Stability of the phase motion in race-track microtrons. (English) Zbl 1376.78006

Summary: We model the phase oscillations of electrons in race-track microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase – the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by an invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. We also clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.


78A55 Technical applications of optics and electromagnetic theory
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
37M05 Simulation of dynamical systems
39A30 Stability theory for difference equations
Full Text: DOI arXiv


[1] Kapitza, S. P.; Melekhin, V. N., The microtron, (1978), Harwood Academic Publishers
[2] Rand, R. E., Recirculating electron accelerators, (1984), Harwood Academic Publishers
[3] Aloev, A. V.; Carrillo, D.; Kubyshin, Yu. A.; Pakhomov, N. I.; Shvedunov, V. I., Electron gun with off-axis beam injection for a race-track microtron, Nucl. Instrum. Methods A, 624, 39-46, (2010)
[4] Yu.A. Kubyshin, et al. Current status of the 12 MeV UPC race-track microtron, in: C. Petit-Jean-Genaz, ed., Proc. PAC-2009 (Vancouver, Canada, 2009), pp. 2775-2779.
[5] Vladimirov, I. Y.; Pakhomov, N. I.; Shvedunov, V. I.; Kubyshin, Y. A.; Rigla, J. P.; Zakharov, V. V., End magnets with rare Earth permanent magnet material for a compact race-track microtron, Eur. Phys. J. Plus, 129, 171, (2014)
[6] Veksler, V. I., A new method of acceleration of relativistic particles, J. Phys., 9, 153-158, (1945)
[7] P. Lidbjörk, Microtrons, in S. Turner, ed., Fifth General Accelerator Physics Course (CERN, 2001) pp. 971-981.
[8] Henderson, C.; Heyman, F. F.; Jennings, R. E., Phase stability of the microtron, Proc. Phys. Soc. B, 66, 41-49, (1953)
[9] Melekhin, V. N., Theory of nonlinear difference equations and resonance instability in phase oscillations in a microtron and of oscillations of rays in open resonators, Sov. Phys. JETP, 34, 702-708, (1972)
[10] Simó, C., Stability of degenerate fixed points of analytic area preserving mappings, Astérisque, 98-99, (1982), Soc. Math. France, Paris
[11] Luque, A.; Villanueva, J., Quasi-periodic frequency analysis using averaging-extrapolation methods, SIAM J. Appl. Dyn. Sist., 13, 1-46, (2013) · Zbl 1338.34088
[12] Vieiro, A., Study of the effect of conservative and weakly dissipative perturbations on symplectic maps and Hamiltonian systems, (2009), Universidad de Barcelona, (Ph.D. Thesis)
[13] Fox, A. M.; de la Llave, R., Barriers to transport and mixing in volume-preserving maps with nonzero flux, Physica D, 295-296, 1-10, (2015) · Zbl 1364.76066
[14] Kubyshin, Yu. A.; Poseryaev, A. P.; Shvedunov, V. I., Longitudinal beam dynamics with phase slip in race-track microtrons, Nucl. Instrum. Methods A, 596, 147-156, (2008)
[15] Mackay, J. M.; Stark, J., Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity, 5, 867-888, (1992) · Zbl 0754.70012
[16] C. Simó, D.V. Treschev, Evolution of the last invariant curve in a family of area preserving maps, preprint 1998.
[17] Broer, H.; Roussarie, R.; Simó, C., Invariant circles in the bogdanov-Takens bifurcation for diffeomorphisms, Ergodic Theory Dynam. Systems, 16, 1147-1172, (1996) · Zbl 0876.58032
[18] Gelfreich, V. G., Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps, Physica D, 136, 266-279, (2000) · Zbl 0942.37016
[19] Simó, C., (Invariant Curves Near Parabolic Points and Regions of Stability, Lecture Notes in Math., 819, (1980), Springer Berlin), 418-424
[20] Moeckel, R., Generic bifurcations of the twist coefficient, Ergodic Theory Dynam. Systems, 10, 185-195, (1990) · Zbl 0734.58021
[21] Devaney, R. L., Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218, 89-113, (1976) · Zbl 0363.58003
[22] Lamb, J. S.W.; Roberts, J. A.G., Time-reversal symmetry in dynamical systems: A survey, Physica D, 112, 1-39, (1998) · Zbl 1194.34072
[23] Katok, A.; Hasselblatt, B., Introduction to the modern theory of dynamical systems, (1995), Cambridge Univ. Press · Zbl 0878.58020
[24] Seara, T. M.; Villanueva, J., On the numerical computation of Diophantine rotation numbers of analytic circle maps, Physica D, 217, 107-120, (2006) · Zbl 1134.37339
