Limit Cycles of Piecewise Smooth Vector Fields Formed by Rigid Systems and Linear Centers

2026/07/27 by Luiz Fernando Gonçalves, Rodrigo Donizete Euzébio, Vitória Chaves Fernandes +1

paper · doi:10.1007/s00332-026-10308-1

Abstract

Abstract We establish sharp upper bounds for the number of limit cycles in a class of piecewise smooth vector fields composed of two subsystems separated by a straight line. One subsystem is a rigid vector field of degree n , while the other is either a linear center or a rigid vector field of degree m . By analyzing the first return map associated with a Bernoulli differential equation, we prove that at most one limit cycle exists when the linear center is located at the origin. When the center is displaced, this upper bound increases to two for any n≥ 1 n ≥ 1 , and it is sharp. For systems formed by two rigid vector fields, we show that uniqueness holds when n=m n = m . In the resonant case m=nk m = n k , we obtain an upper bound of k limit cycles. We also provide a sharper bound related to an open problem on rigid smooth differential systems.

Citations

• On the Limit Cycles of a Class of Discontinuous Piecewise Differential Systems Formed by Two Rigid Centers Governed by Odd Degree Polynomials
• Homoclinic Boundary-Saddle Bifurcations in Planar Nonsmooth Vector Fields
• Some open problems in low dimensional dynamical systems
• Hopf and Homoclinic bifurcations on the sliding vector field of switching systems inR3: A case study in power electronics
• Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems
• Generic bifurcations of low codimension of planar Filippov Systems
• Limit cycles for rigid cubic systems
• One-Parameter Bifurcations in Planar Filippov Systems
• Algebraic Conditions for a Centre or a Focus in Some Simple Systems of Arbitrary Degree

Related

• Limit cycles of piecewise smooth differential systems with nilpotent center and linear saddle
• On the Limit Cycles of a Class of Discontinuous Piecewise Differential Systems Formed by Two Rigid Centers Governed by Odd Degree Polynomials
• Bounding the number of limit cycles in piecewise linear differential systems: methodology and worked examples
• Limit cycles for classes of piecewise smooth differential equations separated by the unit circle
• Limit cycles for some families of smooth and non-smooth planar systems