Poincare-Bendixson Theorem

Consider the system

1. 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.
2. Show that the annular region \(A=\{1<|(x,y)|<\sqrt {2}\}\) is forward invariant.
3. State the Poincaré-Bendixson theorem.
4. Show that the system has at least one non-constant periodic orbit.

Let \(A\) be a real \(d\times d\) matrix. Consider the initial value problem

(3)

where \(h(t)\in \mathbb {R}^d\) is a real \(T\)-periodic continuous function.

1. Define the Poincare Map in this context and give an explicit formula for the Poincare Map in terms of \(A\), \(h\) and \(x_0\).
2. Under what conditions on \(A\) and \(h\) is there is a unique \(T\)-periodic solution of (3)?
3. Compute the derivative of the Poincare map at the periodic solution. What stability property do you conclude for the periodic solution?

1. Suppose the function \(f(t,x)\) is periodic in \(t\), \(f(t+1,x)=f(t,x)\). Define the Poincare map for the differential equation \(\dot {x}=f(t,x)\).

2. Consider the differential equation

   \[ \dot {x}=x(1-x)-h(t)x, \]

   where \(h(t)\ge 0\) is periodic, \(h(t+1)=h(t)\). Under what conditions on \(h(t)\) does this equation have a strictly positive periodic solution?

3. Suppose \(h(t)=a\sin ^2(\pi x)\). What condition on \(a\) guarantees the existence of a positive periodic solution?

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.

Suppose \(A(t)\) is a periodic function, \(A(t+1)=A(t)\) and \(0<A(t)<1\). Prove that the differential equation \(u'=u(A(t)-u)\) has a unique, positive, stable, periodic solution. Prove that the trivial solution \(u(t)=0\) is unstable. Hint: Use the Poincare map.

1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.

2. For the system

   \[ \dot {x}=x-y-x^3,\quad \dot {y}=x+y-y^3, \]

   identify all critical points and their stability, and use the Poincare-Bendixson theorem to show that this system has a periodic orbit in the annular region \(1<x^2+y^2<2\).

1. Consider the dynamical system \[ \begin {aligned} \dot {x} &= -y+x(1-z^2-x^2-y^2),\\ \dot {y} &= x+y(1-z^2-x^2-y^2),\\ \dot {z} &= 0. \end {aligned} \] Determine the invariant sets and attracting set of the system. Give a general definition of the \(\omega \)-limit set for a flow \(\phi (x,t)\) in \(\mathbb {R}^n\), and determine it in the case of a trajectory for which \(|z(0)|<1\).
2. Use the Poincare-Bendixson Theorem and the fact that the planar system \[ \dot {x}=x-y-x^3,\quad \dot {y}=x+y-y^3 \] has only the one critical point at the origin to show that this system has a periodic orbit in the annular region \(A=\{x\in \mathbb {R}^2\mid 1<|x|<\sqrt {2}\}\).
3. 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.
4. Consider the system \[ \dot {x}=x-y-x(x^2+y^2)+\frac {xy}{\sqrt {x^2+y^2}} \] \[ \dot {y}=x+y-y(x^2+y^2)-\frac {x^2}{\sqrt {x^2+y^2}}. \] 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.

Consider the first order differential equation \[ \frac {dx}{dt}=f(x,t), \tag {3} \] where \(f(x,t)\) is periodic in \(t\), \(f(x,t+1)=f(x,t)\) for all \(x\) and \(t\).

1. Define the Poincare map for this differential equation.
2. Suppose \(f(x,t)=x(A(t)-x)\), and \(0<\alpha <A(t)<\beta \). Prove that the differential equation has at least one nontrivial periodic solution.

1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.
2. For the system \[ \dot {x}=x-y-x^3,\qquad \dot {y}=x+y-y^3, \] identify all critical points and their stability, and use the Poincare-Bendixson theorem to show that this system has a periodic orbit in the annular region \(1<x^2+y^2<2\).

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

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

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

1. Define: the periodic solution \(\gamma (t)\) is orbitally stable.
2. Define: the Poincaré Map for the orbit \(\gamma \).
3. \(\gamma (t)=(2\cos 2t,\sin 2t)\) 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 Poincaré Map.

1. Consider the dynamical system \[ \begin {aligned} \dot {x} &= -y+x(1-z^2-x^2-y^2),\\ \dot {y} &= x+y(1-z^2-x^2-y^2),\\ \dot {z} &= 0. \end {aligned} \] Determine the invariant sets and attracting set of the system. Give a general definition of the \(\omega \)-limit set, and determine it in the case of a trajectory for which \(\abs {z(0)}<1\). Sketch the flow.
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 \(A=\{x\in \mathbb {R}^2\mid 1<\abs {x}<\sqrt {2}\}\).

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.

