Rigorous numerics for piecewise-smooth systems: A functional analytic approach based on Chebyshev series

• Marcio Gameiro ^[1] ; Jean-Philippe Lessard ^[2] ; Yann Ricaud ^[2]
  1. [1] Universidade de São Paulo

     Universidade de São Paulo

     Brasil

     
  2. [2] Université Laval, Québec, Canada
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 292, Nº 1 (15 January 2016), 2016, págs. 654-673
• Idioma: inglés
• DOI: 10.1016/j.cam.2015.05.016
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• Resumen

  • In this paper, a rigorous computational method to compute solutions of piecewise-smooth systems using a functional analytic approach based on Chebyshev series is introduced. A general theory, based on the radii polynomial approach, is proposed to compute crossing periodic orbits for continuous and discontinuous (Filippov) piecewise-smooth systems. Explicit analytic estimates to carry the computer-assisted proofs are presented. The method is applied to prove existence of crossing periodic orbits in a model nonlinear Filippov system and in the Chua’s circuit system. A general formulation to compute rigorously crossing connecting orbits for piecewise-smooth systems is also introduced.