Dynamics of Radiation Belt Particles

Abstract

This paper reviews basic concepts of particle dynamics underlying theoretical aspect of radiation belt modeling and data analysis. We outline the theory of adiabatic invariants of quasiperiodic Hamiltonian systems and derive the invariants of particle motion trapped in the radiation belts. We discuss how the nonlinearity of resonant interaction of particles with small-amplitude plasma waves, ubiquitous across the inner magnetosphere, can make particle motion stochastic. Long-term evolution of a stochastic system can be described by the Fokker-Plank (diffusion) equation. We derive the kinetic equation of particle diffusion in the invariant space and discuss its limitations and associated challenges which need to be addressed in forthcoming radiation belt models and data analysis.

Keywords

RBSP mission Radiation belts Quasi-linear diffusion Chaos Particle dynamics 

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References

  1. J.M. Albert, Diffusion by one wave and by many waves. J. Geophys. Res. 115, A00F05 (2010). doi: 10.1029/2009JA014732 Google Scholar
  2. B.J. Anderson, R.E. Denton, S.A. Fuselier, On determining polarization characteristics of ion cyclotron wave magnetic field fluctuations. J. Geophys. Res. 101, 13195 (1996) ADSGoogle Scholar
  3. V.I. Arnold, V. Kozlov, A.I. Neishtadt, Mathematical Methods of Classical Mechanics. Encyclopedia of Mathematical Sciences, vol. 3 (Springer, New York, 2010) Google Scholar
  4. R. Balescu, Transport Processes in Plasmas. 1. Classical Transport Theory (Elsevier, Amsterdam, 1988) Google Scholar
  5. T.J. Birmingham, Pitch angle diffusion in the Jovian magnetodisc. J. Geophys. Res. 89, 2699 (1984) ADSGoogle Scholar
  6. J.B. Blake, W.A. Kolasinski, R.W. Fillius, E.G. Mullen, Injection of electrons and protons with energies of tens of MeV into L<3 on 24 March 1991. Geophys. Res. Lett. 19, 821 (1992) ADSGoogle Scholar
  7. J. Bortnik, R.M. Thorne, T.P. O’Brien, J.C. Green, R.J. Strangeway, Y.Y. Shprits, D.N. Baker, Observation of two distinct, rapid loss mechanisms during the 20 November 2003 radiation belt dropout event. J. Geophys. Res. 111, 12216 (2006). doi: 10.1029/2006JA011802 Google Scholar
  8. J. Büchner, L.M. Zelenyi, Regular and chaotic charged particle motion in magnetotaillike field reversals. I—Basic theory of trapped motion. J. Geophys. Res. 94, 11821 (1989) ADSGoogle Scholar
  9. E.L. Burshtein, V.I. Veksler, A.A. Kolomensky, A stochastic method of accelerating particles, in Some Problems in the Theory of Cyclic Accelerators (Akad. Nauk USSR, Moscow, 1955) Google Scholar
  10. J.R. Cary, A.J. Brizard, Hamiltonian theory of guiding-center motion. Rev. Mod. Phys. 81, 693 (2009) ADSMATHMathSciNetGoogle Scholar
  11. J.R. Cary, D.F. Escande, J.L. Tennyson, Adiabatic-invariant change due to separatrix crossing. Phys. Rev. A 34(5), 256 (1986) Google Scholar
  12. J.R. Cary, D.F. Escande, A.D. Verga, Nonquasilinear diffusion far from the chaotic threshold. Phys. Rev. Lett. 65, 3132 (1990) ADSGoogle Scholar
  13. C. Cattell, J.R. Wygant, K. Goetz, K. Kersten, P.J. Kellogg, T. von Rosenvinge, S.D. Bale, I. Roth, M. Temerin, M.K. Hudson, R.A. Mewaldt, M. Wiedenbeck, M. Maksimovic, R. Ergun, M. Acuna, C.T. Russell, Discovery of very large amplitude whistler-mode waves in Earth’s radiation belts. Geophys. Res. Lett. 35, 01105 (2008). doi: 10.1029/2007GL032009 ADSGoogle Scholar
