Scaling invariance for the escape of particles from a periodically corrugated waveguide

Leonel, Edson D.; Costa, Diogo Ricardo da; Dettmann, Carl P.

doi:10.1016/j.physleta.2011.11.027

Cited by 14 publications

(9 citation statements)

References 29 publications

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“…Because our purpose is to study the transport from W o to W d in the chaotic sea around islands of regular motion, we define elliptical windows that enclosure a thin layer of chaotic sea that surrounds the regions of regular motion. So we use polar variables ρ and θ to represent the coordinates (x,y) of the border of the ellipses as x = ρ cos θ and y = ρ sin θ , where ρ = μσ/ (σ cos θ ) 2 + (μ sin θ ) 2 . The parameters μ and σ represent the semiaxes of the ellipses.…”

Section: B Windows Of Origin and Destination Around Regions Of Regulmentioning

confidence: 99%

“…Escape, transmission, reflection, and recurrence time statistics are some of the approaches employed to study properties of trajectories in phase space of dynamical systems. Escape processes are usually studied in the context where a system presents a natural leak or hole through which a particle or a wave leaves the system [1][2][3][4][5][6]. A leak or hole can be introduced in closed systems in order to study the properties of trajectories in phase space [7][8][9] or to simulate loss in confinement of plasma and measurement devices, e.g., antennas and sensors [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model

et al. 2016

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The chaotic portion of phase space of the simplified Fermi-Ulam model is studied under the context of transport of trajectories in two scenarios: (i) the trajectories are originated from a region distant from the islands of regular motion and are transported to a region located at a high portion of phase space and (ii) the trajectories are originated from chaotic regions around the islands of regular motion and are transported to other regions around islands of regular motion. The transport is investigated in terms of the observables histogram of transport and survival probability. We show that the histogram curves are scaling invariant and we organize the survival probability curves in four kinds of behavior, namely (a) transition from exponential decay to power law decay, (b) transition from exponential decay to stretched exponential decay, (c) transition from an initial fast exponential decay to a slower exponential decay, and (d) a single exponential decay. We show that, depending on choice of the regions of origin and destination, the transport process is weakly affected by the stickiness of trajectories around islands of regular motion.

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Section: B Windows Of Origin and Destination Around Regions Of Regulmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model

et al. 2016

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“…This result is in remarkably well agreement with those obtained for a periodically corrugated waveguide. 30 We obtained also the corresponding value of h max for each curve of Fig. 6(a), as it is displayed in Fig.…”

Section: Scaling Properties For the Transport On The Chaotic Seamentioning

confidence: 99%

Scaling investigation for the dynamics of charged particles in an electric field accelerator

Ladeira¹,

Leonel

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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 e and s represent the amplitude of the time-dependent electric field and the inverse of the magnetic field, respectively. The parameter e defines the nonlinear strength while ps 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 e and s, the phase space presents either regular or mixed structure. For e ¼ 0, the nonlinear term vanishes and the particle's energy is constant in time. For s ¼ 1; 3; 5; …, 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 ensemb...

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“…If the medium between the waveguide boundaries if not uniform the piecewise linear path joining two consecutive reflection points on the lower line or plane is replaced by the geodesic with respect to the metric n ds, where n is the refraction index. The transition from ordered to chaotic motion in 2D billiards and waveguides has been considered [9][10][11] ; the transport and diffusion properties have been extensively analysed [12][13][14][15][16][17][18] . The stability properties of the map depend on the corrugation amplitude.…”

Section: Introductionmentioning

confidence: 99%

Propagation of rays in 2D and 3D waveguides: A stability analysis with Lyapunov and reversibility fast indicators

Gradoni

Panichi

Turchetti

2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Propagation of rays in 2D and 3D corrugated waveguides is performed in the general framework of stability indicators. The analysis of stability is based on the Lyapunov and Reversibility error. It is found that the error growth follows a power law for regular orbits and an exponential law for chaotic orbits. A relation with the Shannon channel capacity is devised and an approximate scaling law found for the capacity increase with the corrugation depth.

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Scaling invariance for the escape of particles from a periodically corrugated waveguide

Cited by 14 publications

References 29 publications

Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model

Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model

Scaling investigation for the dynamics of charged particles in an electric field accelerator

Propagation of rays in 2D and 3D waveguides: A stability analysis with Lyapunov and reversibility fast indicators

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