  Introduced as a model for hyperchaos, the generalized R"ossler system of
dimension N is obtained by linearly coupling N-3 additional degrees of freedom
to the original R"ossler equation. Under variation of a single control
parameter, it is able to exhibit the chaotic hierarchy ranging from fixed
points via limit cycles and tori to chaotic and, finally, hyperchaotic
attractors. By the help of a mode transformation, we reveal a structural
symmetry of the generalized R"ossler system. The latter will allow us to
interpret number, shape, and location in phase space of the observed coexisting
attractors within a common scheme for arbitrary odd dimension N. The appearance
of hyperchaos is explained in terms of interacting coexisting attractors. In a
second part, we investigate the Lyapunov spectra and related properties of the
generalized R"ossler system as a function of the dimension N. We find scaling
properties which are not similar to those found in homogeneous, spatially
extended systems, indicating that the high-dimensional chaotic dynamics of the
generalized R"ossler system fundamentally differs from spatio-temporal chaos.
If the time scale is chosen properly, though, a universal scaling function of
the Lyapunov exponents is found, which is related to the real part of the
eigenvalues of an unstable fixed point.
