Alternate islands of multiple isochronous chains in wave-particle interactions

Abstract

We analyze the dynamics of a relativistic particle moving in a uniform magnetic field and perturbed by a standing electrostatic wave. We show that a pulsed wave produces an infinite number of perturbative terms with the same winding number, which may generate islands in the same region of phase space. As a consequence, the number of isochronous island chains varies as a function of the wave parameters. We observe that in all the resonances, the number of chains is related to the amplitude of the various resonant terms. We determine analytically the position of the periodic points and the number of island chains as a function of the wave number and wave period. Such information is very important when one is concerned with regular particle acceleration, since it is necessary to adjust the initial conditions of the particle to obtain the maximum acceleration. PACS number(s): 05.45.-a, 52.20.Dq, 45.50.DdWave-particle interaction is basically a nonlinear process [1,2] that may present regular and chaotic trajectories in its phase space [3]. This kind of interaction can be found in many areas of physics [1,[4][5][6], and it is used in a wide range of applications as an efficient way for particle heating [1, 7-9] and particle acceleration [1,5,[9][10][11].References [12][13][14] present two cases of particles moving in a uniform magnetic field and perturbed by electrostatic waves. For such systems, appropriate resonant conditions are responsible for a great amount of particle acceleration. References [12,13] determine the parameters values for which the acceleration may be maximum.However, the process of regular acceleration also depends on the trajectory followed by the particles. To attain the condition of maximum acceleration, it is necessary to know the position of the resonances and the number of island chains as a function of the parameters. In this way, it is possible to properly adjust the initial conditions of the particles and make them follow the best trajectory in phase space. Moreover, successive bifurcations changing the number of chains modify the acceleration conditions. Even so, these bifurcations have not been explored yet.In this report, we analyze the onset of different island chains described by the dynamics of a twist system consisting of a relativistic particle moving in a uniform magnetic field. This integrable system is kicked by standing electrostatic pulses [15][16][17], such that it becomes nearintegrable for small amplitudes [1-3, 12, 18].Expanding the pulses in a Fourier-Bessel series, we observe that the wave presents an infinite number of perturbative terms. There are groups of perturbative terms with the same winding number that may generate different islands in the same region of phase space. This superposition alters the number of island chains according to the wave parameters. Using the map of the system, we carry a series of analytical estimates, including the position of the periodic points in phase space and the number of island chains as functions of the wave parameters.

Interação onda-partícula: Ressonâncias, aceleração regular e controle do caos

“…Grande parte dos resultados descritos nesta tese foram publicados em periódicos internacionais e podem ser encontrados nas Refs. [16,17,50,51]. A Ref.…”

“…Em geral, encontra-se na literatura sistemas que apresentam apenas uma cadeia de ilhas nas regiões de ressonância [14,54]. Entretanto, será visto que a maior parte das ressonâncias do sistema em estudo apresenta mais de uma cadeia de ilhas no espaço de fases [50,51]. Observa-se também que o número de cadeias aumenta com o número de onda e com o período da onda eletrostática.…”

“…Para valores suficientemente elevados do período e/ou do número de onda, todas as ressonâncias do sistema possuem mais de uma cadeia de ilhas. Esse fenômeno ocorre pois a onda que perturba o sistema apresenta um número infinito de termos ressonantes com o mesmo número de rotação e que podem gerar ilhas de ressonância na mesma região do espaço de fases [50,51]. A superposição de termos ressonantes com o mesmo número de rotação faz com que o número de cadeias de ilhas varie em função dos parâmetros do sistema.…”

“…Outra característica importante do modelo desenvolvido é que todas as ressonâncias apresentam um número par de ilhas no espaço de fases [50]. Devido a essa propriedade de simetria, todas as cadeias que possuem um número ímpar de ilhas ocorrem aos pares no espaço de fases.…”

“…No Capítulo 3, calcula-se analiticamente a posição dos pontos periódicos das ressonâncias primárias. A partir das expressões obtidas, observa-se que a maior parte das ressonâncias apresenta mais de uma cadeia de ilhas no espaço de fases [50,51].…”