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Mugnaine M, Sales MR, Szezech JD, Viana RL. Dynamics, multistability, and crisis analysis of a sine-circle nontwist map. Phys Rev E 2022; 106:034203. [PMID: 36266788 DOI: 10.1103/physreve.106.034203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/06/2022] [Accepted: 08/03/2022] [Indexed: 06/16/2023]
Abstract
We propose a one-dimensional dynamical system, the sine-circle nontwist map, that can be considered a local approximation of the standard nontwist map and an extension of the paradigmatic sine-circle map. The map depends on three parameters, exhibiting a simple mathematical form but with a rich dynamical behavior. We identify periodic, quasiperiodic, and chaotic solutions for different parameter sets with the Lyapunov exponent and Slater's theorem. From the bifurcation analysis, we determine two bifurcation lines, those that depend on just two of the control parameters, for which the bifurcation that occurs is of the saddle-node type. In order to investigate multistability, we analyze the bifurcation diagrams in the two directions of parameter variation and we observe some regions of hysteresis, representing the coexistence of different attractors. We also analyze different multistable scenarios, as single attractor, coexistence of periodic attractors, coexistence of chaotic and periodic attractors, chaotic behavior, and coexistence of different chaotic bands, by the Lyapunov exponent and the analysis of the domain occupied by the solutions. From the parameter spaces constructed, we observe the prevalence of single attractor and only chaotic behavior scenarios. The multistable scenario is, mostly, formed by different periodic attractors. Lastly, we analyze the crisis in chaotic attractors and we identify the interior and the boundary crisis. From our results, the boundary crisis plays a key role for the extinction of multistability.
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Affiliation(s)
- Michele Mugnaine
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil
| | - Matheus Rolim Sales
- Graduate Program in Science - Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - José Danilo Szezech
- Graduate Program in Science - Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil and Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Ricardo Luiz Viana
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil and Institute of Physics, University of São Paulo, 05508-900 São Paulo, SP, Brazil
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2
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Huggler YH, Hermes JDV, Leonel ED. Application of the Slater criteria to localize invariant tori in Hamiltonian mappings. CHAOS (WOODBURY, N.Y.) 2022; 32:093125. [PMID: 36182398 DOI: 10.1063/5.0103427] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2022] [Accepted: 08/22/2022] [Indexed: 06/16/2023]
Abstract
We investigate the localization of invariant spanning curves for a family of two-dimensional area-preserving mappings described by the dynamical variables I and θ by using Slater's criterion. The Slater theorem says there are three different return times for an irrational translation over a circle in a given interval. The returning time, which measures the number of iterations a map needs to return to a given periodic or quasi periodic region, has three responses along an invariant spanning curve. They are related to a continued fraction expansion used in the translation and obey the Fibonacci sequence. The rotation numbers for such curves are related to a noble number, leading to a devil's staircase structure. The behavior of the rotation number as a function of invariant spanning curves located by Slater's criterion resulted in an expression of a power law in which the absolute value of the exponent is equal to the control parameter γ that controls the speed of the divergence of θ in the limit the action I is sufficiently small.
