Showing papers in "Keldysh Institute Preprints in 2019"

Normal form of a Hamiltonian system with a periodic perturbation

Abstract: A perturbed Hamiltonian system with a time-independent unperturbed part and a time-periodic perturbation is considered near a stationary solution. First, the normal form of an autonomous Hamiltonian function is recalled. Then the normal form of a periodic perturbation is described. This form can always be reduced to an autonomous Hamiltonian function, which makes it possible to compute local families of periodic solutions of the original system. First approximations of some of these families are found by computing the Newton polyhedron of the reduced normal form of the Hamiltonian function. Computer algebra problems arising in these computations are briefly discussed.

Predicting dynamical system evolution with residual neural networks

Abstract: Forecasting time series and time-dependent data is a common problem in many applications. One typical example is solving ordinary differential equation (ODE) systems $\dot{x}=F(x)$. Oftentimes the right hand side function $F(x)$ is not known explicitly and the ODE system is described by solution samples taken at some time points. Hence, ODE solvers cannot be used. In this paper, a data-driven approach to learning the evolution of dynamical systems is considered. We show how by training neural networks with ResNet-like architecture on the solution samples, models can be developed to predict the ODE system solution further in time. By evaluating the proposed approaches on three test ODE systems, we demonstrate that the neural network models are able to reproduce the main dynamics of the systems qualitatively well. Moreover, the predicted solution remains stable for much longer times than for other currently known models.