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Limit cycles created by piecewise linear centers.

Jaume Llibre1, Xiang Zhang2

  • 1Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain.

Chaos (Woodbury, N.Y.)
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Summary
This summary is machine-generated.

This study explores limit cycles in three-piece piecewise linear differential systems. Researchers found these systems can exhibit 1, 2, or up to 3 nested limit cycles, with specific intersection patterns.

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Area of Science:

  • Differential Equations
  • Dynamical Systems Theory
  • Mathematical Physics

Background:

  • Piecewise linear differential systems are increasingly studied due to their relevance in modeling physical phenomena.
  • Limit cycles are crucial in analyzing the behavior of these systems.
  • Previous research predominantly focused on systems with two linear pieces, leaving systems with more pieces less explored.

Purpose of the Study:

  • To investigate the limit cycle behavior of discontinuous piecewise linear differential systems composed of three linear centers.
  • To determine the maximum number of limit cycles possible in such three-piece systems.
  • To analyze the intersection properties of these limit cycles with the system's separating set Σ.

Main Methods:

  • Analysis of discontinuous piecewise linear differential systems in the R² plane.
  • Focus on systems with three arbitrary linear centers.
  • Examination of systems separated by the set Σ = {(x,y)∈R²: y=0 or (x=0 and y≥0)}.

Main Results:

  • Demonstrated that these three-piece systems can possess 1, 2, or a maximum of 3 limit cycles.
  • Established that limit cycles are nested.
  • Showed that limit cycles intersect Σ at either three or four points; systems with three intersection points can yield 3 limit cycles, while those with four yield at most 1.

Conclusions:

  • The study establishes the maximum number of limit cycles for a specific class of three-piece piecewise linear differential systems.
  • The geometric configuration of limit cycle intersections with the separating set is characterized.
  • Findings contribute to a deeper understanding of the complex dynamics in piecewise linear systems beyond the two-piece case.