Chechin G M, Ryabov D S
Department of Physics, Rostov State University, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Mar;69(3 Pt 2):036202. doi: 10.1103/PhysRevE.69.036202. Epub 2004 Mar 11.
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.
存在许多具有二次非线性的著名三维流，它们表现出混沌行为。其中最著名的是洛伦兹系统和罗斯勒系统。J. 斯普罗特通过详尽的计算机搜索，发现了19个具有五项和两个二次非线性或六项和一个非线性的混沌流示例[《物理评论E》50, R647 (1994)]。与这种方法不同，我们利用与对称性相关的考虑来构建任意维度的混沌流类型。讨论基于我们之前致力于具有离散对称性的物理系统的非线性动力学的工作[见《物理D》117, 43 (1998)等]。在此，我们给出了所有具有二次非线性且在32个晶体学点群作用下不变的可能混沌流。这些系统展示了典型的混沌行为以及具有离散对称性的非线性系统的一般动力学性质。特别地，我们发现了一个具有点对称群D2的动力学系统，它似乎比洛伦兹系统和罗斯勒系统更简单、更优美。
