Isochronicity of a class of piecewise continuous oscillators

Mañosas, Francesc; Torres, Pedro

doi:10.1090/S0002-9939-05-07873-1

Isochronicity of a class of piecewise continuous oscillators
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by Francesc Mañosas and Pedro J. Torres PDF

Proc. Amer. Math. Soc. 133 (2005), 3027-3035 Request permission

Abstract:

Motivated by a classical pendulum clock model suggested by Andrade in 1920, we study the equation $\ddot x+g(x)\operatorname {sgn}{\dot x}+x=0$ and prove that for a nonlinear analytic $g$ the origin is never an isochronous focus or an isochronous center.

References

Barbara Alfawicka, Inverse problems connected with periods of oscillations described $\ddot x+g(x)=0$, Ann. Polon. Math. 44 (1984), no. 3, 297–308. MR 817804, DOI 10.4064/ap-44-3-297-308
Barbara Alfawicka, Inverse problem connected with half-period function analytic at the origin, Bull. Polish Acad. Sci. Math. 32 (1984), no. 5-6, 267–274 (English, with Russian summary). MR 785984
J. Andrade, Les frottements et l’isochronisme, C. R. Acad. Sci. Paris, (1920), 664-665.
Carmen Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), no. 3, 310–321. MR 903390, DOI 10.1016/0022-0396(87)90122-7
Anna Cima, Armengol Gasull, and Francesc Mañosas, Period function for a class of Hamiltonian systems, J. Differential Equations 168 (2000), no. 1, 180–199. Special issue in celebration of Jack K. Hale’s 70th birthday, Part 1 (Atlanta, GA/Lisbon, 1998). MR 1801350, DOI 10.1006/jdeq.2000.3912
A. Cima, F. Mañosas, and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Differential Equations 157 (1999), no. 2, 373–413. MR 1713265, DOI 10.1006/jdeq.1999.3635
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, DOI 10.3934/dcds.2000.6.609
Klaus Deimling and Paul Szilágyi, Periodic solutions of dry friction problems, Z. Angew. Math. Phys. 45 (1994), no. 1, 53–60. MR 1259526, DOI 10.1007/BF00942846
Lubomir Gavrilov, Isochronicity of plane polynomial Hamiltonian systems, Nonlinearity 10 (1997), no. 2, 433–448. MR 1438261, DOI 10.1088/0951-7715/10/2/008
R.C. Hibbeler, Mechanics for engineers, MacMillan Publishing Company, 1985.
Minoru 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 141834, DOI 10.1007/BF00253926
Minoru Urabe, Potential forces which yield periodic motions of a fixed period, J. Math. Mech. 10 (1961), 569–578. MR 0123060