[25] Khinchin, A. Ya., Continued fractions, (1964), The University of Chicago Press · Zbl 0117.28601
[26] Simó, C.; Vieiro, A., Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps, Nonlinearity, 22, 1191-1245, (2009) · Zbl 1181.37077
[27] Simó, C.; Vieiro, A., Dynamics in chaotic zones of area preserving maps: close to the separatrix and global instability zones, Physica D, 240, 732-753, (2011) · Zbl 1217.37051
[28] Simó, C.; Vieiro, A., Some remarks on the abundance of stable periodic orbits inside homoclinic lobes, Physica D, 240, 1936-1953, (2011) · Zbl 1244.37039
[29] Hénon, M., Numerical study of quadratic area-preserving mappings, Quart. Appl. Math., 27, 291-312, (1969) · Zbl 0191.45403
[30] Shapiro, L.; Stockman, G., Computer vision, (2002), Prentice-Hall
[31] Miguel, N.; Simó, C.; Vieiro, A., From the Hénon conservative map to the chirikov standard map for large parameter values, Regul. Chao. Dyn., 18, 469-489, (2013) · Zbl 1417.37191
[32] Giovannazzi, M., Stability domain of planar symplectic maps using invariant manifolds, Phys. Rev. E, 53, 6403-6412, (1996)
[33] Meiss, J. D., Symplectic maps, variational principles, and transport, Rev. Modern Phys., 64, 795-848, (1992) · Zbl 1160.37302
[34] Fontich, E.; Simó, C., The splitting of separatrices for analytic diffeomorphisms, Ergodic Theory Dynam. Systems, 10, 295-318, (1990) · Zbl 0706.58061
[35] MacKay, R. S.; Meiss, J. D.; Percival, I. C., Transport in Hamiltonian systems, Physica D, 13, 55-81, (1984) · Zbl 0585.58039
[36] Ushiki, S., Sur LES liaisons-cols des systèmes dynamiques analytiques, C. R. Acad. Sci., Paris Ser. A, 291, 447-449, (1980) · Zbl 0477.58026
[37] Gelfreich, V. G.; Lazutkin, V. F.; Svanidze, N. V., A refined formula for the separatrix splitting for the standard map, Physica D, 71, 82-101, (1994) · Zbl 0812.70017
[38] Martín, P.; Sauzin, D.; Seara, T. M., Exponentially small splitting of separatrices in the perturbed mcmillan map, Discrete Contin. Dyn. Syst., 31, 301-372, (2011) · Zbl 1230.37070
[39] Delshams, A.; Ramírez-Ros, R., Singular separatrix splitting and the Melnikov method: an experimental study, Exp. Math., 8, 29-48, (1999) · Zbl 0932.37012
[40] Ramírez-Ros, R., Exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables, Physica D, 210, 149-179, (2005) · Zbl 1152.37336
[41] Gelfreich, V.; Simó, C., High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete Contin. Dyn. Syst., 10, 681-698, (2008) · Zbl 1169.37013
[42] Simó, C., Analytical and numerical computation of invariant manifolds, (Benest, D.; Froeschlé, C., Modern Methods in Celestial Mechanics, (1990), Editions Frontières Gif-sur-Yvette), 285-330
[43] Moutsinas, G., Splitting of separatrices in area-preserving maps close to \(1 : 3\) resonance, (2016), U. Warwick, (Ph.D. Thesis)
[44] Gelfreich, V. G.; Lazutkin, V. F.; Tabanov, M. B., Exponentially small splittings in Hamiltonian systems, Chaos, 1, 137-142, (1991) · Zbl 0899.58016
[45] Melekhin, V. N., Phase dynamics of particles in a microtron and the problem of stochastic instability of nonlinear systems, Sov. Phys. JETP, 41, 803-808, (1976)
[46] Wiedemann, H., Particle accelerator physics, (2003), Springer
[47] Melekhin, V. N.; Luganskii, L. B., High-current microtron instability, Sov. Phys.-Tech. Phys., 1930-1931, (1970)
[48] Bykov, V. P., Investigation of electron bunches in a microtron, Sov. Phys. JETP, 13, 1169-1174, (1961)
[49] A. Jankowiak, et al. Commissioning and operation of the 1.5 GeV Harmonic Double Sided Microtron at Mainz University, in: C. Petit-Jean-Genaz, ed., Proc. EPAC-2008 (Genoa, Italy, 2008), pp. 51-55.
[50] Levi-Civita, T., Sopra alcuni criteri di instabilità, Ann. Mat. Ser. III, 5, 221-307, (1901) · JFM 32.0720.01
[51] Larreal, O., Cálculo de la escisión de separatrices y regiones de estabilidad usando precisión múltiple: el microtrón y la singularidad Hopf-cero, (2011), U. Politècnica de Catalunya, (Ph.D. Thesis)
[52] Siegel, C. L.; Moser, J. K., Lectures on celestial mechanics, (1995), Springer-Verlag
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.