Shape of attractors for three-dimensional dissipative dynamical systems

We introduce a method to bound attractors of dissipative dynamical systems in phase and parameter spaces. The method is based on the determination of families of transversal surfaces (surfaces crossed by the flow in only one direction). This technique yields very restrictive geometric bounds in phase space for the attractors. It also gives ranges of parameters of the system for which no chaotic behavior is possible. We illustrate our method on different three-dimensional dissipative systems.