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Away from $x=\pm 1$, the sign of $\dot{x}=x^2-1$ is fixed within each of the three invariant strips, so $x(t)$ is strictly monotone. On $x=\pm 1$, the equation is $\dot{y}=\pm ky$, whose nonconstant solutions are exponential and hence not periodic. Therefore the monotone coordinate excludes periodic orbits, and there are