Chapter 11

The nonlinear second-order differential equation $$ m x^{\prime \prime}+k x+k_{1} x^{3}=0 $$ for \(k>0\), represents a general model for the free, undamped oscillations of a mass \(m\) attached to a spring. If \(k_{1}>0\), the spring is called hard (see Example 1 in Section 3.11). Determine the nature of the solutions to \(x^{\prime \prime}+x+x^{3}=0\) in a neighborhood of \((0,0)\).

(a) Show that \((0,0)\) is an isolated critical point of the plane autonomous system $$ \begin{aligned} &x^{\prime}=x^{4}-2 x y^{3} \\ &y^{\prime}=2 x^{3} y-y^{4} \end{aligned} $$ but that linearization gives no useful information about the nature of this critical point. (b) Use the phase-plane method to show that \(x^{3}+y^{3}=3 c x y\). This classic curve is called a folium of Descartes. Parametric equations for a folium are $$ x=\frac{3 c t}{1+t^{3}}, y=\frac{3 c t^{2}}{1+t^{3}} $$ [Hint: The differential equation in \(x\) and \(y\) is homogeneous.] (c) Use a graphing utility or a numerical solver to obtain solution curves. Based on your phase portrait, would you classify the critical point as stable or unstable? Would you classify the critical point as a node, saddle point, center, or spiral point? Explain.