91Ó°ÊÓ

(a) Use the Poincaré-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 \(A=\left\\{x \in \mathbf{R}^{2} \mid\right.\) \(1<|\mathbf{x}|<\sqrt{2}\\}\). Hint: Convert to polar coordinates and show that for all \(\epsilon>0, \dot{r}<0\) on the circle \(r=\sqrt{2}+\epsilon\) and \(\dot{r}>0\) on \(r=1-\epsilon\); then use the Poincaré-Bendixson theorem to show that this implies that there is a limit cycle in $$ \bar{A}=\left\\{\mathbf{x} \in \mathbf{R}^{2}|1 \leq| \mathbf{x} \mid \leq \sqrt{2}\right\\} $$ and then show that no limit cycle can have a point in common with either one of the circles \(r=1\) or \(r=\sqrt{2}\). (b) Show that there is at least one stable limit cycle in \(A\). (In fact, this system has exactly one limit cycle in \(A\) and it is stable. Cf. Problem 3 in Section 3.9.) This limit cycle and the annular region \(A\) are shown in Figure 2 .

Show that $$ \dot{x}=y \quad \dot{y}=-x+\left(1-x^{2}-y^{2}\right) y $$ has a unique stable limit cycle which is the \(\omega\)-limit set of every trajectory except the critical point at the origin. Hint: Compute \(\dot{r}\).

Consider the Lorenz system $$ \begin{aligned} &\dot{x}=\sigma(y-x) \\ &\dot{y}=\rho x-y-x z \\ &z=x y-\beta z \end{aligned} $$ with \(\sigma>0, \rho>0\) and \(\beta>0\) (a) Show that this system is invariant under the transformation \((x, y \cdot z, t) \rightarrow(-x,-y, z, t)\) (b) Show that the \(z\)-axis is invariant under the flow of this system and that it consists of three trajectories. (c) Show that this system has equilibrium points at the origin and at \((\pm \sqrt{\beta(\rho-1)}, \pm \sqrt{\beta(\rho-1)}, \rho-1)\) for \(\rho>1\). For \(\rho>1\) show that there is a one-dimensional unstable manifold \(W^{u}(0)\) at the origin. (d) For \(\rho \in(0,1)\) use the Liapunov function \(V(x, y, z)=\rho x^{2}+\sigma y^{2}+\) \(\sigma z^{2}\) to show that the origin is globally stable; i.e., for \(\rho \in(0,1)\), the origin is the \(\omega\)-limit set of every trajectory of this system.

Let \(z=x+i y, \bar{z}=x-i y\) and show that the vector fields in the complex plane defined by $$ \dot{z}=z^{k} \quad \text { and } \quad \dot{z}=\bar{z}^{k} $$ have unique critical points at \(z=0\) with indices \(k\) and \(-k\) respectively. Hint: Write \(\dot{x}=\operatorname{Re}\left(z^{k}\right) ; \dot{y}=\operatorname{Im}\left(z^{k}\right)\) and let \(z=r e^{i \theta} .\) Sketch the phase portrait near the origin in those cases with indices \(\pm 2\) and \(\pm 3\)