论文标题
退化平衡附近的动力系统的不整合性
Nonintegrability of dynamical systems near degenerate equilibria
论文作者
论文摘要
我们证明，当jacobian矩阵具有零和纯粹想象的特征值或两个不遵循的Imigential Imimenviliary EigeNervalib的纯净的eigennate Pairs时，在额外的弱条件下，在Bogoyavlenskij的情况下，在Bogoyavlenskij感官中，在Bogoyavlenskij感官上，差异方程式的一般三维系统百分比在bogoyavlenskij的近位平衡中是真实的不可分割的。为此，我们将它们的整合性降低到相应的繁殖性繁殖形式的正常形式和简单平面系统的整合性，并使用一种新颖的方法来证明平面系统的分析性不整合性。我们的结果还意味着，在弱条件下，一般的三维和四维系统分别表现出折叠 - hopf和双HOPF codimension-two分叉，这是无分析的。为了证明这些结果，我们为Rossler系统和耦合的Van der Pol振荡器提供了两个示例。
We prove that general three- or four-dimensional systems %of differential equations are real-analytically nonintegrable near degenerate equilibria in the Bogoyavlenskij sense under additional weak conditions when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of the corresponding Poincare-Dulac normal forms and further to that of simple planar systems, and use a novel approach for proving the analytic nonintegrability of planar systems. Our result also implies that general three- and four-dimensional systems exhibiting fold-Hopf and double-Hopf codimension-two bifurcations, respectively, are real-analytically nonintegrable under the weak conditions. To demonstrate these results, we give two examples for the Rossler system and coupled van der Pol oscillators.
