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The planar system $\dot{x}=y+(1-x^2-y^2)x$, $\dot{y}=-x+(1-x^2-y^2)y$ has polar coordinates $\dot{r}=r(1-r^2)$, $\dot{\theta }=-1$. Every nonzero orbit approaches the clockwise unit-circle limit cycle, of period $2\pi$. For $r(t_0)=r_0>0$,Interior orbits approach the origin backward in time, but exterior orbits have backward finite-time blow-up of an ordinary differential equation at $t=t_0+\frac{1}{2}\log (1-r_0^{-2})$. Thus forward convergence to a limit cycle does not imply existence for all negative times.

The Van der Pol equation is the nonlinear oscillatorin dimensionless variables. Its damping is negative at small amplitude and positive when $|x|>1$. For the quadratic oscillator energy $E=({\dot{x}}^2+x^2)/2$,Thus small oscillations gain energy while sufficiently large excursions dissipate it. This balances growth and saturation, producing the familiar attracting limit cycle of this system.

When $\mu =0$, the equation reduces to the simple harmonic oscillator. For small positive $\mu$, averaging $x\approx A\cos t$ gives $⟨\dot{E}⟩\approx \mu (A^2/2-A^4/8)$, predicting a stable amplitude near two. This amplitude is an approximation in the weakly nonlinear regime, not an exact sinusoidal solution. The equation is a standard model for self-sustained oscillation and, for large $\mu$, relaxation oscillations.