Topological Degree and Fixed Point Methods

Consider the differential equation \(\ddot {x}+\eta \dot {x}+x^3-x=0\), with \(0<\eta <2\sqrt {2}\) a constant.

1. Sketch a phase portrait for this equation. Identify all fixed points and their type.
2. Find a first integral of the motion if \(\eta = 0\). Make a sketch of these integral curves in the \(x-\dot {x}\) plane.
3. Suppose that at time \(t=0\), \(\frac {\dot {x}^2}{2}+\frac {1}{4}x^4-\frac {1}{2}x^2<0\), and \(0<x<\sqrt {2}\). Determine \(\lim _{t\to \infty }x(t)\), and explain how to prove this.

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 \(\|x\|=\max _{0\le t\le T}|x(t)|\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\,ds \] for \(x \in X\). Define \(V=\{x\in X\mid \|x-x_0\|\le \epsilon \}\) 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. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\), show that \[ |K(x(t))-K(y(t))|\le L_0t\|x-y\|. \] Hence, show that \[ \|K(x)-K(y)\|\le L_0T\|x-y\|,\quad x,y\in V. \]
2. 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 \[ \frac {dx}{dt}=f(x),\quad x(0)=x_0. \]
3. Show that the solution of the equation \[ \dot {x}=x^2,\quad x(0)=x_0 \] blows up in finite time. Does this contradict the result of part (b)?

Let \(p(t)\), \(q(t)\) and \(r(t)\) be continuous functions defined for \(t\in \mathbb {R}\) and \(t_0,x_0,x_1\in \mathbb {R}\). Consider the initial value problem \[ \begin {cases} \ddot {x}+p(t)\dot {x}+q(t)x=r(t)\\ x(t_0)=x_0,\qquad \dot {x}(t_0)=x_1. \end {cases} \tag {1} \]

1. Let \(X\) be a Banach Space with norm \(\norm {\bullet }\). Let \(C\subset X\) be a closed subset. State the Contraction Mapping Theorem (otherwise known as the Banach Fixed Point Theorem) for a map \(K:C\to X\).
2. Carefully reformulate (1) as a fixed point problem. Prove that the fixed point problem is equivalent to the local existence problem for (1).
3. Using the Contraction Mapping Theorem, show that there is a \(\delta >0\) and a unique function \(x(t)\in C^2([t_0,t_0+\delta ])\) that satisfies your fixed point problem, and hence (1), for \(t_0\le t\le t_0+\delta \).

Let \(D\subset \mathbb {R}\times \mathbb {R}^n\) be an open set, \((t_0,x_0)\in D\) a point and \(f(t,x)\in C(D,\mathbb {R}^n)\) be a map.

1. Let \(X\) be a Banach Space with norm \(\norm {\bullet }\). Let \(C\subset X\) be a closed subset. State the Contraction Mapping Theorem (otherwise known as the Banach Fixed Point Theorem) for a map \(K:C\to X\).
2. Define what it means for a map \(f(t,x)\in C(D,\mathbb {R}^n)\) to be locally Lipschitz with respect to \(x\)
3. Carefully reformulate the Initial Value Problems as a fixed point problem. Prove that the fixed point problem is equivalent to the local existence problem for the (IVP) \[ \begin {cases} \dfrac {dx}{dt}=f(t,x),\\ x(t_0)=x_0. \end {cases} \tag {IVP} \]
4. Assume \(f(t,x)\in C(D,\mathbb {R}^n)\) is locally Lipschitz with respect to \(x\). Using the Contraction Mapping Theorem, show that there is a \(\epsilon >0\) and a unique function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem (IVP) for \(t_0\le t\le t_0+\epsilon \).

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are proportional to concentration of protein and the messenger RNA from which it is translated, respectively, and \(\mu >0\) is a rate constant \[ \begin {aligned} \dot {x} &= y-\mu x,\\ \dot {y} &= \frac {x^2}{1+x^2}-y. \end {aligned} \]

1. Show that the system has three fixed points when \(\mu <\mu _c\) and one when \(\mu >\mu _c\) where \(\mu _c\) is to be determined.
2. What is the nature of the bifurcation at \(\mu =\mu _c\)?

Consider a Brusselator system for chemical reactants \(x,y\) given by \[ \begin {aligned} \dot {x} &= 1-4x+x^2y,\\ \dot {y} &= 3x-x^2y \end {aligned} \]

1. Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and \(y=10-x\) is a forward invariant subset. [Hint: need to show that if \(n\) is the outer normal to a given edge of the quadrilateral, then \(n\cdot (\dot {x},\dot {y})\le 0\).]
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are proportional to concentration of protein and the messenger RNA from which it is translated, respectively, and \(\mu >0\) is a rate constant \[ \begin {aligned} \dot {x} &= y-\mu x,\\ \dot {y} &= \frac {x^2}{1+x^2}-y. \end {aligned} \]

1. Show that the system has three fixed points when \(\mu <\mu _c\) and one when \(\mu >\mu _c\) where \(\mu _c\) is to be determined.
2. What is the nature of the bifurcation at \(\mu =\mu _c\)?

Consider a Brusselator system for chemical reactants \(x,y\) given by \[ \begin {aligned} \dot {x} &= 1-4x+x^2y,\\ \dot {y} &= 3x-x^2y \end {aligned} \]

1. Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and \(y=10-x\) is a forward invariant subset.
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

Consider a modified predator-prey system for populations \(x,y\) given by \[ \begin {aligned} \dot {x} &= x\cdot \left (1-\frac {x}{4}-\frac {y}{1+x}\right ),\\ \dot {y} &= y\cdot \left (\frac {x}{1+x}-\frac 12\right ) \end {aligned} \]

1. Show that the triangle with vertices \((0,0)\), \((5,0)\) and \((0,5)\) is a forward invariant subset.
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

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.

