Bistability and hidden attractors in the paradigmatic Rössler'76 system

In this paper, the dynamics of the paradigmatic Rössler system is investigated in a yet unexplored region of its three-dimensional parameter space. We prove a necessary condition in this space for which the Rössler system can be chaotic. By using standard numerical tools, like bifurcation diagrams, Poincaré sections, and first-return maps, we highlight both asymptotically stable limit cycles and chaotic attractors. Lyapunov exponents are used to verify the chaotic behavior while random numerical procedures and various plane cross sections of the basins of attraction of the coexisting attractors prove that both limit cycles and chaotic attractors are hidden. We thus obtain previously unknown examples of bistability in the Rössler system, where a point attractor coexists with either a hidden limit cycle attractor or a hidden chaotic attractor.