  14. Y. Chen, G.D. Reeves, R.H. Friedel, The energization of relativistic electrons in the outer Van Allen radiation belt. Nat. Phys. 3, 614 (2007) Google Scholar
  15. B.V. Chirikov, Resonance processes in magnetic traps. J. Nucl. Energy, Part C Plasma Phys. Accel. Thermonucl. Res. 1, 253 (1960) ADSGoogle Scholar
  16. B.V. Chirikov, A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263 (1979) ADSMathSciNetGoogle Scholar
  17. J.M. Cornwall, Scattering of energetic trapped electrons by very-low-frequency waves. J. Geophys. Res. 69, 1251 (1964) ADSGoogle Scholar
  18. J.M. Cornwall, Cyclotron instabilities and electromagnetic emission in the ultra low frequency and very low frequency ranges. J. Geophys. Res. 70, 61 (1965) ADSGoogle Scholar
  19. R.E. Denton, B.J. Anderson, G. Ho, D.C. Hamilton, Effects of wave superposition on the polarization of electromagnetic ion cyclotron waves. J. Geophys. Res. 4, 271 (1996) Google Scholar
  20. A.J. Dessler, R. Karplus, Some properties of the Van Allen radiation. Phys. Rev. Lett. 4, 271 (1960) ADSGoogle Scholar
  21. W.E. Drummond, D. Pines, Non-linear Stability of Plasma Oscillations, in General Atomic, GA-2386 (1961) Google Scholar
  22. J.W. Dungey, Loss of Van Allen electrons due to whistlers. Planet. Space Sci. 11, 591 (1963) ADSGoogle Scholar
  23. J.W. Dungey, Effects of the electromagnetic perturbations on particles trapped in the radiation belts. Space Sci. Rev. 4, 199 (1965) ADSGoogle Scholar
  24. S.R. Elkington, M.K. Hudson, A.A. Chan, Acceleration of relativistic electrons via drift-resonant interaction with toroidal-mode Pc-5 ULF oscillations. Geophys. Res. Lett. 26, 3273 (1999) ADSGoogle Scholar
  25. S.R. Elkington, M.K. Hudson, A.A. Chan, Resonant acceleration and diffusion of outer zone electrons in an asymmetric geomagnetic field. J. Geophys. Res. 108, 1116 (2003) Google Scholar
  26. R.E. Erlandson, A.Y. Ukhorskiy, Observations of electromagnetic ion cyclotron waves during geomagnetic storms: wave occurrence and pitch angle scattering. J. Geophys. Res. 106, 3883 (2001) ADSGoogle Scholar
  27. C.-G. Falthammar, Effects of time-dependent electric fields on geomagnetically trapped radiation. J. Geophys. Res. 70, 2503 (1965) ADSMathSciNetGoogle Scholar
  28. J.L. Gannon, X. Li, M. Temerin, Parametric study of shock-induced transport and energization of relativistic electrons in the magnetosphere. J. Geophys. Res. 110, 12206 (2005). doi: 10.1029/2004JA010679 Google Scholar
  29. H. Goldstein, Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, 1980) MATHGoogle Scholar
  30. L. Gomberoff, R. Neira, Convective growth rate of ion-cyclotron waves in a H+–He+ and H+–He+–O+ plasma. J. Geophys. Res. 88, 2170 (1983) ADSGoogle Scholar
  31. J.C. Green, M.G. Kivelson, Relativistic electrons in the outer radiation belt: differentiating between acceleration mechanisms. J. Geophys. Res. 109, 03213 (2004). doi: 10.1029/2003JA010153 Google Scholar
  32. P. Helander, L. Kjellberg, Simulation of nonquasilinear diffusion. Phys. Plasmas 1, 210 (1994) ADSGoogle Scholar
  33. R.B. Horne, Plasma astrophysics: acceleration of killer electrons. Nat. Phys. 3, 590 (2007) Google Scholar
  34. R.B. Horne, R.M. Thorne, Convective instabilities of electromagnetic ion-cyclotron waves in the outer magnetosphere. J. Geophys. Res. 99, 17259 (1994) ADSGoogle Scholar
