Tame topology and non-integrability of dynamical systems (Q6621128)

The paper discusses issues of nonintegrability of dynamical systems in terms of differential Galois theory. The essence of the algebraic approach is that nonintegrability can be proved by establishing the noncommutativity of the identity component of the Galois group of the variational equation associated with it. This task is sometimes easy to fulfill. The paper contains an introduction to known results in this context and a brief overview of recent results. At the same time, it examines an example of the following gradient dynamical system:\N\[\N\dot{\xi}=\nabla F,\qquad F(\xi,\eta)=\dfrac{1}{3}\xi^{3}+\dfrac{1}{2}\xi^{2}+(\xi+\eta)^{2}\eta^{2}+\dfrac{1}{4}\eta^{4}, \tag{1}\N\]\Nwith associated foliation\N\[\N\dfrac{d\eta}{d\xi}=\dfrac{\partial F/\partial\eta}{\partial F/\partial\xi}=\dfrac{2(\xi+\eta)\eta^{2}+2(\xi+\eta)^{2}\eta+\eta^{3}}{\xi^{2}+\xi+2(\xi+\eta)\eta^{2}}.\tag{2}\N\]\N\NThe authors show that the linearized second variational equation of (2) along a particular solution \((\eta=0)\) has noncommutative identity component of its Galois group. Hence the dynamical system (1) is not meromorphically integrable. Thus they ``have provided an example of a nonintegrable system that has tame topology, hence it seems unreasonable to put ``chaos'' and ``non-integrability'' on equal footing''.\N\NFor the entire collection see [Zbl 1547.32001].