Two-parametric unfoldings for planar invisible double-tangency singularities.

Area of Science:

• Dynamical Systems Theory
• Bifurcation Theory
• Piecewise Linear Systems

Background:

• Planar discontinuous piecewise linear systems with specific boundary conditions are analyzed.
• A seven-parametric canonical form for these systems is established.
• The study focuses on systems with one tangency point in each region.

Purpose of the Study:

• To investigate bifurcations in a family of planar discontinuous piecewise linear systems.
• To provide a two-parametric unfolding for invisible fold-fold and focus-fold singularities.
• To introduce and characterize the pseudo-Bautin (pB) bifurcation.

Main Methods:

• Utilizing a seven-parametric canonical form for the system family.
• Selecting distance between tangency points and the first Lyapunov coefficient as bifurcation parameters.
• Analyzing bifurcation diagrams to identify curves of pseudo-Hopf and saddle-node bifurcations.

Main Results:

• A two-parametric unfolding for invisible fold-fold and focus-fold singularities is presented.
• The study identifies curves of pseudo-Hopf and saddle-node bifurcation points for crossing limit cycles.
• A novel phenomenon, termed pseudo-Bautin (pB) bifurcation, is described.

Conclusions:

• The pseudo-Bautin (pB) bifurcation exhibits dynamical behavior analogous to the Bautin bifurcation in smooth systems.
• The findings contribute to the understanding of bifurcations in discontinuous dynamical systems.
• This research offers a framework for analyzing complex dynamics in piecewise linear models.