1. Show how to convert the ODE \[ \dot {x}=f_1(x,y),\qquad \dot {y}=f_2(x,y). \] into polar coordinates.
2. Consider the dynamical system \[ \begin {aligned} \dot {x} &= -y+x(1-z^2-x^2-y^2),\\ \dot {y} &= x+y(1-z^2-x^2-y^2),\\ \dot {z} &= 0. \end {aligned} \] Determine the invariant sets and attracting set of the system. Give a general definition of the \(\omega \)-limit set for a flow \(\phi (x,t)\) in \(\mathbb {R}^n\), and determine it in the case of a trajectory for which \(\abs {z(0)}<1\).
3. 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 \(A=\{x\in \mathbb {R}^2\mid 1<\abs {x}<\sqrt {2}\}\).

Consider a nonconstant \(T\)-periodic orbit \(\gamma (t)\) of the smooth autonomous differential equation \[ \dot {x}=f(x),\qquad x\in \mathbb {R}^n. \]

1. Describe the Poincaré Return Map associated to \(\gamma \).
2. A \(2\pi \)-periodic solution of the planar system \[ \begin {aligned} \dot {x} &= x-(x+y)\sqrt {x^2+y^2},\\ \dot {y} &= y+(x-y)\sqrt {x^2+y^2} \end {aligned} \] is given by \(\gamma (t)=(\cos t,\sin t)\). Find a Poincaré Return Map associated to \(\gamma \).
3. Find the derivative of the Poincaré Return Map from part (b) at \(\gamma \).
4. Using (c), determine whether the solution (b) is stable and define the sense of stability you mean.

1. Let \(G\) be an open set. Define the Sobolev spaces \(H^m(G)\), \(m=1,2,\cdots \), and \(H_0^m(G)\), \(m=1,2,\cdots \). Indicate why these spaces are Hilbert spaces.
2. Let \(G\) be a bounded open set. State and prove Poincaré’s inequality for functions \(u\in H_0^1(G)\) and thus verify that \(H_0^1(G)\) is also a Hilbert space with respect to the inner product \[ (u,v)_{H_0^1(G)}=\int _G \nabla u\cdot \overline {\nabla v}\,dx. \]

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.

Let \(U=\{x\in \mathbb {R}^2:0<\lvert x\rvert <1\}\), and denote \(\Gamma =\{\lvert x\rvert =1\}\). Consider \[ \begin {aligned} -\Delta u &= f, && \text {in } U,\\ u &= 0, && \text {on } \{x=0\},\\ \frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma , \end {aligned} \] where \(\eta \) is the unit outward normal, and \(f\in L^2(U)\).

Prove a Poincaré-type inequality for this problem, and use it to show that there exists a unique weak solution \(u\) in an appropriate Hilbert space.

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.

Problem. State and proof Poincaré’s inequality for \(H^1_0(\Omega )\) and use it to obtain two equivalent inner products for this space.

Let \(G\) be a bounded open set in \(\mathbb {R}^N\) with smooth boundary. State and indicate a proof of Poincaré’s inequality for functions \(u\in H^1_0(G)\). Using the inequality show that \[ \norm {u}_0^2=\int _G \lvert \nabla u\rvert ^2\,dx \] defines a norm equivalent to the \(H^1\) norm. Consider the form \[ a(u,v)=\int _G \nabla u\cdot \overline {\nabla v} +\int _G c(x)\cdot \nabla u\,\overline {v}\,dx, \tag {7} \] where the components of the vector function \(c\) belong to \(L^\infty (G)\).

Provide a condition on the function \(c\) in order that \(a\) is a sesquilinear form satisfying the conditions of the Lax-Milgram theorem. Determine a boundary value problem solved by the Lax-Milgram solution.

1. Prove Bendixson’s Criterion: Let \(\mathbf {f}\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(\mathbf {f}\) is not identically zero and is either nonnegative in \(E\) or nonpositive in \(E\). Then \(\dot {x}=\mathbf {f}(x)\) has no closed orbit which is entirely in \(E\).
2. Extend the above proof to cover Dulac’s Criterion: Let \(B(x_1,x_2)\) and \(\mathbf {f}\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(B\mathbf {f}\) is not identically zero and is either nonnegative in \(E\) or nonpositive in \(E\). Then \(\dot {x}=\mathbf {f}(x)\) has no closed orbit which is entirely in \(E\).
3. Consider the nondimensionalized Lotka-Volterra competition equations \[ \begin {aligned} \dot {u}&=ru(1-u-\alpha _1v),\\ \dot {v}&=v(1-v-\alpha _2u), \end {aligned} \] where \(r\), \(\alpha _1\) and \(\alpha _2\) are positive parameters. Apply Dulac’s criterion to this system for \(u,v>0\). (HINT: try \(B(u,v)=1/uv\).) Does Bendixson’s criteria give a similar result?
4. Draw a biological conclusion.

Consider the system \[ \begin {aligned} \dot {x}_1&=x_2,\\ \dot {x}_1&=-x_1+x_2(1-x_1^2-2x_2^2). \end {aligned} \] Prove, by means of the Poincaré-Bendixson theorem, that there exists at least one periodic orbit in the annulus \(1/2<x_1^2+x_2^2<1\).