  35. R.B. Horne, R.M. Thorne, Potential waves for relativistic electron scattering and stochastic acceleration during magnetic storms. Geophys. Res. Lett. 25, 3011 (1998) ADSGoogle Scholar
  36. R.B. Horne, R.M. Thorne, S.A. Glauert, N.P. Meredith, D. Pokhotelov, O. Santolík, Electron acceleration in the Van Allen radiation belts by fast magnetosonic waves. Geophys. Res. Lett. 34, 17107 (2007). doi: 10.1029/2007GL030267 ADSGoogle Scholar
  37. M.K. Hudson, S.R. Elkington, J.G. Lyon, V.A. Marchenko, I. Roth, M. Temerin, J.B. Blake, M.S. Gussenhoven, J.R. Wygant, Simulations of radiation belt formation during storm sudden commencements. J. Geophys. Res. 102, 14087 (1997) ADSGoogle Scholar
  38. M.K. Hudson, B.T. Kress, H.-R. Mueller, J.A. Zastrow, J.B. Blake, Relationship of the Van Allen radiation belts to solar wind drivers. J. Atmos. Sol.-Terr. Phys. 70, 708 (2008) ADSGoogle Scholar
  39. U. Jaekel, R. Schlickeiser, The Fokker-Planck coefficients of cosmic ray transport in random electromagnetic fields. J. Phys. G, Nucl. Part. Phys. 18, 1089 (1992) ADSGoogle Scholar
  40. J.R. Jokipii, P.J. Coleman, Cosmic-ray diffusion tensor and its variation observed with Mariner 4. J. Geophys. Res. 73, 5495 (1968) ADSGoogle Scholar
  41. R. Keller, K.H. Schmitter, Beam storage with stochastic acceleration and improvement of a synchrocyclotron beam, in CERN Rept. 58-13, Geneva, Switzerland (1958) Google Scholar
  42. P.J. Kellog, Van Allen radiation of solar origin. Nature 183, 1295 (1959) ADSGoogle Scholar
  43. P.J. Kellog, C.A. Cattell, K. Goetz, S.J. Monson, L.B. Wilson III, Electron trapping and charge transport by large amplitude whistlers. Geophys. Res. Lett. 37, 09224 (2010). doi: 10.1029/2010JA015919 Google Scholar
  44. P.J. Kellog, C.A. Cattell, K. Goetz, S.J. Monson, L.B. Wilson III, Large amplitude whistlers in the magnetosphere observed with Wind–Waves. J. Geophys. Res. 116, 09224 (2011). doi: 10.1029/2010JA015919 Google Scholar
  45. C.F. Kennel, F. Engelmann, Velocity space diffusion from weak plasma turbulence in a magnetic field. Phys. Fluids 9, 2377 (1966) ADSGoogle Scholar
  46. C.F. Kennel, H.E. Petscheck, Limit on stably trapped particle fluxes. J. Geophys. Res. 71, 427 (1966) Google Scholar
  47. K. Kersten, C.A. Cattell, A. Breneman, K. Goetz, P.J. Kellogg, J.R. Wygant, L.B. Wilson III, J.B. Blake, M.D. Looper, I. Roth, Observation of relativistic electron microbursts in conjunction with intense radiation belt whistler-mode waves. Geophys. Res. Lett. 38, 08107 (2011). doi: 10.1029/2011GL046810 ADSGoogle Scholar
  48. B.T. Kress, M.K. Hudson, M.D. Looper, J. Albert, J.G. Lyon, C.C. Goodrich, Global MHD test particle simulations of >10 MeV radiation belt electrons during storm sudden commencement. J. Geophys. Res. 112, 09215 (2007). doi: 10.1029/2006JA012218 Google Scholar
  49. B.T. Kress, M.K. Hudson, M.D. Looper, J.G. Lyon, C.C. Goodrich, Global MHD test particle simulations of solar energetic electron trapping in the Earth’s radiation belts. J. Atmos. Sol.-Terr. Phys. 70, 1727 (2008) ADSGoogle Scholar
  50. M. Kruskal, Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic. J. Math. Phys. 23, 742 (1962) MathSciNetGoogle Scholar
  51. L. Landau, E. Lifshitz, The Classical Theory of Fields, vol. 2 (Pergamon, London, 1959) Google Scholar
  52. L. Landau, E. Lifshitz, Mechanics, vol. 1, 3rd edn. (Pergamon, New York, 1976) Google Scholar
