scholarly journals A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems

2018 ◽  
Vol 382 (35) ◽  
pp. 2439-2444 ◽  
Author(s):  
D.J.W. Simpson
Keyword(s):  
Dynamical Systems ◽  
Piecewise Smooth

scholarly journals Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

2016 ◽  
Vol 260 (7) ◽  
pp. 6108-6129 ◽  
Author(s):  
Márcio R.A. Gouveia ◽  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Claudio Pessoa
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Normal Forms ◽  
Piecewise Smooth

scholarly journals On Contraction of Piecewise Smooth Dynamical Systems

2011 ◽  
Vol 44 (1) ◽  
pp. 13299-13304 ◽  
Author(s):  
Giovanni Russo ◽  
Mario di Bernardo
Keyword(s):  
Dynamical Systems ◽  
Piecewise Smooth

scholarly journals On Singular "Semifocus" Type Point Bifurcations of Piecewise Smooth Dynamical System

2018 ◽  
pp. 57-70
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Piecewise Smooth ◽  
Dimensional Manifold ◽  
Smooth Vector ◽  
Weak Focus ◽  
Smooth Vector Field ◽  
Small Parameters ◽  
Line Of Discontinuity ◽  
Piecewise Smooth Vector Field

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

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