^{1}and Edson D. Leonel

^{2,3}

### Abstract

Some dynamical properties of an ensemble of trajectories of individual (non-interacting) classical particles of mass *m* and charge *q* interacting with a time-dependent electric field and suffering the action of a constant magnetic field are studied. Depending on both the amplitude of oscillation of the electric field and the intensity of the magnetic field, the phase space of the model can either exhibit: (i) regular behavior or (ii) a mixed structure, with periodic islands of regular motion, chaotic seas characterized by positive Lyapunov exponents, and invariant Kolmogorov–Arnold–Moser curves preventing the particle to reach unbounded energy. We define an escape window in the chaotic sea and study the transport properties for chaotic orbits along the phase space by the use of scaling formalism. Our results show that the escape distribution and the survival probability obey homogeneous functions characterized by critical exponents and present universal behavior under appropriate scaling transformations. We show the survival probability decays exponentially for small iterations changing to a slower power law decay for large time, therefore, characterizing clearly the effects of stickiness of the islands and invariant tori. For the range of parameters used, our results show that the crossover from fast to slow decay obeys a power law and the behavior of survival orbits is scaling invariant.

The formalism of escape is used to study the dynamics and hence the transport of charged particles in an accelerator. The model consists of a periodically time dependent electric field limited to a certain region in space that furnishes or absorbs energy of a particle. After the particle leaves the electric field region, a constant magnetic field impels the particle to move in a circular trajectory. This magnetic field is responsible for bringing the particle back to the electric field region leaving the energy unchanged. The control parameters and represent the amplitude of the time-dependent electric field and the inverse of the magnetic field, respectively. The parameter defines the nonlinear strength while defines the time that the particle spends in the magnetic region. The dynamics can be described by a two-dimensional nonlinear mapping. Depending on both and , the phase space presents either regular or mixed structure. For , the nonlinear term vanishes and the particle's energy is constant in time. For , there is a strong correlation between the frequency of oscillation of the electric field and the intensity of the magnetic field, and as a consequence, the motion of the particle is regular. For different control parameters than those discussed, the phase space presents mixed structure with islands of regular motion surrounded by chaotic seas and invariant KAM curves that prevent the particle from acquiring unbounded energy gain. We study the escape of particles through a hole in the phase space placed in the velocity axis by considering the position of the lowest energy KAM curve. An ensemble of trajectories evolves in time from low energy until each reaches the escaping hole or a maximum time. This formalism is used to study the transport of particles though the chaotic region of the phase space. We obtain both the histogram of exit times as well as the survival probability as functions of

*n*(number of egresses from the region of the electric field). The survival probability decays exponentially for short

*n*and, due to the existence of sticky domains, it decays as a power law for large

*n*. Both the histogram of escape and the survival probability are described using scaling approach.

D.G.L. thanks to FAPEMIG, CNPq, and FAPESP. E.D.L. thanks to FAPESP, CNPq, and FUNDUNESP, Brazilian agencies. The authors acknowledge Carl P. Dettmann for a careful reading of the paper. This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP).

I. INTRODUCTION

II. THE MODEL AND PHASE SPACE PROPERTIES

III. SCALING PROPERTIES FOR THE TRANSPORT ON THE CHAOTIC SEA

IV. CONCLUSIONS

### Key Topics

- Electric fields
- 38.0
- Phase space methods
- 22.0
- Magnetic fields
- 14.0
- Transport properties
- 7.0
- Particle velocity
- 6.0

## Figures

Pieces of two possible orbits in the proposed model. The upper trajectory illustrates a situation where the particle enters into and leaves from the electric field region at *x* = 0. The lower trajectory illustrates a situation where the particle enters into the electric field at the left side and leaves this region at *x* = *d*.

Pieces of two possible orbits in the proposed model. The upper trajectory illustrates a situation where the particle enters into and leaves from the electric field region at *x* = 0. The lower trajectory illustrates a situation where the particle enters into the electric field at the left side and leaves this region at *x* = *d*.

These plots display the phase space of the system for different control parameters: (a) and , (b) and , (c) and and (d) and .

These plots display the phase space of the system for different control parameters: (a) and , (b) and , (c) and and (d) and .

These plots illustrate the average Lyapunov exponent as function of: (a) for constant and; (b) as function of for fixed .

These plots illustrate the average Lyapunov exponent as function of: (a) for constant and; (b) as function of for fixed .

This figure shows the position of the lowest energy invariant KAM curves for different values of and considering: (a) ; (b) ; and (c) .

This figure shows the position of the lowest energy invariant KAM curves for different values of and considering: (a) ; (b) ; and (c) .

The figure shows the average of the absolute values of the velocity of the KAM curves as function of for three different values of . The best fits furnish with .

The figure shows the average of the absolute values of the velocity of the KAM curves as function of for three different values of . The best fits furnish with .

(a) shows the plot of the normalized number of escaping orbits at the iteration *n* for different values of and fixed . (b) displays the overlap of all curves shown in (a) onto a single plot, after a suitable rescaling of the axis.

(a) shows the plot of the normalized number of escaping orbits at the iteration *n* for different values of and fixed . (b) displays the overlap of all curves shown in (a) onto a single plot, after a suitable rescaling of the axis.

(a) illustrates the crossover as function of . The best fit furnishes with *z* = −0.98 ± 0.02. (b) illustrates the maximum values of *h* as function of . The best fit furnishes with .

(a) illustrates the crossover as function of . The best fit furnishes with *z* = −0.98 ± 0.02. (b) illustrates the maximum values of *h* as function of . The best fit furnishes with .

(a) displays the normalized number of remaining orbits (do not escape) as function of *n* and different values of ; (b) illustrates the decay of the quantity showing evidently two regimes.

(a) displays the normalized number of remaining orbits (do not escape) as function of *n* and different values of ; (b) illustrates the decay of the quantity showing evidently two regimes.

(a) shows the numerical data of crossover of . The best fit furnishes with . Appropriate scaling transformation overlaps the curves onto a single plot, as displayed in (b).

(a) shows the numerical data of crossover of . The best fit furnishes with . Appropriate scaling transformation overlaps the curves onto a single plot, as displayed in (b).

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