A simple model of wave-particle interaction is studied in its self-consistent form, that is, where the particles are allowed to feedback on the wave dynamics. We focus on the configurations of locked solutions (equilibria) and how the energy-momentum exchange mechanism induces chaos in the model. As we explore the system, we analyze the mathematical structure that gives rise to locked states and how the model’s non-linearity enables multiple equilibrium amplitudes for waves. We also explain the predominance of regularity as we vary the control parameters and the mechanism behind the emergence of chaos under limited parameter choices.
REFERENCES
1.
Y.
Elskens
and D.
Escande
, Microscopic Dynamics of Plasmas and Chaos
(IOP Publishing
, Bristol
, 2003
).2.
D. F.
Escande
and Y.
Elskens
, “Microscopic dynamics of plasmas and chaos: The wave-particle interaction paradigm
,” Plasma Phys. Control Fusion
45
, A115
–A124
(2003
). 3.
R.
Balescu
, Statistical Mechanics of Charged Particles
(Wiley-Interscience
, London
, 1963
).4.
F.
Doveil
, A.
Macor
, and K.
Auhmani
, “Wave-particle interaction investigated in a travelling wave tube
,” Plasma Phys. Control Fusion
47
, A261
–A271
(2005
). 5.
M. C.
de Sousa
, F.
Doveil
, Y.
Elskens
, and I. L.
Caldas
, “Wave-particle interactions in a long traveling wave tube with upgraded helix
,” Phys. Plasmas
27
(9
), 093108
(2020
).6.
A.
Macor
, F.
Doveil
, and Y.
Elskens
, “Electron climbing a “devil’s staircase” in wave-particle interaction
,” Phys. Rev. Lett.
95
, 264102
(2006
).7.
8.
G. R.
Smith
and N. R.
Pereira
, “Phase-locked particle motion in a large-amplitude plasma wave
,” Phys. Fluids
21
, 2253
–2262
(1978
). 9.
N.
Besse
, Y.
Elskens
, D. F.
Escande
, and P.
Bertrand
, “Validity of quasilinear theory: Refutations and new numerical confirmation
,” Plasma Phys. Control. Fusion
53
(025012
), 36
(2010
).10.
T. M.
O’Neil
, J. H.
Winfrey
, and J. H.
Malmberg
, “Nonlinear interaction of a small cold beam and a plasma
,” Phys. Fluids
14
(6
), 1204
–1212
(1971
).11.
I. N.
Onishchenko
, A. R.
Linetskii
, N. G.
Matsiborko
, V. D.
Shapiro
, and V. I.
Shevchenko
, “Contribution to the nonlinear theory of excitation of a monochromatic plasma wave by an electron beam
,” JETP Lett.
12
(8
), 281
–285
(1970
).12.
H. E.
Mynick
and A. K.
Kaufman
, “Soluble theory of nonlinear beam-plasma interaction
,” Phys. Fluids
21
, 653
–663
(1978
). 13.
J. V.
Gomes
, M. C.
de Sousa
, R. L.
Viana
, I. L.
Caldas
, and Y.
Elskens
, “Low-dimensional chaos in the single wave model for self-consistent wave-particle Hamiltonian
,” Chaos
31
, 083104
(2021
). 14.
J. C.
Adam
, G.
Laval
, and I.
Mendonça
, “Time-dependent nonlinear Langmuir waves
,” Phys. Fluids
24
, 260
–267
(1981
). 15.
D.
del Castillo-Negrete
, “Dynamics and self-consistent chaos in a mean field Hamiltonian model,” in Dynamics and Thermodynamics of Systems with Long-Range Interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo, and M. Wilkens (Springer, Berlin, 2002), pp. 407–436.16.
B. V.
Chirikov
, “A universal instability of many-dimensional oscillator systems
,” Phys. Rep.
52
(5
), 263
–379
(1979
). 17.
D. F.
Escande
, “Stochasticity in classical Hamiltonian systems: Universal aspects
,” Phys. Rep.
121
(3–4
), 165
–261
(1985
). 18.
C.
Skokos
, T.
Bountis
, C. G.
Antonopoulos
, and M. N.
Vrahatis
, “Detecting order and chaos in Hamiltonian systems by the SALI method
,” J. Phys. A: Math. Gen.
37
, 6269
–6284
(2004
). 19.
G. A.
Gottwald
, C. H.
Skokos
, and J.
Laskar
, Chaos Detection and Predictability
(Springer-Verlag
, Berlin
, 2015
).20.
Y.
Elskens
and D. F.
Escande
, “Slowly pulsating separatrices sweep homoclinic tangles where islands must be small—An extension of classical adiabatic theory
,” Nonlinearity
4
, 615
–667
(1991
). 21.
A. I.
Neishtadt
, V. V.
Sidorenko
, and D. V.
Treschev
, “Stable periodic motions in the problem on passage through a separatrix
,” Chaos
7
(2
), 1
–11
(1997
).22.
A. I.
Neishtadt
, C.
Simó
, D.
Treschev
, and A.
Vasiliev
, “Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems
,” Discrete Contin. Dyn. Syst. Ser. B
10
(2/3
), 621
–650
(2008
).© 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
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