Chapter 11

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. $$ \begin{aligned} &x^{\prime}=x y-3 y-4 \\ &y^{\prime}=y^{2}-x^{2} \end{aligned} $$

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

In Problems, for the given linear dynamical system (taken from Exercises 10.2) (a) find the general solution and determine whether there are periodic solutions, (b) find the solution satisfying the given initial condition, and, (c) with the aid of a graphing utility, plot the solution in part (b) and indicate the direction in which the curve is traversed. $$ \begin{aligned} &x^{\prime}=x+2 y \\ &y^{\prime}=4 x+3 y, \mathbf{X}(0)=(2,-2)(\text { Problem } 1, \text { Exercises } 10.2) \end{aligned} $$

Determine conditions on the real constant \(\mu\) so that \((0,0)\) is a center for the linear system $$ \begin{aligned} &x^{\prime}=-\mu x+y \\ &y^{\prime}=-x+\mu y \end{aligned} $$

Use the phase-plane method to show that the solutions of the nonlinear second- order differential equation $$ x^{n}=-2 x \sqrt{\left(x^{\prime}\right)^{2}+1} $$ that satisfy \(x(0)=x_{0}\) and \(x^{\prime}(0)=0\) are periodic.

In the analysis of free, damped motion in Section \(3.8\) we assumed that the damping force was proportional to the velocity \(x^{\prime}\). Frequently the magnitude of this damping force is proportional to the square of the velocity, and the new differential equation becomes $$ x^{\prime \prime}=-\frac{\beta}{m} x^{\prime}\left|x^{\prime}\right|-\frac{k}{m} x . $$ (a) Write the second-order differential equation as a plane aatonomous system, and find all critical points. (b) The system is called overdamped when \((0,0)\) is a stable node and is called underdamped when \((0,0)\) is a stable spiral point. Physical considerations suggest that \((0,0)\) must be an asymptotically stable critical point. Show that the system is necessarily underdamped. [Hint: \(d / d y(y|y|)=2|y|\).]

Use the Poincaré-Bendixson theorem to show that the secondorder nonlinear differential equation $$ x^{\prime \prime}=x^{\prime}\left[1-3 x^{2}-2\left(x^{\prime}\right)^{2}\right]-x $$ has at least one periodic solution. [Hint: Find an invariant annular region for the corresponding plane autonomous system.]

Determine a condition on the real constant \(\mu\) so that \((0,0)\) is a stable spiral point of the linear system $$ \begin{aligned} &x^{\prime}=y \\ &y^{\prime}=-x+\mu y . \end{aligned} $$

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. $$ \begin{aligned} &x^{\prime}=x\left(1-x^{2}-3 y^{2}\right) \\ &y^{\prime}=y\left(3-x^{2}-3 y^{2}\right) \end{aligned} $$

Let \(\mathbf{X}=\mathbf{X}(t)\) be the solution of the plane autonomous system $$ \begin{aligned} &x^{\prime}=y \\ &y^{\prime}=-x-\left(1-x^{2}\right) y \end{aligned} $$ that satisfies \(\mathbf{X}(0)=\left(x_{0}, y_{0}\right) .\) Show that if \(x_{0}^{2}+y_{0}^{2}<1\), then \(\lim _{t \rightarrow \infty} \mathbf{X}(t)=(0,0) .\left[\right.\) Hint \(:\) Select \(r<1\) with \(x_{0}^{2}+y_{0}^{2}

In Problems, for the given linear dynamical system (taken from Exercises 10.2) (a) find the general solution and determine whether there are periodic solutions, (b) find the solution satisfying the given initial condition, and, (c) with the aid of a graphing utility, plot the solution in part (b) and indicate the direction in which the curve is traversed. $$ \begin{aligned} &x^{\prime}=4 x-5 y \\ &y^{\prime}=5 x-4 y, \mathbf{X}(0)=(4,5)(\text { Problem } 39, \text { Exercises } 10.2) \end{aligned} $$

An undamped oscillation satisfies a nonlinear second-order differential equation of the form \(x^{\prime \prime}+f(x)=0\), where \(f(0)=0\) and \(x f(x)>0\) for \(x \neq 0\) and \(-d

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. \(x^{\prime}=x\left(10-x-\frac{1}{2} y\right)\) \(y^{\prime}=y(16-y-x)\)

Show that the plane autonomous system $$ \begin{aligned} &x^{\prime}=4 x+2 y-2 x^{2} \\ &y^{\prime}=4 x-3 y+4 x y \end{aligned} $$ has no periodic solutions.

