ON LOCAL BIFURCATIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH A PIECE-SMOOTH RIGHT-HAND SIDE // BSU Bulletin. Mathematics, Informatics. - 2023. №4. . - С. 3-13.

ON LOCAL BIFURCATIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH A PIECE-SMOOTH RIGHT-HAND SIDE

DOI: 10.18101/2304-5728-2023-4-3-13 UDK: 517.925

The paper considers dynamical systems on the plane, given by second- order autonomous differential equations, with right-hand sides depending on one and two parameters, discontinuous on the “zero velocity line” y = 0 and smooth outside it. At zero values of the parameters, it is assumed that the origin of coordinates is a stable equilibrium "linked" from the foci of smooth dynamical systems specified in the half- planes y < 0 and y > 0. Bifurcation diagrams are described for generic one-parameter and two-parameter families of such systems. In particular, it is shown that in a one- parameter family, a single (stable) limit cycle can be born from the equilibrium, and in a two-parameter family, from one to two limit cycles can be born from an the equilibrium.

second-order differential equation, phase plane, piecewise-smooth dynamical system, equilibrium, bifurcation diagram, limit cycle.