Question

\begin{equation} \frac{dx}{dt} = y =: F(x, y) \end{equation} \begin{equation} \frac{dy}{dt} = x - x^3 - \Phi y =: G(x, y) \end{equation} Let $(x_e, y_e)$ denote an equilibrium point and \begin{equation} a := \frac{\partial F}{\partial x}(x_e, y_e) \qquad b := \frac{\partial F}{\partial y}(x_e, y_e) \end{equation} \begin{equation} c := \frac{\partial G}{\partial x}(x_e, y_e) \qquad d := \frac{\partial G}{\partial y}(x_e, y_e) \end{equation} \begin{equation} p := -(a + d) \end{equation} \begin{equation} q := ad - bc \end{equation} Fill in the table below, assuming that $0 < \Phi < 4.$ \begin{tabular}{|c|c|c|c|c|c|c|} \hline Equilibrium & a & b & c & d & p & q \\ Point, & & & & & & \\ $(x_e, y_e)$ & & & & & & \\ \hline & & & & & & \\ \hline & & & & & & \\ \hline & & & & & & \\ \hline \end{tabular} Stability properties, like unstable saddle, stable spiral, center etc.

          \begin{equation} \frac{dx}{dt} = y =: F(x, y) \end{equation} \begin{equation} \frac{dy}{dt} = x - x^3 - \Phi y =: G(x, y) \end{equation} Let $(x_e, y_e)$ denote an equilibrium point and \begin{equation} a := \frac{\partial F}{\partial x}(x_e, y_e) \qquad b := \frac{\partial F}{\partial y}(x_e, y_e) \end{equation} \begin{equation} c := \frac{\partial G}{\partial x}(x_e, y_e) \qquad d := \frac{\partial G}{\partial y}(x_e, y_e) \end{equation} \begin{equation} p := -(a + d) \end{equation} \begin{equation} q := ad - bc \end{equation} Fill in the table below, assuming that $0 < \Phi < 4.$ \begin{tabular}{|c|c|c|c|c|c|c|} \hline Equilibrium & a & b & c & d & p & q \\ Point, & & & & & & \\ $(x_e, y_e)$ & & & & & & \\ \hline & & & & & & \\ \hline & & & & & & \\ \hline & & & & & & \\ \hline \end{tabular} Stability properties, like unstable saddle, stable spiral, center etc.

(dx)/(dt) = y =: F(x, y)

(dy)/(dt) = x - x^3 - Φ y =: G(x, y)
Let (xe, ye) denote an equilibrium point and
a := (∂ F)/(∂ x)(xe, ye) b := (∂ F)/(∂ y)(xe, ye)

c := (∂ G)/(∂ x)(xe, ye) d := (∂ G)/(∂ y)(xe, ye)

p := -(a + d)

q := ad - bc
Fill in the table below, assuming that 0 < Φ < 4. Equilibrium a b c d p q
Point,
(xe, ye)

Stability properties, like unstable saddle, stable spiral, center etc.

Added by Brenda B.

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