Chapter 11

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=3 x^{2}-4 y \\ &y^{\prime}=x-y \end{aligned} $$

In Problems, use the Dulac negative criterion to show that the given plane autonomous system has no periodic solutions. Experiment with simple functions of the form \(\delta(x, y)=a x^{2}+b y^{2}, e^{a x+b y}\), or \(x^{a} y^{b}\) $$ \begin{aligned} &x^{\prime}=-x^{3}+4 x y \\ &y^{\prime}=-5 x^{2}-y^{2} \end{aligned} $$

A predator-preyinteraction is describedby the Lotka-Volterra model $$ \begin{aligned} &x^{\prime}=-0.1 x+0.02 x y \\ &y^{\prime}=0.2 y-0.025 x y . \end{aligned} $$ (a) Find the critical point in the first quadrant, and use a numerical solver to sketch some population cycles. (b) Estimate the period of the periodic solutions that are close to the critical point in part (a).

In Problems \(3-10\), without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d A}{d t}=k \sqrt{A}(K-\sqrt{A}), A>0 $$

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x^{3}-y \\ &y^{\prime}=x-y^{3} \end{aligned} $$

Without referring back to the text. Fill in the blank or answer true/false. If a plane autonomous system has no critical points in an annular invariant region \(R\), then there is at least one periodic solution in \(R\). _____.

Use the Dulac negative criterion to show that the given plane autonomous system has no periodic solutions. Experiment with simple functions of the form \(\delta(x, y)=a x^{2}+b y^{2}, e^{a x+b y}\), or \(x^{a} y^{b}\). $$ \begin{aligned} &x^{\prime}=-x^{3}+4 x y \\ &y^{\prime}=-5 x^{2}-y^{2} \end{aligned} $$

A predator-prey interaction is described by the Lotka-Volterra model $$ \begin{aligned} &x^{\prime}=-0.1 x+0.02 x y \\ &y^{\prime}=0.2 y-0.025 x y \end{aligned} $$ (a) Find the critical point in the first quadrant, and use a numerical solver to sketch some population cycles. (b) Estimate the period of the periodic solutions that are close to the critical point in part (a).

Without solving explicitly, classify the critical points of the given first- order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$\frac{d A}{d t}=k \sqrt{A}(K-\sqrt{A}), A>0$$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x^{3}-y \\ &y^{\prime}=x-y^{3} \end{aligned} $$

In Problems \(11-20\), classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. \(x^{\prime}=1-2 x y\) \(y^{\prime}=2 x y-y\)

A competitive interaction is described by the Lotka-Volterra competition model $$ \begin{aligned} &x^{\prime}=0.08 x(20-0.4 x-0.3 y) \\ &y^{\prime}=0.06 y(10-0.1 y-0.3 x). \end{aligned} $$ Find and classify all critical points of the system.

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x\left(10-x-\frac{1}{2} y\right) \\ &y^{\prime}=y(16-y-x) \end{aligned} $$

Solve the following nonlinear plane autonomous system by switching to polar coordinates, and describe the geometric behavior of the solution that satisfies the given initial condition. $$ \begin{aligned} &x^{\prime}=-y-x\left(\sqrt{x^{2}+y^{2}}\right)^{3} \\ &y^{\prime}=x-y\left(\sqrt{x^{2}+y^{2}}\right)^{3}, \quad \mathbf{X}(0)=(1,0) \end{aligned} $$

Show that the plane autonomous system $$ \begin{aligned} &x^{\prime}=x\left(1-x^{2}-3 y^{2}\right) \\ &y^{\prime}=y\left(3-x^{2}-3 y^{2}\right) \end{aligned} $$ has no periodic solutions in an elliptical region about the origin.

Classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$ \begin{aligned} &x^{\prime}=1-2 x y \\ &y^{\prime}=2 x y-y \end{aligned} $$

Classify the critical point \((0,0)\) of the given linear system by computing the trace \(\tau\) and determinant \(\Delta\) and using Figure 11.2.12. $$ \begin{aligned} &x^{\prime}=-5 x+3 y \\ &y^{\prime}=-2 x+5 y \end{aligned} $$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x\left(10-x-\frac{1}{2} y\right) \\ &y^{\prime}=y(16-y-x) \end{aligned} $$

Classify the critical point \((0,0)\) of the given linear system by computing the trace \(\tau\) and determinant \(\Delta\) and using Figure \(11.2 .12 .\) $$ \begin{aligned} &x^{\prime}=-5 x+3 y \\ &y^{\prime}=-7 x+4 y \end{aligned} $$

In Problems \(11-20\), classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. \(x^{\prime}=x^{2}-y^{2}-1\) \(y^{\prime}=2 y\)

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-2 x+y+10 \\ &y^{\prime}=2 x-y-15 \frac{y}{y+5} \end{aligned} $$

Discuss the geometric nature of the solutions to the linear system \(\mathbf{X}^{\prime}=\mathbf{A X}\) given the general solution. (a) \(\mathbf{X}(t)=c_{1}\left(\begin{array}{l}1 \\ 1\end{array}\right) e^{-t}+c_{2}\left(\begin{array}{r}1 \\ -2\end{array}\right) e^{-2 t}\) (b) \(\mathbf{X}(t)=c_{1}\left(\begin{array}{r}1 \\ -1\end{array}\right) e^{-t}+c_{2}\left(\begin{array}{l}1 \\ 2\end{array}\right) e^{2 t}\)

Classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$ \begin{aligned} &x^{\prime}=x^{2}-y^{2}-1 \\ &y^{\prime}=2 y \end{aligned} $$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-2 x+y+10 \\ &y^{\prime}=2 x-y-15 \frac{y}{y+5} \end{aligned} $$

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x^{2} e^{y} \\ &y^{\prime}=y\left(e^{x}-1\right) \end{aligned} $$

Classify the critical point \((0,0)\) of the given linear system by computing the trace \(\tau\) and determinant \(\Delta\). (a) \(\begin{aligned} x^{\prime} &=-3 x+4 y \\ y^{\prime} &=-5 x+3 y \end{aligned}\) (b) \(\begin{aligned} x^{\prime} &=-3 x+2 y \\ y^{\prime} &=-2 x+y \end{aligned}\)

Classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$ \begin{aligned} &x^{\prime}=y-x^{2}+2 \\ &y^{\prime}=x^{2}-x y \end{aligned} $$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x^{2} e^{y} \\ &y^{\prime}=y\left(e^{x}-1\right) \end{aligned} $$

In Problems, find a circular invariant region for the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-y-x e^{x+y} \\ &y^{\prime}=x-y e^{x+y} \end{aligned} $$

Classify the critical point \((0,0)\) of the given linear system by computing the trace \(\tau\) and determinant \(\Delta\) and using Figure \(11.2 .12 .\) $$ \begin{aligned} &x^{\prime}=\frac{3}{2} x+\frac{1}{4} y \\ &y^{\prime}=-x+\frac{1}{2} y \end{aligned} $$

In Problems \(11-20\), classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. \(x^{\prime}=2 x-y^{2}\) \(y^{\prime}=-y+x y\)

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=\sin y \\ &y^{\prime}=e^{x-y}-1 \end{aligned} $$

Find and classify (if possible) the critical points of the plane autonomous system $$ \begin{aligned} &x^{\prime}=x+x y-3 x^{2} \\ &y^{\prime}=4 y-2 x y-y^{2} \end{aligned} $$ Does this system have any periodic solutions in the first quadrant?

Classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$ \begin{aligned} &x^{\prime}=2 x-y^{2} \\ &y^{\prime}=-y+x y \end{aligned} $$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} x^{\prime} &=\sin y \\ y^{\prime} &=e^{x-y}-1 \end{aligned} $$

In Problems, find a circular invariant region for the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-x+y+x y \\ &y^{\prime}=x-y-x^{2}-y^{3} \end{aligned} $$

In Problems \(11-20\), classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. \(x^{\prime}=-3 x+y^{2}+2\) \(y^{\prime}=x^{2}-y^{2}\)

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x\left(1-x^{2}-3 y^{2}\right) \\ &y^{\prime}=y\left(3-x^{2}-3 y^{2}\right) \end{aligned} $$

Find a circular invariant region for the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-x+y+x y \\ &y^{\prime}=x-y-x^{2}-y^{3} \end{aligned} $$

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x\left(1-x^{2}-3 y^{2}\right) \\ &y^{\prime}=y\left(3-x^{2}-3 y^{2}\right) \end{aligned} $$

If we assume that a damping force acts in a direction opposite to the motion of a pendulum and with a magnitude directly proportional to the angular velocity \(d \theta / d t\), the displacement angle \(\theta\) for the pendulum satisfies the nonlinear second-order differential equation $$ m l \frac{d^{2} \theta}{d t^{2}}=-m g \sin \theta-\beta \frac{d \theta}{d t}. $$ (a) Write the second-order differential equation as a plane autonomous system, and find all critical points. (b) Find a condition on \(m, l\), and \(\beta\) that will make \((0,0)\) a stable spiral point.

In Problems \(11-20\), classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. \(x^{\prime}=x y-3 y-4\) \(y^{\prime}=y^{2}-x^{2}\)

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-x\left(4-y^{2}\right) \\ &y^{\prime}=4 y\left(1-x^{2}\right) \end{aligned} $$

Without solving explicitly, classify (if possible) the critical points of the autonomous first-order differential equation \(x^{\prime}=\left(x^{2}-1\right) e^{-x / 2}\) as asymptotically stable or unstable.