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. $$ \begin{aligned} &x^{\prime}=x\left(10-x-\frac{1}{2} y\right) \\ &y^{\prime}=y(16-y-x) \end{aligned} $$

Show that \((0,0)\) is always an unstable critical point of the linear system $$ \begin{aligned} &x^{\prime}=\mu x+y \\ &y^{\prime}=-x+y \end{aligned} $$ where \(\mu\) is a real constant and \(\mu \neq-1\). When is \((0,0)\) an unstable saddle point? When is \((0,0)\) an unstable spiral point?

Investigate global stability for the system $$ \begin{aligned} &x^{\prime}=y-x \\ &y^{\prime}=-x-y^{3} \end{aligned} $$

The Lotka-Volterra predator-prey model assumes that, in the absence of predators, the number of prey grows exponentially. If we make the alternative assumption that the prey population grows logistically, the new system is $$ \begin{aligned} &x^{\prime}=-a x+b x y \\ &y^{\prime}=-c x y+\frac{r}{K} y(K-y), \end{aligned} $$ where \(a, b, c, r\), and \(K\) are positive and \(K>a / b\). (a) Show that the system has critical points at \((0,0),(0, K)\), and \((\hat{x}, \hat{y})\), where \(\hat{y}=a l b\) and \(c \hat{x}=\frac{r}{K}(K-\hat{y})\). (b) Show that the critical points at \((0,0)\) and \((0, K)\) are saddle points, whereas the critical point at \((\hat{x}, \hat{y})\) is either a stable node or a stable spiral point. (c) Show that \((\hat{x}, \hat{y})\) is a stable spiral point if \(\hat{y}<\frac{4 b K^{2}}{r+4 b K}\). Explain why this case will occur when the carrying capacity \(K\) of the prey is large.

In Problems, for the given linear dynamical system (taken from Exercises 10.2) (a) find the general solution and determine whether there are periodic solutions, (b) find the solution satisfying the given initial condition, and, (c) with the aid of a graphing utility, plot the solution in part (b) and indicate the direction in which the curve is traversed. $$ \begin{aligned} &x^{\prime}=x+y \\ &y^{\prime}=-2 x-y, \mathbf{X}(0)=(-2,2)(\text { Problem } 36, \text { Exercises } 10.2) \end{aligned} $$

Let \(\mathbf{X}=\mathbf{X}(t)\) be the response of the linear dynamical system $$ \begin{aligned} &x^{\prime}=\alpha x-\beta y \\ &y^{\prime}=\beta x+\alpha y \end{aligned} $$ that satisfies the initial condition \(\mathbf{X}(0)=\mathbf{X}_{0} .\) Determine conditions on the real constants \(\alpha\) and \(\beta\) that will ensure \(\lim _{t \rightarrow \infty} \mathbf{X}(t)=(0,0) . \operatorname{Can}(0,0)\) be a node or saddle 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. \(x^{\prime}=-2 x+y+10\) \(y^{\prime}=2 x-y-15 \frac{y}{y+5}\)

Use the Poincaré-Bendixson theorem to show that the plane autonomous system $$ \begin{aligned} &x^{\prime}=\epsilon x+y-x\left(x^{2}+y^{2}\right) \\ &y^{\prime}=-x+\epsilon y-y\left(x^{2}+y^{2}\right) \end{aligned} $$ has at least one periodic solution when \(\epsilon>0\). What occurs when \(\epsilon<0\) ?

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. $$ \begin{aligned} &x^{\prime}=-2 x+y+10 \\ &y^{\prime}=2 x-y-15 \frac{y}{y+5} \end{aligned} $$

The nonlinear system $$ \begin{aligned} &x^{\prime}=\alpha_{1+y} x-x \\ &y^{\prime}=-\frac{y}{1+y} x-y+\beta \end{aligned} $$ arises in a model for the growth of microorganisms in a chemostat, a simple laboratory device in which a nutrient from a supply source flows into a growth chamber. In the system, \(x\) denotes the concentration of the microorganisms in the growth chamber, \(y\) denotes the concentration of nutrients, and \(\alpha>1\) and \(\beta>0\) are constants that can be adjusted by the experimenter. Find conditions on \(\alpha\) and \(\beta\) that ensure that the system has a single critical point \((\hat{x}, \hat{y})\) in the first quadrant, and investigate the stability of this critical point.