  53. L.D. Landau, Kinetic equation for the case of Coulomb interaction. Zh. Eksp. Teor. Fiz. 7, 203 (1937) MATHGoogle Scholar
  54. L.J. Lanzerotti, A. Hasegawa, C.G. Maclennan, Drift mirror instability in the magnetosphere: particle and field oscillations and electron heating. J. Geophys. Res. 74, 5565 (1969) ADSGoogle Scholar
  55. X. Li, M. Temerin, J.R. Wygant, M.K. Hudson, J.B. Blake, Simulation of the prompt energization and transport of radiation belt particles during the March 24, 1991 SSC. Geophys. Res. Lett. 20, 2423 (1993) ADSGoogle Scholar
  56. A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics. Applied Mathematical Sciences, vol. 38 (Springer, New York, 1983) Google Scholar
  57. M.D. Looper, J.B. Blake, R.A. Mewaldt, Response of the inner radiation belt to the violent Sun–Earth connection events of October–November 2003. Geophys. Res. Lett. 32, L03S06 (2005). doi: 10.1029/2004GL021502 Google Scholar
  58. C.E. McIlwain, Ring current effects on trapped particles. J. Geophys. Res. 71, 3623 (1966) ADSGoogle Scholar
  59. N.P. Meredith, R.B. Horne, R.R. Anderson, Substorm dependence of chorus amplitudes: implications for the acceleration of electrons to relativistic energies. J. Geophys. Res. 106, 13165 (2001). doi: 10.1029/2000JA900156 ADSGoogle Scholar
  60. R.M. Millan, K.B. Yando, J.C. Green, A.Y. Ukhorskiy, Spatial distribution of relativistic electron precipitation during a radiation belt depletion event. Geophys. Res. Lett. 37, 20103 (2010). doi: 10.1029/2010GL044919 ADSGoogle Scholar
  61. A.I. Neishtadt, On change in adiabatic invariant at a passage through a separatrix. Fiz. Plazmy 12, 992 (1986) Google Scholar
  62. T.G. Northrop, The Adiabatic Motion of Charged Particles (Interscience, New York, 1963) MATHGoogle Scholar
  63. T.G. Northrop, E. Teller, Stability of the adiabatic motion of charged particles in the Earth’s field. Phys. Rev. 117, 215 (1960) ADSMathSciNetGoogle Scholar
  64. F.W.J. Olver, Asymptotics and Special Functions (Academic Press, New York, 1974) Google Scholar
  65. M.K. Öztürk, R.A. Wolf, Bifurcation of drift shells near the dayside magnetopause. J. Geophys. Res. 112, 07207 (2007). doi: 10.1029/2006JA012102 Google Scholar
  66. A.B. Rechester, R.B. White, Calculation of turbulent diffusion for the Chirikov-Taylor model. Phys. Rev. Lett. 44, 1586 (1980) ADSMathSciNetGoogle Scholar
  67. J.G. Roederer, Experimental evidence on radial diffusion of geomagnetically trapped particles, in Earth’s Particles and Fields, ed. by B.M. McCormac (1968), p. 143 Google Scholar
  68. J.G. Roederer, Dynamics of geomagnetically trapped radiation, in Physics and Chemistry in Space, ed. by J.G. Roederer, J. Zahringer, vol. 2 (Springer, Berlin, 1970) Google Scholar
  69. Y.A. Romanov, G.F. Filippov, The interaction of fast electron beams with longitudinal plasma waves. Sov. Phys. JETP 13, 87 (1961) Google Scholar
  70. R.Z. Sagdeev, D.A. Usikov, G.M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic, New York, 1988) MATHGoogle Scholar
  71. M. Schulz, L.J. Lanzerotti, Particle diffusion in the radiation belts, in Physics and Chemistry in Space, vol. 7 (Springer, New York, 1974) Google Scholar
  72. V.P. Shabansky, Some processes in magnetosphere. Space Sci. Rev. 12(3), 299 (1971) ADSGoogle Scholar
  73. D.R. Shklyar, H. Matsumoto, Oblique whistler-mode waves in the inhomogeneous magnetospheric plasma: resonant interactions with energetic charged particles. Surv. Geophys. 30, 55 (2009) ADSGoogle Scholar
