ul. Chernyshevskogo 112, Ufa, 450008 Russia
Institute of mathematics USC RAS
Kalyakin L. A.
Asymptotics of Dynamical Saddle-node Bifurcations
2022, Vol. 18, no. 1, pp. 119-135
Dynamical bifurcations occur in one-parameter families of dynamical systems, when the parameter is slow time. In this paper we consider a system of two nonlinear differential equations with slowly varying right-hand sides. We study the dynamical saddle-node bifurcations that occur at a critical instant. In a neighborhood of this instant the solution has a narrow transition layer, which looks like a smooth jump from one equilibrium to another. The main result is asymptotics for a solution with respect to the small parameter in the transition layer. The asymptotics is constructed by the matching method with three time scales. The matching of the asymptotics allows us to find the delay of the loss of stability near the critical instant.
Kalyakin L. A.
Analysis of a Mathematical Model for Nuclear Spins in an Antiferromagnet
2018, Vol. 14, no. 2, pp. 217-234
This paper is concerned with a system of three nonlinear differential equations, which is a mathematical model for a system of nuclear spins in an antiferromagnet. The model has arisen in recent physical studies and differs from the well-known and well-understood Landau – Lifshitz and Bloch models in the manner of incorporating dissipation effects. It is established that the system under consideration is related to the Landau – Lifshitz system by the passage to the limit only on one invariant sphere. The initial equations contain three dimensionless parameters. Equilibrium points and their stability are examined depending on these parameters. The position of the bifurcation surface is found in the parameter space. It is proved that the corresponding equilibrium is of saddle-node type. Exact statements are illustrated by results of numerical experiments.