Rössler system ($\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+z(x-c)$)

The Rössler system is the three-variable autonomous differential equationIts only nonlinear term is the quadratic product $xz$. The planar rotation in the first two variables interacts with the third variable through that term, allowing behaviour much richer than a linear harmonic oscillator. Otto Rössler introduced the model as a simple setting for continuous-time chaos.^[ref]

Its equilibrium points can be calculated without integrating an orbit. The first two equations give $y=-z$ and $x=az$; the third then reduces to $a{z}^2-cz+b=0$. When $a\ne 0$, the real equilibria therefore correspond toprovided $c^2-4ab\ge 0$. A repeated quadratic root gives one equilibrium; a negative discriminant gives none. The case $a=0$ is obtained directly from the linear equation $-cz+b=0$, without dividing by $a$.

The Jacobian matrix at a general state iswith divergence $a+x-c$. These formulas test local stability and local volume contraction. Their signs alone do not establish the global folding and return required for a Rössler attractor.

A Rössler attractor is a chaotic attracting invariant set occurring in a Rössler system at suitable parameter values. The model was designed to combine continuous-time motion with a folded Poincaré map.^[ref]

The distinction between local and global information is essential. An unstable spiral can stretch nearby trajectories apart during one part of an excursion. To sustain bounded irregular motion, the trajectories must also return through the global phase portrait. A local Jacobian matrix describes the stretching near a point; it does not determine where trajectories go after leaving that neighborhood.

Numerically, a useful diagnostic is to record consecutive intersections with a transverse section and plot the resulting Poincaré map. A folded shape suggests how stretching and return interact, but a finite point cloud does not itself prove an invariant set, an attracting neighborhood or a rigorous definition of chaos. The parameter values, section and discarded transient should accompany such a computation.