  74. Y.Y. Shprits, R.M. Thorne, R. Friedel, G.D. Reeves, J. Fennell, D.N. Baker, S.G. Kanekal, Outward radial diffusion driven by losses at magnetopause. J. Geophys. Res. 111, 11214 (2006). doi: 10.1029/2006JA011657 Google Scholar
  75. Y.Y. Shprits, S.R. Elkington, N.P. Meredith, D.A. Subbotin, Review of modeling of losses and sources of relativistic electrons in the outer radiation belt I: radial transport. J. Atmos. Sol.-Terr. Phys. 70, 1679 (2008a). doi: 10.1016/j.jastp.2008.06.008 ADSGoogle Scholar
  76. Y.Y. Shprits, S.R. Elkington, N.P. Meredith, D.A. Subbotin, Review of modeling of losses and sources of relativistic electrons in the outer radiation belt II: local acceleration and loss. J. Atmos. Sol.-Terr. Phys. 70, 1694 (2008b). doi: 10.1016/j.jastp.2008.06.014 ADSGoogle Scholar
  77. M.I. Sitnov, N.A. Tsyganenko, A.Y. Ukhorskiy, P.C. Brandt, Dynamical data-based modeling of the storm-time geomagnetic field with enhanced spatial resolution. J. Geophys. Res. 113, 07218 (2008). doi: 10.1029/2007JA013003 Google Scholar
  78. D.V. Sivukhin, Motion of charged particles in electromagnetic fields in the drift approximation. Rev. Plasma Phys. 1, 1 (1965) ADSGoogle Scholar
  79. G.R. Smith, A.N. Kaufman, Stochastic acceleration by an obliquely propagating wave—an example of overlapping resonances. Phys. Fluids 21, 2230 (1978) ADSGoogle Scholar
  80. D.J. Southwood, J.W. Dungey, R.L. Eherington, Bounce resonant interaction between pulsations and trapped particles. Planet. Space Sci. 17, 349 (1969) ADSGoogle Scholar
  81. T.W. Speiser, Particle trajectories in a model current sheet, based on the open model of the magnetosphere, with applications to auroral particles. J. Geophys. Res. 70, 1717 (1965) ADSGoogle Scholar
  82. D. Summers, R.M. Thorne, F. Xiao, Relativistic theory of wave particle resonant diffusion with application to electron acceleration in the magnetosphere. J. Geophys. Res. 103, 20487 (1998) ADSGoogle Scholar
  83. K. Takahashi, A.Y. Ukhorskiy, Solar wind control of Pc5 pulsation power at geosynchronous orbit. J. Geophys. Res. 112, 11205 (2007). doi: 10.1029/2007JA012483 Google Scholar
  84. R.M. Thorne, Radiation belt dynamics: the importance of wave-particle interactions. Geophys. Res. Lett. 37, 22107 (2010). doi: 10.1029/2010GL044990 ADSGoogle Scholar
  85. R.M. Thorne, C.F. Kennel, Relativistic electron precipitation during magnetic storm main phase. J. Geophys. Res. 76, 4446 (1971) ADSGoogle Scholar
  86. V.Y. Trakhtengerts, Stationary states of the Earth’s outer radiation zone. Geomagn. Aeron. 6, 827 (1966) Google Scholar
  87. N.A. Tsyganenko, Modeling the global magnetic field of the large-scale Birkeland current systems. J. Geophys. Res. 101, 27187 (1996) ADSGoogle Scholar
  88. N.A. Tsyganenko, M.I. Sitnov, Modeling the dynamics of the inner magnetosphere during strong geomagnetic storms. J. Geophys. Res. 110, 03208 (2005). doi: 10.1029/2004JA010798 Google Scholar
  89. N.A. Tsyganenko, M.I. Sitnov, Magnetospheric configurations from a high-resolution data-based magnetic field model. J. Geophys. Res. 112, 06225 (2007). doi: 10.1029/2007JA012260 Google Scholar
  90. D.L. Turner, Y. Shprits, M. Hartinger, V. Angelopoulos, Explaining sudden losses of outer radiation belt electrons during geomagnetic storms. Nat. Phys. (2012). doi: 10.1038/nphys2185 Google Scholar
