Three-dimensional chaotic flows with discrete symmetries

There are a number of well-known three-dimensional flows with quadratic nonlinearities, which demonstrate a chaotic behavior. The most popular among them are Lorenz and Rössler systems. Using an exhaustive computer search, J. Sprott found 19 examples of chaotic flows with either five terms and two quadratic nonlinearities or six terms and one nonlinearity [Phys. Rev. E 50, R647 (1994)]. In contrast to this approach, we use symmetry-related considerations to construct types of chaotic flows with an arbitrary dimension. The discussion is based on our previous work devoted to nonlinear dynamics of the physical systems with discrete symmetries [see Physica D 117, 43 (1998), etc.]. Here, we present all possible chaotic flows with quadratic nonlinearities which are invariant under the action of 32 point groups of crystallographic symmetry. These systems demonstrate a typical chaotic behavior as well as general dynamical properties of nonlinear systems with discrete symmetries. In particular, we found a dynamical system with the point symmetry group D2 which seems to be more simple and more elegant than those by Lorenz and Rössler.