一个比Rssler系统代数结构更为简单的三阶混沌系统及其混沌抑制

A Third-order Chaotic System Whose Algebraic Structure Simpler than the Rssler System and Its Chaos Suppression

混沌的发现与研究大都集中在非线性项不只一项,或者非线性项仅有一项时其为平方项或立方项,或者系统状态方程代数项较多的系统;而类似于Rssler系统的简单混沌系统的发现与研究较少。提出了一个结构比Rssler混沌系统代数结构更为简单的三阶混沌系统,该系统方程为y'″+cy″+by'+ay+yy'=0。通过Lyapunov指数谱图证实了该系统取特定值时为混沌状态;系统分岔图展示了该系统随所定参数变化走向混沌的道路;状态方程给出了系统有唯一的平衡点;这表明所论系统是一种不同于Genesio-Tesi系统与Coullet系统的新混沌系统。给出系统在平衡点稳定应满足的条件。通过散度计算可知系统轨迹是以与系数c有关的指数形式收敛的。将所论系统改造为参数未知Genesio-Tesi系统后,借助一种自适应控制律可以对所论系统的混沌进行抑制。

The discovery and research of the chaos are mostly concentrated in the following systems： some of these systems contain more than one nonlinear terms, or the nonlinear term is quadratic term or cubic term when the system contains only one nonlinear term, or the state equations of some systems contain too many algebraic terms, however, the chaos discovered and researched in the simple systems who are similar to the Rossler system is rare. This paper proposes a third-order nonlinear chaotic system whose algebraic structure is simpler than the Rossler system, and the system is y＂＂＂＋cy＂＋by＂＋ ay＋ yy＂ =0. There is chaotic phenomenon in this system by Lyapunov exponent spectra. Bifurcation diagrams of this system portray the path of chaos with the change of parameters. The state equation of this system gives a unique equilibrium point. From three aspects discussed it can be seen that the system discussed is a new chaotic system which is different from the Genesio-Tesi system and Coullet system. The condition for stability of the system in the equilibrium point is also given. Calculation of divergence shows that the system is convergent in a form of exponent related to coefficient c. Suppression of the chaos can be worked out with the help of an adaptive law after the system transformed to the unknown parameter Genesio-Tesi system.