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 / 3 / Solution

The Jacobian matrix of the Rössler system isAt the origin the vertical homogeneous perturbation obeys $\delta \dot{z}=-c\delta z$, so it decays as $e^{-ct}$. Nearby, $\dot{z}\simeq a-cz$ also relaxes the trajectory toward $z\simeq a/c$, close to the $x$–$y$ plane. The small fixed-point value of $z$ weakly couples planar errors back into $z$; this is neglected in the requested approximation.

The planar block has trace $a$ and determinant $1$, withFor $0<a\ll 1$, this is an unstable spiral. Writing $X=x-x_{-}$ and $Y=y-y_{-}$, its linearized trajectories have an envelope proportional to $e^{at/2}$ and angular frequency approximately $1$. They rotate counterclockwise: at $X>0,Y=0$, $\dot{Y}=X>0$. Fast vertical contraction occurs on scale $1/c$, much shorter than the planar growth scale $2/a$.

Confinement weakens as $x$ approaches $c$ from below; the approximate vertical relaxation time is $1/(c-x)$, and the quasi-steady value $a/(c-x)$ ceases to be small. For $x>c$, the coefficient of $z$ becomes positive, so vertical departures are amplified. The plane $z=0$ is never exactly invariant because $\dot{z}=a$ there.