Finite-dimensional dynamics of systems of evolutionary differential equations with many spatial variables

The main ideas of the theory of finite-dimensional dynamics were formulated in the 2000s in the works of B.S. Kruglikov, V.V. Lychagin and O.V. Lychagina. These papers also found finite-dimensional dynamics of the Kolmogorov-Petrovsky-Piskunov and Korteweg-de Vries equations. This theory is a natural development of the theory of dynamical systems. Finite-dimensional dynamics make it possible to find families of solutions depending on a finite number of parameters among all solutions of evolutionary differential equations. Namely, finite-dimensional submanifolds are constructed in the space of smooth functions that are invariant under the flow given by the evolution equation. This removes the question of the existence of solutions, since such submanifolds consist of solutions to ordinary differential equations, and, moreover, gives a constructive method for finding them. Note that finite-dimensional dynamics can exist for equations that do not have symmetries. The talk will present the results obtained by us for systems of evolutionary equations, including those with many spatial variables.