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Isochronicity of a class of piecewise continuous oscillators

Authors: Francesc Mañosas and Pedro J. Torres
Journal: Proc. Amer. Math. Soc. 133 (2005), 3027-3035
MSC (2000): Primary 34C05, 34C15
Published electronically: March 31, 2005
MathSciNet review: 2159782
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Abstract: Motivated by a classical pendulum clock model suggested by Andrade in 1920, we study the equation $\ddot x+g(x)\mathop{sgn}\nolimits{\dot x}+x=0$ and prove that for a nonlinear analytic $g$ the origin is never an isochronous focus or an isochronous center.

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  • 1. B. Alfawicka, Inverse problems connected with periods of oscillations described $\ddot x+g(x)=0$. Ann. Polon. Math. 44 (1984), no. 3, 297-308. MR 0817804 (87d:34015)
  • 2. B. Alfawicka, Inverse problem connected with half-period function analytic at the origin. Bull. Polish Acad. Sci. Math. 32 (1984), no. 5-6, 267-274.MR 0785984 (86e:34028)
  • 3. J. Andrade, Les frottements et l'isochronisme, C. R. Acad. Sci. Paris, (1920), 664-665.
  • 4. C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields. J. Differential Equations 69 (1987), no. 3, 310-321.MR 0903390 (88i:58050)
  • 5. A. Cima, A. Gasull and F. Mañosas, Period function for a class of Hamiltonian systems. J. Differential Equations 168 (2000), no. 1, 180-199.MR 1801350 (2001m:34068)
  • 6. A. Cima, F. Mañosas and J. Villadelprat, Isochronism for several classes of Hamiltonian systems, J. Differential Equations, 157 (1999), 373-413.MR 1713265 (2000h:34073)
  • 7. B. Coll, A. Gasull and R. Prohens, Center-focus and isochronous center problems for discontinuous differential equations. Discrete Contin. Dynam. Systems 6 (2000), no. 3, 609-624. MR 1757390 (2001f:34053)
  • 8. K. Deimling and P. Szilágyi, Periodic solutions of dry friction problems, Z. angew. Math. Phys. 45 (1994), 53-60. MR 1259526 (94m:34099)
  • 9. L. Gavrilov, Isochronism of plane polynomial Hamiltonian systems. Nonlinearity 10 (1997) 433-448.MR 1438261 (98b:58143)
  • 10. R.C. Hibbeler, Mechanics for engineers, MacMillan Publishing Company, 1985.
  • 11. M. Urabe, The potential force yielding a periodic motion whose period is an arbitrary continuous function of the amplitude of the velocity. Arch. Rational Mech. Anal. 11 (1962), 27-33 MR 0141834 (25:5231)
  • 12. M. Urabe, Potential forces which yield periodic motions of a fixed period. J. Math. Mech. 10 (1961), 569-578. MR 0123060 (23:A391)

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Additional Information

Francesc Mañosas
Affiliation: Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Pedro J. Torres
Affiliation: Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

Keywords: Isochronous, center, focus
Received by editor(s): March 1, 2004
Received by editor(s) in revised form: May 27, 2004
Published electronically: March 31, 2005
Additional Notes: The first author was partially supported by DGES No. BFM2002-04236-C02-2, BFM2002-01344 and the CONACIT grant number 2001SGR-00173.
The second author was partially supported by D.G.I. BFM2002-01308, Ministerio Ciencia y Tecnología, Spain
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2005 American Mathematical Society

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