(2\cos 2t,\sin 2t)\) is a periodic solution to \[ \begin {cases} \dot {x}=-4y+x\left (1-\dfrac {x^2}{4}-y^2\right ),\\ \dot {y}=x+y\left (1-\dfrac {x^2}{4}-y^2\right ). \end {cases} \] Determine the orbital stability of \(\gamma (t)\) by computing the derivative of its Poincare Map.

Find the stable/center/unstable manifolds of the system