（2+1）维Boussinesq方程的混沌行为与控制

【作者】 李素梅；

【导师】 林国广；

【作者基本信息】 云南大学 ， 基础数学， 2010， 硕士

【摘要】 从1963年洛伦兹发表《决定论非周期流》至今,非线性系统科学得到了飞速的发展,于是更深一步揭示了非线性系统共同的性质、基本特征和运动规律.非线性科学于近二十多年可以得到迅猛发展,其中一个非常重要的原因就是:在描述很多大自然现象时,不管是线性还是非线性动力系统中都存在混沌运动.可以说混沌运动无处不在,许多物理学、力学中的问题均能化为带有周期小扰动项的具有同宿轨或者异宿轨的二阶常微分方程.对于此类系统,判断其混沌不变集在Smale马蹄变换意义下是否存在,一般可以用Melnikov方法.本文利用Melnikov方法研究了带扰动项的(2+1)维Boussinesq方程及相关问题解的分支、周期性态、同宿运动和混沌行为,并用正反馈和耦合反馈方法对混沌进行了控制.此系统混沌现象的存在可由次谐不稳定性与同宿横截点的存在得到,而该系统的周期分岔与分岔的Melnikov序列亦进一步获得.不断完善发展以Boussinesq方程为基础的浅水波模型,可使其更加准确、有效地模拟浅水域的各种自然现象.更多还原

【Abstract】 Since Lorenz published "Decision on the non-periodic flow" in 1963, non-linear system scientific access to the rapid development, which further reveals the common nature of basic characteristics, the law of sport non-linear system. The found of chaotic motion was extremely important in describing the various types of natural phenomena in power systems, thus nonlinear science can be development rapidly in the last twenty years. That is chaos everywhere, many physical, mechanical problems in the can into a small disturbance with a period with homoclinic orbit or heteroclinic orbit of the second order ordinary differential equations. For such systems can usually be handled by Melnikov method to determine whether there is in the sense of Smale horseshoes in chaotic invariant set.Bifurcation, periodic, homoclinic and chaotic behaviour for (2+1)-Dmensional Boussinesq equation and related issues are studied under perturbation using the Melnikov method, and use positive feedback and coupling feedback method of chaos to control. Chaos arising from subharmonic instability and homoclinic crossings are observed. Both period-doubling bifurcation and the Melnikov sequence of subharmonic bifurcation are found and lead chaotic behaviour. Continuously improve the development of Boussinesq quation-based model with shallow water wave to make it more accurate and effective simulation of natural phenomena in shallow waters.更多还原