Show that the nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}+\mathbf{F}\) has a unique critical point \(\mathbf{X}_{1}\) when \(\Delta=\operatorname{det} \mathbf{A} \neq 0\). Conclude that if \(\mathbf{X}=\mathbf{X}(t)\) is a solution to the nonhomogeneous system, \(\tau<0\) and \(\Delta>0\), then \(\lim _{t \rightarrow \infty} \mathbf{X}(t)=\mathbf{X}_{1} .[\) Hint \(:\) \(\left.\mathbf{X}(t)=\mathbf{X}_{c}(t)+\mathbf{X}_{1} \cdot\right]\)

In Problems 21-26, classify (if possible) each critical point of the given second-order differential equation as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$ \theta^{\prime \prime}=(\cos \theta-0.5) \sin \theta, \quad|\theta|<\pi $$

In Problems, for the given linear dynamical system (taken from Exercises 10.2) (a) find the general solution and determine whether there are periodic solutions, (b) find the solution satisfying the given initial condition, and, (c) with the aid of a graphing utility, plot the solution in part (b) and indicate the direction in which the curve is traversed. $$ \begin{aligned} &x^{\prime}=x-8 y \\ &y^{\prime}=x-3 y, \mathbf{X}(0)=(2,1)(\text { Problem } 40, \text { Exercises } 10.2) \end{aligned} $$

In Problems, solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). $$ \begin{aligned} &x^{\prime}=-y-x\left(x^{2}+y^{2}\right)^{2} \\ &y^{\prime}=x-y\left(x^{2}+y^{2}\right)^{2}, \mathbf{X}(0)=(4,0) \end{aligned} $$

A nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A X}+\mathbf{F}\) is given. (a) In each case determine the unique critical point \(\mathbf{X}_{1}\). (b) Use a numerical solver to determine the nature of the critical point in part (a). (c) Investigate the relationship between \(\mathbf{X}_{1}\) and the critical point \((0,0)\) of the homogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A X}\). $$ \begin{aligned} &x^{\prime}=2 x+3 y-6 \\ &y^{\prime}=-x-2 y+5 \end{aligned} $$

Solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). \(x^{\prime}=-y-x\left(x^{2}+y^{2}\right)^{2}\) $$ y^{\prime}=x-y\left(x^{2}+y^{2}\right)^{2}, \mathbf{X}(0)=(4,0) $$

In Problems, solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). $$ \begin{aligned} &x^{\prime}=y+x\left(x^{2}+y^{2}\right) \\ &y^{\prime}=-x+y\left(x^{2}+y^{2}\right), \mathbf{X}(0)=(4,0) \end{aligned} $$

A nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A X}+\mathbf{F}\) is given. (a) In each case determine the unique critical point \(\mathbf{X}_{1}\). (b) Use a numerical solver to determine the nature of the critical point in part (a). (c) Investigate the relationship between \(\mathbf{X}_{1}\) and the critical point \((0,0)\) of the homogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A X}\). $$ \begin{aligned} &x^{\prime}=-5 x+9 y+13 \\ &y^{\prime}=-x-11 y-23 \end{aligned} $$

Solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). \(x^{\prime}=y+x\left(x^{2}+y^{2}\right)\) $$ y^{\prime}=-x+y\left(x^{2}+y^{2}\right), \mathbf{X}(0)=(4,0) $$

In Problems, solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). $$ \begin{aligned} &x^{\prime}=-y+x\left(1-x^{2}-y^{2}\right) \\ &y^{\prime}=x+y\left(1-x^{2}-y^{2}\right), \mathbf{X}(0)=(1,0) ; \mathbf{X}(0)=(2,0) \end{aligned} $$

A nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}+\mathbf{F}\) is given (a) In each case determine the unique critical point \(\mathbf{X}_{1}\). (b) Use a numerical solver to determine the nature of the critical point in part (a). (c) Investigate the relationship between \(\mathbf{X}_{1}\) and the critical point \((0,0)\) of the homogeneous linear system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\) $$ \begin{aligned} &x^{\prime}=0.1 x-0.2 y+0.35 \\ &y^{\prime}=0.1 x+0.1 y-0.25 \end{aligned} $$

Classify (if possible) each critical point of the given second-order differential equation as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$x^{\prime \prime}+x=\epsilon x^{3}\( for \)\epsilon>0$$

Solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). \(x^{\prime}=-y+x\left(1-x^{2}-y^{2}\right)\) $$ y^{\prime}=x+y\left(1-x^{2}-y^{2}\right), \mathbf{X}(0)=(1,0) ; \mathbf{X}(0)=(2,0) $$

A nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}+\mathbf{F}\) is given (a) In each case determine the unique critical point \(\mathbf{X}_{1}\). (b) Use a numerical solver to determine the nature of the critical point in part (a). (c) Investigate the relationship between \(\mathbf{X}_{1}\) and the critical point \((0,0)\) of the homogeneous linear system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\) $$ \begin{aligned} &x^{\prime}=3 x-2 y-1 \\ &y^{\prime}=5 x-3 y-2 \end{aligned} $$

In Problems 21-26, classify (if possible) each critical point of the given second-order differential equation as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$ x^{n}+x-\epsilon x|x|=0 \text { for } \epsilon>0\left[\text { Hint }: \frac{d}{d x} x|x|=2|x| .\right] $$

Solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). \(x^{\prime}=y-\frac{x}{\sqrt{x^{2}+y^{2}}}\left(4-x^{2}-y^{2}\right)\) $$ \begin{aligned} &y^{\prime}=-x-\frac{x}{\sqrt{x^{2}+y^{2}}}\left(4-x^{2}-y^{2}\right) \\ &\mathbf{X}(0)=(1,0) ; \mathbf{X}(0)=(2,0) \end{aligned} $$

Show that the dynamical system $$ \begin{aligned} &x^{\prime}=-\alpha x+x y \\ &y^{\prime}=1-\beta y-x^{2} \end{aligned} $$ has a unique critical point when \(\alpha \beta>1\) and that this critical point is stable when \(\beta>0\).

(a) Show that the plane autonomous system $$ \begin{aligned} &x^{\prime}=-x+y-x^{3} \\ &y^{\prime}=-x-y+y^{2} \end{aligned} $$ has two critical points by sketching the graphs of \(-x+y-x^{3}=0\) and \(-x-y+y^{2}=0\). Classify the critical point at \((0,0)\). (b) Show that the second critical point \(\mathbf{X}_{1}=(0.88054,1.56327)\) is a saddle point.

If a plane autonomous system has a periodic solution, then there must be at least one critical point inside the curve generated by the solution. In Problems 27-30, use this fact together with a numerical solver to investigate the possibility of periodic solutions. $$ \begin{aligned} &x^{\prime}=x y \\ &y^{\prime}=-1-x^{2}-y^{2} \end{aligned} $$

If \(z=f(x, y)\) is a function with continuous first partial derivatives in a region \(R\), then a flow \(\mathbf{V}(x, y)=(P(x, y), Q(x, y))\) in \(R\) may be defined by letting \(P(x, y)=-\frac{\partial f}{\partial y}(x, y)\) and \(Q(x, y)=\frac{\partial f}{\partial x}(x, y) .\) Show that if \(\mathbf{X}(t)=(x(t), y(t))\) is a solution of the plane autonomous system $$ \begin{aligned} &x^{\prime}=P(x, y) \\ &y^{\prime}=Q(x, y) \end{aligned} $$ then \(f(x(t), y(t))=c\) for some constant \(c .\) Thus a solution curve lies on the level curves of \(f\). [Hint: Use the Chain Rule to compute \(\left.\frac{d}{d t} f(x(t), y(t)) .\right]\)

Use the phase-plane method to show that \((0,0)\) is a center of the nonlinear second-order differential equation \(x^{\prime \prime}+2 x^{3}=0\).

Use the phase-plane method to show that the solution to the nonlinear second- order differential equation \(x^{\prime \prime}+2 x-x^{2}=0\) that satisfies \(x(0)=1\) and \(x^{\prime}(0)=0\) is periodic.

(a) Find the critical points of the plane autonomous system $$ \begin{aligned} &x^{\prime}=2 x y \\ &y^{\prime}=1-x^{2}+y^{2} \end{aligned} $$ and show that linearization gives no information about the nature of these critical points. (b) Use the phase-planemethod to show that the critical points in part (a) are both centers. [Hint: Let \(u=y^{2} / x\), and show that \(\left.(x-c)^{2}+y^{2}=c^{2}-1 .\right]\)

Theoriginistheonlycritical point of thenonlinear second-order differential equation \(x^{\prime \prime}+\left(x^{\prime}\right)^{2}+x=0\). (a) Show that the phase-plane method leads to the Bernoulli differential equation dyld \(x=-y-x y^{-1}\). (b) Show that the solution satisfying \(x(0)=\frac{1}{2}\) and \(x^{\prime}(0)=0\) is not periodic.

Theorigin is the only critical point of the nonlinear second-order differential equation \(x^{\prime \prime}+\left(x^{\prime}\right)^{2}+x=0\). (a) Show that the phase-plane method leads to the Bemoulli differential equation \(d y / d x=-y-x y^{-1}\). (b) Show that the solution satisfying \(x(0)=\frac{1}{2}\) and \(x^{\prime}(0)=0\) is not periodic.

When a nonlinear capacitor is present in an \(L R C\)-series circuit, the voltage drop is no longer given by \(q / C\) but is more accurately described by \(\alpha q+\beta q^{3}\), where \(\alpha\) and \(\beta\) are constants and \(\alpha>0\). Differential equation (34) of Section \(3.8\) for the free circuit is then replaced by $$ L \frac{d^{2} q}{d t^{2}}+R \frac{d q}{d t}+\alpha q+\beta q^{3}=0 $$ Find and classify all critical points of this nonlinear differential equation. [Hint: Divide into the two cases \(\beta>0\) and \(\beta<0\).]