1. Give the definitions of Poincare and Liapunov stability. Show that solutions of the system \(\dot {x}=y\), \(\dot {y}=0\) are Poincare but not Liapunov stable.
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.

Contraction Mapping Theorem. Show that the problem \[ \begin {aligned} \dot {x}(t) &= f(x(t))+\frac {1}{a}\int _0^a x(s)\,ds,\quad t\in (0,a),\\ x(0) &= 0, \end {aligned} \] admits a unique solution \(x\in C([0,a];\mathbb {R}^n)\) for \(a>0\) sufficiently small. Assume \(f:\mathbb {R}^n\to \mathbb {R}^n\) satisfies \(\lvert f(x)-f(y)\rvert \le M\lvert x-y\rvert \) for all \(x,y\in \mathbb {R}^n\).

Contraction Mapping. Consider the initial value problem \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\) satisfies for all \(x,y\in \mathbb {R}^n\) \[ \lvert f(x)-f(y)\rvert \le (1+\lvert x\rvert +\lvert y\rvert )\lvert x-y\rvert . \] Use the contraction mapping principle to find a condition on \(\lvert x_0\rvert \) which guarantees local existence and uniqueness of solutions. How does the local time interval upon which existence is obtained depend on \(\lvert x_0\rvert \)?

Linearization. Assume \(g\in C^\infty (\mathbb {R}^n)\) has a strict local minimum at the origin. Consider the second-order system \[ \ddot {x}+B\dot {x}+\nabla g(x)=0, \] where \(x(t)\in \mathbb {R}^n\) and \(B\in \mathbb {R}^{n\times n}\) is a symmetric positive definite matrix.

1. Rewrite the system as a first-order system.
2. Write the linearization of the system at the origin.
3. Prove that the origin is an asymptotically stable fixed point.

Consider the system of ordinary differential equations \[ u'=-\lvert u\rvert ^2u+h(t), \tag {1} \] where \[ h:[0,1]\to \mathbb {R}^N \] is a continuous function and \(\lvert \cdot \rvert \) is the Euclidean norm in \(\mathbb {R}^N\). Use Brouwer’s fixed point theorem to show that (1) has a solution \(u\in C^1[0,1]\) such that \[ u(0)=u(1). \] Use this result to conclude that if \[ h:\mathbb {R}\to \mathbb {R}^N \] is continuous and periodic of period 1, then (1) has a solution of period 1.

(Hint: Consider equation (1) subject to initial data \(u_0:\lvert u_0\rvert \le r\), where \(r\) is large (in relation to \(h\)) and examine solution curves by computing \(\frac {d}{dt}\lvert u\rvert ^2\) for solutions \(u\).) In relation to the above, what can you say about the equation \[ u'=\lvert u\rvert ^2u+h(t). \tag {2} \]

Let \(A\) be a real \(N\times N\) matrix, all of whose eigenvalues have negative real parts. Let \[ h:\mathbb {R}\to \mathbb {R}^N \] be a continuous function which is periodic of period \(T\). Let \[ \begin {aligned} (t,v)&\longmapsto u(t,v),\\ \mathbb {R}\times \mathbb {R}^N&\longrightarrow \mathbb {R}^N \end {aligned} \] denote the unique solution of \[ u'=Au+h(t),\qquad u(0)=v. \tag {3} \] For given \(T>0\), define \[ S_T:\mathbb {R}^N\to \mathbb {R}^N \] by \[ S_T(v)=u(T,v). \]

1. Use the variation of constants formula (and what you know about stability theory for the linear system \(u'=Au\)) to show that \(S_{nT}\) is a contraction mapping, for some positive integer \(n\), sufficiently large.
2. Show that \[ S_{nT}=\underbrace {S_T\circ \cdots \circ S_T}_{n}, \] the \(n\)-fold composition of \(S_T\) with itself.
3. Show that if \(S_{nT}\) has a unique fixed point, then so does \(S_T\).
4. Apply this to show that if \(h\) is a continuous periodic function of period \(T\), then (3) has a unique \(T\)-periodic solution.

1. State the theorem of Rellich-Kondrachov for bounded open sets \(G\subset \mathbb {R}^N\).
2. Let \[ f:\overline {G}\times \mathbb {R}\to \mathbb {R} \] be a continuous bounded function such that \[ \lvert f(x,u)\rvert \le M,\qquad (x,u)\in \overline {G}\times \mathbb {R} \] where \(M\) is a constant. Show that the boundary value problem: \[ -\Delta u=f(x,u),\qquad u\in H^1_0(G), \tag {8} \] has a solution by converting (8) into an equivalent fixed point problem in \(L^2(G)\) and by applying Schauder’s fixed point theorem (give reasons why its hypotheses hold!).

Define the mapping \[ y\in \mathbb {R}^N\mapsto F_t(y)\in \mathbb {R}^N \tag {2.5} \] by \[ F_t(y)=x(t,y). \tag {2.6} \] where \(x(t,y)\) solves \[ \begin {aligned} x'&=f(t,x),\\ x(0,y)&=y. \end {aligned} \tag {2.7} \] Give reasons that \(F_t\) is well-defined for each small \(t\) and well-defined for all \(t\in \mathbb {R}\), whenever \(\lvert y\rvert \le R\).

Show that for each \(T\ge 0\) there exists \(y_T\), \(\lvert y_T\rvert \le R\) such that \[ F_T(y_T)=y_T. \tag {2.8} \]

What can one say if the condition (2.4) is replaced by \[ x\cdot f(t,x)>0,\quad t\in \mathbb {R},\quad \lvert x\rvert =R. \tag {2.9} \]

Convert problem (2.13) into an equivalent fixed point problem in an appropriate Banach space.