American Institute of Mathematical Sciences

Advanced
July  2000, 6(3): 609-624. doi: 10.3934/dcds.2000.6.609

Center-focus and isochronous center problems for discontinuous differential equations

B. Coll 1, , A. Gasull 2, and  R. Prohens 1,

1. 

Dept. de Matemàtiques i Informàtica, Universitat de les Illes Balears, Facultat de ciències, 07071, Palma de Mallorca, Spain, Spain
2. 

Dept. de Matemàtiques, Universitat Autónoma de Barcelona, Edifici C, 08193 Bel-laterra, Barcelona, Spain

Received  October 1999 Revised  April 2000 Published  April 2000

The study of the center focus problem and the isochronicity problem for differential equations with a line of discontinuities is usually done by computing the whole return map as the composition of the two maps associated to the two smooth differential equations. This leads to large formulas which usually are treated with algebraic manipulators. In this paper we approach to this problem from a more theoretical point of view. The results that we obtain relate the order of degeneracy of the critical point of the discontinuous differential equations with the order of degeneracy of the two smooth component differential equations. Finally we apply them to some families of examples.
Keywords: discontinuous differential equation, isochronicity., Center point.
Mathematics Subject Classification: 34C25, 58F21, 58F2.
Citation: B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609
[1]

Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
[2]

Cristopher Hermosilla. Stratified discontinuous differential equations and sufficient conditions for robustness. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4415-4437. doi: 10.3934/dcds.2015.35.4415
[3]

Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009
[4]

Jaume Llibre, Roland Rabanal. Center conditions for a class of planar rigid polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1075-1090. doi: 10.3934/dcds.2015.35.1075
[5]

Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177
[6]

Jun Shen, Kening Lu, Bixiang Wang. Convergence and center manifolds for differential equations driven by colored noise. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4797-4840. doi: 10.3934/dcds.2019196
[7]

Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
[8]

Giuseppe Maria Coclite, Lorenzo di Ruvo. Discontinuous solutions for the generalized short pulse equation. Evolution Equations & Control Theory, 2019, 8 (4) : 737-753. doi: 10.3934/eect.2019036
[9]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136
[10]

Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
[11]

Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150
[12]

Amin Boumenir, Vu Kim Tuan, Nguyen Hoang. The recovery of a parabolic equation from measurements at a single point. Evolution Equations & Control Theory, 2018, 7 (2) : 197-216. doi: 10.3934/eect.2018010
[13]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[14]

Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118
[15]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029
[16]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139
[17]

Flavia Antonacci, Marco Degiovanni. On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 833-842. doi: 10.3934/dcds.2006.15.833
[18]

Anya Désilles, Hélène Frankowska. Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions. Networks & Heterogeneous Media, 2013, 8 (3) : 727-744. doi: 10.3934/nhm.2013.8.727
[19]

Zuowei Cai, Jianhua Huang, Lihong Huang. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3591-3614. doi: 10.3934/dcdsb.2017181
[20]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences