91Ó°ÊÓ

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 P}{d t}=P(a-b P)\left(1-c P^{-1}\right), P>0, a

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 \(x^{a} y^{b}\) $$ \begin{aligned} &x^{\prime}=-2 x+x y \\ &y^{\prime}=2 y-x^{2} \end{aligned} $$

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 \(x^{a} y^{b}\). $$ \begin{aligned} &x^{\prime}=-2 x+x y \\ &y^{\prime}=2 y-x^{2} \end{aligned} $$

Classify the critical point \((0,0)\) of the given linear system by computing the trace \(\tau\) and determinant \(\Delta\) and using Figure \(11.2 .12 .\) $$ \begin{aligned} &x^{\prime}=-5 x+3 y \\ &y^{\prime}=2 x+7 y \end{aligned} $$

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 P}{d t}=P(a-b P)\left(1-c P^{-1}\right), P>0, a

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

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 \(x^{a} y^{b}\). $$ \begin{aligned} &x^{\prime}=-x^{3}+4 x y \\ &y^{\prime}=-5 x^{2}-y^{2} \end{aligned} $$

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 \(R\). _____.

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