Bifurcation Study Research Articles

Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

Stability, instability, and bifurcation phenomena in non-autonomous differential equations

There is a vast body of literature devoted to the study ofbifurcation phenomena in autonomous systems of differentialequations. However, there is currently no well-developed theorythat treats similar questions for the non-autonomous case.Inspired in part by the theory of pullback attractors, we discussgeneralizations of various autonomous concepts of stability,instability, and invariance. Then, by means of relatively simpleexamples, we illustrate how the idea of a bifurcation as a changein the structure and stability of invariant sets remains afruitful concept in the non-autonomous case.