APPLICATION OF DIFFERENTIAL GEOMETRY METHODS TO SLOW-FAST DYNAMIC SYSTEMS

The slow-fast autonomous two-dimensional dynamic systems are considered. Slow-fast dynamic systems describe various physical, mechanical, and other phenomena in which the gradual evolutionary accumulation of small changes over time leads to a sudden transition of the system to a new dynamic regime. The study of slow-fast dynamic systems is connected with finding the analytical equation of a slow invariant manifold, that is, a manifold where the phase trajectories of the system change slowly. This is due to the fact that the trajectory of the slow manifold divides the phase space of the dynamic system into slow and fast regions, that is, regions where the dynamics of the system slows down and accelerates, respectively. The work considers geometric approaches to finding the slow manifold of such systems, which allow writing the analytical equation of the slow manifold rather simply. The flow curve method is based on the concepts and methods of differential geometry and mechanics. Using the concept of the curvature of a curve allows one to determine the analytical equation of a slow manifold regardless of the "slow eigenvalues" of the system. The flow curvature method is applied to a slow-fast dynamic system. The analytical equation of the slow manifold of the considered system and the slow and fast regions of the phase space are determined. It was established that the equations of the slow invariant manifold of the dynamic system, obtained by the flow curvature method and the geometric singular perturbation method, are completely identical in the first order of approximation with a small parameter .