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By Otto Rössler. Physics Letters A 57(5), pages 397–398.

Author-uploaded record: www.researchgate.net/publication/223112399_An_Equation_for_Continuous_Chaos

Otto Rössler introduced a simple continuous-time model of deterministic chaos in 1976. The Rössler system is commonly writtenOnly the product $xz$ is nonlinear. Nevertheless, suitable parameters produce the folding and stretching associated with the Rössler attractor.^[ref]

The example makes an important modelling point: a small number of equations and nonlinear terms can generate complicated motion. The existence of a chaotic regime does not imply that every parameter choice or every initial condition in the model is chaotic.

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.