Past exam of the flows, fluctuations and complexity and symmetry and relativity paper of Part B of the physics course of the University of Oxford / 2011 / 3 / 2 / Solution

At an equilibrium point of the Rössler system, $y=-z$, $x=az$, and $a{z}^2-cz+a=0$. HenceThere are two distinct real equilibrium points only if $c>2a$; they merge if $c=2a$ and there are none if $c<2a$. Positivity of $a,c$ alone does not guarantee two. Under the later limits, the nearer one is $(x_{-},y_{-},z_{-})\simeq (a^2/c,-a/c,a/c)$, while $(x_{+},y_{+},z_{+})\simeq (c,-c/a,c/a)$.