Hyperchaos in the generalized R"ossler system

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.