= Solution
A <critical point of a dynamical system> is a point where both components of the <vector field> vanish. For a continuously differentiable <vector field>, strictly negative real parts of both <eigenvalues> of its <Jacobian matrix> imply <asymptotic stability> of the equilibrium. If an isolated critical point is a <center of a planar dynamical system> and the <eigenvalues> are distinct, they must be a nonzero purely imaginary conjugate pair. Purely imaginary <eigenvalues> alone do not in general prove nonlinear stability; the <nonlinear ordinary differential equation> system must still be examined.
Here the critical points are $(0,0)$ and $(\pm1,0)$, with Jacobian $\begin{pmatrix}0&1\\1-3x^2&0\end{pmatrix}$. At zero its <eigenvalues> are $1,-1$, giving an unstable <saddle equilibrium>. The unstable direction is $(1,1)$ and the stable direction is $(1,-1)$. Thus trajectories leaving the origin are tangent to $y=x$, and those returning are tangent to $y=-x$, with both opposite rays included.
An exact <first integral> is
$$
H(x,y)=\frac{y^2}{2}-\frac{x^2}{2}+\frac{x^4}{4},\qquad
\frac{dH}{dt}=(-x+x^3)y+y(x-x^3)=0.
$$
The two homoclinic loops belong to $H=0$:
$$
y^2=x^2\left(1-\frac{x^2}{2}\right),\qquad |x|\le\sqrt2.
$$
They confirm the incoming and outgoing tangent directions. For example the right loop is $x=\sqrt2\operatorname{sech}(t-t_0)$, $y=-\sqrt2\operatorname{sech}(t-t_0)\tanh(t-t_0)$; changing both signs gives the left loop.
\Image[/past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2004-double-well-phase-portrait.png]
{title=Periodic energy curves and the two homoclinic loops of the nonlinear planar system}
At $(1,0)$ put $x=1+X$, $y=Y$. The exact equations are $X'=Y$, $Y'=-2X-3X^2-X^3$, so the <linearization of a dynamical system> gives $X''+2X=0$. Its nonconstant solutions have
$$
\boxed{T=\frac{2\pi}{\sqrt2}=\sqrt2\pi,\qquad2X^2+Y^2=\text{constant}.}
$$
The same linear classification holds at $(-1,0)$. For the <nonlinear ordinary differential equation> system, $H$ has strict local minima $-1/4$ at both points. Levels $-1/4<H<0$ are two families of smooth closed ovals, proving that both are stable centers, rather than asymptotically stable equilibria. These give actual <periodic orbits> of the <nonlinear ordinary differential equation> system, with small-amplitude periods tending to $\sqrt2\pi$.
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