Limit Sets and Global Bifurcation Structure in Planar Control Models with Large Hysteresis
The present paper addresses a problem that may be of considerable interest to a broad audience since the systems considered here operate according to a switching protocol involving two distinct dynamical regimes. Starting from an initial condition, the evolution follows a first vector field until a selected state variable $y$ reaches a lower threshold $C_1$. At this moment, the dynamics switches to a second vector field. The second regime remains active until the same variable attains an upper threshold $C_2>C_1$, when the first vector field is restored. This alternating procedure is then repeated indefinitely giving rise to a piecewise smooth vector field. A complete characterization of the $\omega$-limit sets is obtained for every admissible combination of parameters and all initial condition. The analysis is carried out by combining explicit solutions of the vector fields with geometric arguments and the first return map. Beyond the classification of limit sets, the paper describes the global bifurcation structure of the family. As the parameters vary, the system undergoes qualitative transitions between distinct asymptotic regimes, including the birth and disappearance of periodic orbits, changes in their stability, the occurrence of continuum of periodic trajectories in degenerate situations, and the replacement of bounded dynamics by monotone zig-zag motions or unbounded trajectories. The corresponding bifurcation diagrams provide a complete qualitative description of the asymptotic dynamics of the model.