Abstract
We present a geometric characterization of the nonlinear smooth functions \(V:\mathbb R \rightarrow \mathbb R \) for which the origin is a global isochronous center for the scalar equation \(\ddot{x}=-V^{\prime }(x)\). We revisit Stillinger and Dorignac isochronous potentials \(V\) and show a new simple explicit family. Implicit examples are easily produced.
Similar content being viewed by others
References
Asorey, M., Cariñena, J.F., Marmo, G., Perelomov, A.: Isoperiodic classical systems and their quantum counterparts. Ann. Phys. 322, 1444–1465 (2007)
Bolotin, S., MacKay, R.S.: Isochronous potentials. In: Vázquez, L., et al., (eds.) Proceedings of the 3rd conference on localization and energy transfer in nonlinear systems, World Sci., pp. 217–224 (2003)
Calogero, F.: Isochronous systems. Oxford University Press, Oxford (2008)
Chalykh, O.A., Veselov, A.P.: A remark on rational isochronous potentials. J. Nonlinear Math. Phys. 12(suppl 1), 179183 (2005)
Cima, A., Mañosas, F., Villadelprat, J.: Isochronicity for several classes of Hamiltonian systems. J. Differ. Equ. 157, 373–413 (1999)
Cima, A., Gasull, A., Mañosas, F.: New periodic recurrences with applications. J. Math. Anal. Appl. 382, 418–425 (2011)
Dorignac, J.: On the quantum spectrum of isochronous potentials. J. Phys. A Math. Gen. 38(27), 6183–6210 (2005)
Koukles, I., Piskounov, N.: Sur les vibrations tautochrones dans les systèmes conservatifs et non conservatifs. C. R. Acad. Sci. URSS 17(9), 417–475 (1937)
Mamode, M.: Some remarks on nonlinear oscillators: period, action, semiclassical quantization and Gibbs ensembles. J. Phys. A Math. Theor. 43(50), Article ID 505101 (2010)
Shisha, O., Mehr, C.B.: On involutions. J. Natl. Bur. Stand. 71B, 19–20 (1967)
Stillinger, F.H., Stillinger, D.K.: Pseudoharmonic oscillators and inadequacy of semiclassical quantization. J. Phys. Chem. 93, (6890–6892) (1989)
Strelcyn, J.-M.: On Chouikha’s isochronicity, criterion. Ar**v:1201.6503 (2012)
Urabe, M.: Potential forces which yield periodic motions of fixed period. J. Math. Mech. 10, 569–578 (1961)
Urabe, M.: The potential force yielding a periodic motion whose period is an arbitrary continuous function of the amplitude of the velocity. Arch. Ration. Mech. Anal. 11, 26–33 (1962)
Wiener, J., Watkins, W.: A glimpse into the wonderland of involutions. Mo. J. Math. Sci. 14, 175–185 (2002)
Zampieri, G.: On the periodic oscillations of $\ddot{x}=g(x)$. J. Differ. Equ. 78, 74–88 (1989)
Zampieri, G.: Completely integrable Hamiltonian systems with weak Lyapunov instability or isochrony. Commun. Math. Phys. 303, 73–87 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Jorge Lewowicz for his 75th birthday.
Rights and permissions
About this article
Cite this article
Gorni, G., Zampieri, G. Global Isochronous Potentials. Qual. Theory Dyn. Syst. 12, 407–416 (2013). https://doi.org/10.1007/s12346-013-0097-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12346-013-0097-1