Collective behaviour of an ensemble of forced Duffing oscillators near the 1:1 resonance. (English) Zbl 0900.70321
Summary: Preserving the beam size (or equivalently the emittance) is a primary aim of accelerator physics. Here we study the effect on the emittance of power supply ripple in the guiding magnetic fields. The collective behaviour of forced Duffing oscillators is the natural model to understand this phenomenon. We consider the case when the external frequency is near the linearized natural frequency and nonlinearity and forcing are small. The method of averaging reduces the problem to an autonomous system. A coarse grained long time limit of the phase space density and the rate of approach to this limit are discussed in terms of the autonomous system. The equilibrium density and the rate of approach to equilibrium depend crucially on the detuning parameter, e.g. the equilibration time for phase space averages is much shorter above the bifurcation value of the detuning parameter than it is below. We find the frequencies which lead to the largest emittance growth in three different forcing regimes (weak, moderate and relatively large) and also characterize the dependence of emittance growth on forcing amplitude in these regimes.
MSC:
| 70K40 | Forced motions for nonlinear problems in mechanics |
| 70K30 | Nonlinear resonances for nonlinear problems in mechanics |
| 78A35 | Motion of charged particles |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
Keywords:
accelerator physics; guiding magnetic fields; external frequency; linearized natural frequency; method of averaging; autonomous system; long time limit of phase space density; equilibrium density; detuning parameterReferences:
| [1] | Warnock, R. L.; Ruth, R. R., Phys. Rev. Lett., 56, 188 (1992) · Zbl 0753.70013 |
| [2] | Dumas, H. S.; Laskar, J., Phys. Rev. Lett., 70, 2975 (1993) |
| [3] | A. Bazzani et al., CERN report 94-02.; A. Bazzani et al., CERN report 94-02. |
| [4] | Chao, A., Phys. Rev. Lett., 61, 2752 (1988) |
| [5] | Lee, S. Y., Phys. Rev. Lett., 67, 3768 (1991) |
| [6] | Syphers, M., Phys. Rev. Lett., 71, 719 (1993) |
| [7] | Fischer, W., CERN-SL/94-43(AP) (1994) |
| [8] | Brüning, O., DESY 94-085 (1994) |
| [9] | Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vertor Fields (1983), Springer: Springer New York · Zbl 0515.34001 |
| [10] | H.G. Hereward, Landau Damping by Nonlinearity, CERN/MPS/DL 69-11.; H.G. Hereward, Landau Damping by Nonlinearity, CERN/MPS/DL 69-11. |
| [11] | Courant, E. D.; Snyder, H. S., Ann. Phys., 3, 1 (1958) |
| [12] | Chao, A. W., Physics of Collective Instabilities in High Energy Accelerators (1993), John Wiley: John Wiley New York |
| [13] | Sen, T.; Ellison, J.; Kaufmann, S. K., SSCL-Preprint-561 (July 1994) |
| [14] | Shampine, L. F.; Watts, H. A., (Forsythe, G. E.; Malcolm, M. A.; Moler, C. B., Computer Methods for Mathematical Applications (1977), Prentice Hall: Prentice Hall Englewood Cliffs, NJ) · Zbl 0361.65002 |
| [15] | Murdock, J. A., Perturbations: Theory and Methods (1991), Wiley: Wiley New York · Zbl 0810.34047 |
| [16] | Cogburn, R.; Ellison, J. A., Comm. Math. Phys., 149, 97 (1992) · Zbl 0755.60095 |
| [17] | Pathria, R. K., Statistical Mechanics (1972), Pergamon Press: Pergamon Press Oxford · Zbl 0862.00007 |
| [18] | Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in Fortran (1992), Cambridge · Zbl 0778.65002 |
| [19] | Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists (1971), Springer: Springer New York · Zbl 0213.16602 |
| [20] | Ellison, J. A.; Guinn, T., Phys. Rev. B, 13, 1880 (1976) |
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