= Focus
{disambiguate=dynamical systems}
= Focus
{synonym}
= Focus equilibrium
{synonym}
A <focus> is a planar equilibrium whose <linearization> has nonreal complex-conjugate <eigenvalues> with nonzero <real part>. The imaginary part produces local rotation, while the <real part> determines growth or decay of its envelope. A <stable focus> has negative <real part>; a focus with positive <real part> repels nearby trajectories.
The linear system $\dot x=-\alpha x-\omega y$, $\dot y=\omega x-\alpha y$ with $\alpha>0$ and $\omega\ne0$ has radius $r(t)=r(0)e^{-\alpha t}$ and angular advance $\omega t$. It therefore spirals inward, combining two measurable scales: rotation period $2\pi/|\omega|$ and decay <time> $1/\alpha$. A damped mechanical oscillator can exhibit this geometry in its position-velocity plane. <Eigenvalues> with zero <real part> require a separate nonlinear analysis and are excluded from this hyperbolic focus definition.
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