91Ó°ÊÓ

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$\begin{array}{l}{J=\left[ \begin{array}{cc}{2 x_{1}} & {-\sin x_{2}} \\\ {\cos x_{1}} & {0}\end{array}\right]} \\ {\text { (i) } \hat{x}_{1}=1, \hat{x}_{2}=-\pi} \\ {\text { (ii) } \hat{x}_{1}=1, \hat{x}_{2}=\pi}\end{array}$$

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{rr}{1} & {2} \\ {-2} & {1}\end{array}\right]\)

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{-\frac{3}{2}} & {\frac{1}{2}} \\ {\frac{1}{2}} & {-\frac{3}{2}}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {2}\end{array}\right]\)

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{\frac{1}{2}} & {-\frac{3}{2}} \\ {-\frac{3}{2}} & {\frac{1}{2}}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{1} \\ {2}\end{array}\right]\)

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{rr}{1} & {2} \\ {2} & {-1}\end{array}\right]\)

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$J=\left[ \begin{array}{cc}{-\frac{1}{1+x_{2}}} & {\frac{x_{1}}{\left(1+x_{2}\right)^{2}}} \\\ {-1+\frac{x_{2}}{\left(1+x_{1}\right)^{2}}} & {-\frac{1}{1+x_{1}}}\end{array}\right]$$ $$\begin{array}{l}{\text { (i) } \hat{x}_{1}=-2, \hat{x}_{2}=-2} \\ {\text { (ii) } \hat{x}_{1}=\frac{1}{2}, \hat{x}_{2}=-\frac{3}{4}}\end{array}$$

Given the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) , construct the phase plane, including the nullclines. Does the equilibrium look like a saddle, a node, or a spiral? \(A=\left[ \begin{array}{ll}{-1} & {2} \\ {-3} & {0}\end{array}\right]\)

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{1} & {0} \\ {4} & {-1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{3} \\ {2}\end{array}\right]\)

Find all equilibria and determine their local stability properties. $$x^{\prime}=x(3-x-y), \quad y^{\prime}=y(2-x-y)$$

Find all equilibria and determine their local stability properties. $$p^{\prime}=p(1-p-q), \quad q^{\prime}=q(2-3 p-q)$$