A new simple proof for Lum-Chua's conjecture

Abstract

The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel characterization for Poincaré half-maps in planar linear systems. This proof is very short and straightforward, because this characterization avoids the inherent flaws of the usual methods to study piecewise linear systems (the appearance of large case-by-case analysis due to the unnecessary discrimination between the different spectra of the involved matrices, to deal with transcendental equations forced by the implicit occurrence of flight time, ...). In addition, the application of the characterization allow us to prove that if a limit cycle exists, then it is hyperbolic and its stability is determined by a simple relationship between the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle and this simple expression for its stability have not been pointed out before.

References in corpus (1)

• Integral characterization for Poincaré half-maps in planar linear systems

Cited by in corpus (6)

• Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line
• Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region
• On cyclicity in discontinuous piecewise linear near-Hamiltonian differential systems with three zones having a saddle in the central one
• A succinct characterization of period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness
• Properties of Poincaré half-maps for planar linear systems and some direct applications to periodic orbits of piecewise systems
• Integral characterization for Poincaré half-maps in planar linear systems