Integrable deformations of a class of Rikitake dynamical systems are constructed by deforming their underlying Lie–Poisson Hamiltonian structures, which are considered linearizations of Poisson–Lie structures on certain (dual) Lie groups. By taking into account that there exists a one-to one correspondence between Poisson–Lie groups and Lie bialgebra structures, a number of deformed Poisson coalgebras can be obtained, which allow the construction of integrable deformations of coupled Rikitake systems. Moreover, the integrals of the motion for these coupled systems can be explicitly obtained by means of the deformed coproduct map. The same procedure can be also applied when the initial system is bi-Hamiltonian with respect to two different Lie–Poisson algebras. In this case, to preserve a bi-Hamiltonian structure under deformation, a common Lie bialgebra structure for the two Lie–Poisson structures has to be found. Coupled dynamical systems arising from this bi-Hamiltonian deformation scheme are also presented, and the use of collective ‘cluster variables’, turns out to be enlightening in order to analyse their dynamical behaviour. As a general feature, the approach here presented provides a novel connection between Lie bialgebras and integrable dynamical systems.

中文翻译：

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Rikitake 系统、李双代数和双哈密顿结构的可积变形


一类 Rikitake 动力系统的可积变形是通过变形其基础李-泊松哈密顿结构来构造的，这些结构被认为是某些（对偶）李群上泊松-李结构的线性化。考虑到泊松-李群和李双代数结构之间存在一一对应关系，可以获得许多变形泊松余代数，从而可以构造耦合 Rikitake 系统的可积变形。此外，这些耦合系统的运动积分可以通过变形余积图明确获得。当初始系统是关于两个不同的李-泊松代数的双哈密顿系统时，也可以应用相同的过程。在这种情况下，为了在变形下保留双哈密顿结构，必须找到两个李-泊松结构的共同李双代数结构。还提出了由这种双哈密尔顿变形方案产生的耦合动力系统，并且集体“簇变量”的使用对于分析它们的动力行为具有启发性。作为一般特征，这里提出的方法提供了李双代数和可积动力系统之间的新颖联系。