  91. B.A. Tverskoy, Dynamics of the radiation belts of the Earth, 2. Geomagn. Aeron. 4, 436 (1964) Google Scholar
  92. B.A. Tverskoy, Main mechanisms in the formation of the Earth’s radiation belts. Rev. Geophys. Space Phys. 7, 219 (1969) ADSGoogle Scholar
  93. A.Y. Ukhorskiy, M.I. Sitnov, Radial transport in the outer radiation belt due to global magnetospheric compressions. J. Atmos. Sol.-Terr. Phys. 70, 1714 (2008) ADSGoogle Scholar
  94. A.Y. Ukhorskiy, K. Takahashi, B.J. Anderson, H. Korth, The impact of toroidal ULF waves on the outer radiation belt electrons. J. Geophys. Res. 110, 10202 (2005). doi: 10.1029/2005JA011017 Google Scholar
  95. A.Y. Ukhorskiy, K. Takahashi, B.J. Anderson, N.A. Tsyganenko, The impact of ULF oscillations in solar wind dynamic pressure on the outer radiation belt electrons. Geophys. Res. Lett. 33, 06111 (2006a). doi: 10.1029/2005GL024380 ADSGoogle Scholar
  96. A.Y. Ukhorskiy, B.J. Anderson, P.C. Brandt, N.A. Tsyganenko, Storm-time evolution of the outer radiation belt: transport and losses. J. Geophys. Res. 111, A11S03 (2006b). doi: 10.1029/2006JA011690 Google Scholar
  97. A.Y. Ukhorskiy, M.I. Sitnov, K. Takahashi, B.J. Anderson, Radial transport of radiation belt electrons due to stormtime pc5 waves. Ann. Geophys. 27, 2173 (2009) ADSGoogle Scholar
  98. A.Y. Ukhorskiy, Y.Y. Shprits, B.J. Anderson, K. Takahashi, R.M. Thorne, Rapid scattering of radiation belt electrons by storm-time EMIC waves. Geophys. Res. Lett. 37, 09101 (2010). doi: 10.1029/2010GL042906 ADSGoogle Scholar
  99. A.Y. Ukhorskiy, M.I. Sitnov, R.M. Millan, B.T. Kress, The role of drift orbit bifurcations in energization and loss of electrons in the outer radiation belt. J. Geophys. Res. 116, 09208 (2011). doi: 10.1029/2011JA016623 Google Scholar
  100. J.A. Van Allen, The geomagnetically trapped corpuscular radiation. J. Geophys. Res. 64, 1683 (1959) ADSGoogle Scholar
  101. A.A. Vedenov, E.P. Velikhov, R.Z. Sagdeev, Stability of plasma. Usp. Fiz. Nauk 73, 701 (1961) Google Scholar
  102. S.N. Vernov, A.E. Chudakov, P.V. Vakulov, Y.I. Logachev, Study of terrestrial corpuscular radiation and cosmic rays during flight of the cosmic rocket. Dokl. Akad. Nauk SSSR 125, 304 (1959) Google Scholar
  103. Y. Wan, S. Sazykin, R.A. Wolf, M.K. Öztürk, Drift shell bifurcation near the dayside magnetopause in realistic magnetospheric magnetic fields. J. Geophys. Res. 115, 10205 (2010). doi: 10.1029/2010JA015395 Google Scholar
  104. L.B.I. Wilson, C.A. Cattell, P.J. Kellogg, J.R. Wygant, K. Goetz, A. Breneman, K. Kersten, The properties of large amplitude whistler mode waves in the magnetosphere: propagation and relationship with geomagnetic activity. Geophys. Res. Lett. 38, 17107 (2011). doi: 10.1029/2011GL048671 ADSGoogle Scholar
  105. J.R. Wygant, F. Mozer, M. Temerin, J. Blake, N. Maynard, H. Singer, M. Smiddy, Large amplitude electric and magnetic field signatures in the inner magnetosphere during injection of 15 MeV electron drift echoes. Geophys. Res. Lett. 21, 1739 (1994) ADSGoogle Scholar
  106. G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461 (2002) ADSMATHMathSciNetGoogle Scholar
  107. G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, New York, 2005) MATHGoogle Scholar

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© The Author(s) 2012

Authors and Affiliations

  1. 1.Applied Physics LaboratoryJohns Hopkins UniversityLaurelUSA

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