Let X X(t) be the solution of the plane autonomous system x y y x (1 x2 )y that

Chapter 11, Problem 19

(choose chapter or problem)

Let X=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 \(\lim _{t \rightarrow \infty} \mathbf{X}(t)=(0,0)\). [Hint: Select r<1 with \(x_{0}^{2}+y_{0}^{2}<r^{2}\) and first show that the circular region R defined by \(x^{2}+y^{2} \leq r^{2}\) is an invariant region.]