Let \(\Omega \) be a bounded open set. Define the Sobolev space \(H^{1,2}_0(\Omega )\). Indicate why this is a Hilbert space. State and prove Poincaré’s inequality for functions \(u\in H^{1,2}_0(\Omega )\) and thus verify that \(H^{1,2}_0(\Omega )\) is also a Hilbert space with respect to the inner product \[ (u,v)=\int _\Omega \nabla u\cdot \overline {\nabla v}\,dx. \]

Let \(\Omega \) be a bounded domain in \(\mathbb {R}^n\). Consider \[ \begin {aligned} \Delta u&=f &&\text {in }\Omega ,\\ u&=0 &&\text {on }\partial \Omega \end {aligned} \] where \(f\in L^2(\Omega )\).

1. Define the concept of a weak \(H^{1,2}_0(\Omega )\) solution to the above problem.
2. Use Poincaré’s inequality to prove existence of a weak \(H^{1,2}_0(\Omega )\) solution.
3. Assume that \(\Omega \) is smooth. Prove that any two solutions in \(C^2(\Omega )\cap C^1(\overline {\Omega })\) must agree. How would you prove this result for any two solutions in \(C^2(\Omega )\cap C(\overline {\Omega })\)?

Let \(G\) be a bounded open set. State and prove Poincaré’s inequality for functions \(u\in H^1_0(G)\) and thus verify that \(H^1_0(G)\) is also a Hilbert space with respect to the inner product \[ (u,v)_{H^1_0(G)}=\int _G \nabla u\cdot \overline {\nabla v}\,dx. \]

Consider the system \[ x'=f(x) \tag {4.1} \] where \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\), and suppose that (4.1) has a periodic solution \(\gamma (t)\) of least period \(T\).

1. Define what it means to say that \(\gamma (t)\) is orbitally asymptotically stable with asymptotic phase.
2. Let \[ \xi '=A(t)\xi \tag {4.2} \] be the variational equation (the linearization of (4.1)) relative to \(\gamma (t)\). State the Floquet representation for the fundamental matrix solution of (4.2) and discuss the stability of \(\gamma (t)\) in terms of properties of the fundamental matrix solution.
3. Consider the system \[ \begin {aligned} x' &= x-y-x(x^2+y^2),\\ y' &= x+y-y(x^2+y^2). \end {aligned} \] Find a nontrivial periodic solution of this equation, and prove (in the easiest way possible) that this periodic solution is orbitally asymptotically stable with asymptotic phase. Compute the asymptotic phase explicitly.
4. Compute the Poincare map associated with the periodic orbit you found in (c).

State the Poincaré inequality for functions \(u\in H^1_0(\Omega )\).

Let \(G\) be a bounded open set. State and prove Poincaré’s inequality for functions \(u\in W_0^{1,2}(G)\) and thus verify that \(W_0^{1,2}(G)\) is also a Hilbert space with respect to the inner product \[ (u,v)_{W_0^{1,2}(G)}=\int _G \nabla u\cdot \overline {\nabla v}\,dx. \]

Let \(r^2=x^2+y^2\), \(u=r^4-3r^2+1\) and consider the planar system \(x'=-y+xu\), \(y'=x+yu\). Prove by means of the Poincaré–Bendixson theorem that there are two periodic orbits, one in \(0<r<1\) and another in \(1<r<3\).

Let \(G\) be an open set in \(\mathbb {R}^n\) with \[ \sup \{\lvert x_1\rvert :(x_1,x_2,\ldots ,x_n)\in G\}=K<+\infty . \] Then \[ \lVert \varphi \rVert _{L^2(G)}\le 2K\lVert \partial _1\varphi \rVert _{L^2(G)}, \quad \forall \varphi \in H_0^1(G). \tag {6} \] \[ \left (\text {Hint: For }\varphi \in C_0^\infty (G)\text { compute } \frac {\partial }{\partial x_1}(x_1\lvert \varphi (x)\rvert ^2)\right . \] \[ \left .\text {and use the divergence theorem.}\right ) \] Use (6) to obtain Poincaré’s inequality for functions \(\varphi \in H_0^1(G)\), whenever \(G\) is a bounded domain.

Consider the Dirichlet problem \[ \Delta u=f\quad \text {in }\Omega , \] \[ u=0\quad \text {on }\partial \Omega , \] where \(\Omega \subset \mathbb {R}^n\) is a bounded open set and \(f\in L_2(\Omega )\).

1. Define the concept of a weak \(H_0^1(\Omega )\) solution of the above problem.
2. Use the Poincaré inequality, \[ \int _\Omega u^2\,dx\le c\int _\Omega \lvert \nabla u\rvert ^2\,dx \quad \text {for all }u\in H_0^1(\Omega ), \] to prove the existence of a weak \(H_0^1(\Omega )\) solution.