Unstable manifolds for the hyperchaotic Roessler system

We analyse stability of the generalized four-variable Roessler oscillating system depending on selected control parameters, by using analytic and Hurwitz-Routh methods. In contrast to the usual three-dimensional Roessler and Lorenz systems, we show that always there exists at least one unstable direction, and the number of positive local Lyapunov exponents may be different for both fixed points. We have found two new types of Hopf bifurcation, in which the dimension of the unstable manifold can be increased or reduced by two. Hence there are many possibilities for hyperchaotic unstable manifolds of various dimensions. We have also calculated various ranges of the control parameters for which different unstable manifolds can be obtained. This allows a better characterization of stability of the attractors in the hyperchaotic regime