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Affiliation(s)
- Yoná H Huggler
- Department of Physics, São Paulo State University-UNESP, Av. 24A, 1515 Bela Vista, 13506-900 Rio Claro, SP, Brazil
| | - Joelson D V Hermes
- Department of Physics, São Paulo State University-UNESP, Av. 24A, 1515 Bela Vista, 13506-900 Rio Claro, SP, Brazil
| | - Edson D Leonel
- Department of Physics, São Paulo State University-UNESP, Av. 24A, 1515 Bela Vista, 13506-900 Rio Claro, SP, Brazil
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3
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Mugnaine M, Batista AM, Caldas IL, Szezech JD, de Carvalho RE, Viana RL. Curry-Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems. CHAOS (WOODBURY, N.Y.) 2021; 31:023125. [PMID: 33653060 DOI: 10.1063/5.0035303] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2020] [Accepted: 01/22/2021] [Indexed: 06/12/2023]
Abstract
The routes to chaos play an important role in predictions about the transitions from regular to irregular behavior in nonlinear dynamical systems, such as electrical oscillators, chemical reactions, biomedical rhythms, and nonlinear wave coupling. Of special interest are dissipative systems obtained by adding a dissipation term in a given Hamiltonian system. If the latter satisfies the so-called twist property, the corresponding dissipative version can be called a "dissipative twist system." Transitions to chaos in these systems are well established; for instance, the Curry-Yorke route describes the transition from a quasiperiodic attractor on torus to chaos passing by a chaotic banded attractor. In this paper, we study the transitions from an attractor on torus to chaotic motion in dissipative nontwist systems. We choose the dissipative standard nontwist map, which is a non-conservative version of the standard nontwist map. In our simulations, we observe the same transition to chaos that happens in twist systems, known as a soft one, where the quasiperiodic attractor becomes wrinkled and then chaotic through the Curry-Yorke route. By the Lyapunov exponent, we study the nature of the orbits for a different set of parameters, and we observe that quasiperiodic motion and periodic and chaotic behavior are possible in the system. We observe that they can coexist in the phase space, implying in multistability. The different coexistence scenarios were studied by the basin entropy and by the boundary basin entropy.
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Affiliation(s)
- Michele Mugnaine
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil
| | - Antonio M Batista
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, 05508-900 São Paulo, SP, Brazil
| | - José D Szezech
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Ricardo Egydio de Carvalho
- Department of Statistics, Applied Mathematics and Computer Science, Institute of Geosciences and Exact Sciences-IGCE, São Paulo State University (UNESP), 13506-900 Rio Claro, SP, Brazil
| | - Ricardo L Viana
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil
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Mugnaine M, Batista AM, Caldas IL, Szezech JD, Viana RL. Ratchet current in nontwist Hamiltonian systems. CHAOS (WOODBURY, N.Y.) 2020; 30:093141. [PMID: 33003918 DOI: 10.1063/5.0022073] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2020] [Accepted: 09/10/2020] [Indexed: 06/11/2023]
Abstract
Non-monotonic area-preserving maps violate the twist condition locally in phase space, giving rise to shearless invariant barriers surrounded by twin island chains in these regions of phase space. For the extended standard nontwist map, with two resonant perturbations with distinct wave numbers, we investigate the presence of such barriers and their associated island chains and compare our results with those that have been reported for the standard nontwist map with only one perturbation. Furthermore, we determine in the control parameter space the existence of the shearless barrier and the influence of the additional wave number on this condition. We show that only for odd second wave numbers are the twin island chains symmetrical. Moreover, for even wave numbers, the lack of symmetry between the chains of twin islands generates a ratchet effect that implies a directed transport in the phase space.
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Affiliation(s)
- Michele Mugnaine
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil
| | - Antonio M Batista
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, 05508-900 São Paulo, SP, Brazil
| | - José D Szezech
- Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Ricardo L Viana
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil
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5
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Kato LK, de Carvalho RE. Transport barriers with shearless attractors. Phys Rev E 2019; 99:032218. [PMID: 30999446 DOI: 10.1103/physreve.99.032218] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2019] [Indexed: 06/09/2023]
Abstract
We present a mechanism to generate a sequence of shearless curves or attractors to form a band of transport barriers. We consider the labyrinthic nontwist standard map to prepare a scenario with three shearless curves. Dissipation is introduced and three shearless attractors coexist, very close to each other. In both cases a collective transport barrier is formed.
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Affiliation(s)
- L Kimi Kato
- São Paulo State University-UNESP Institute of Geosciences and Exact Sciences, IGCE, Avenue 24A 1515, Bela Vista, Rio Claro SP, Brazil
| | - R Egydio de Carvalho
- São Paulo State University-UNESP Institute of Geosciences and Exact Sciences, IGCE, Avenue 24A 1515, Bela Vista, Rio Claro SP, Brazil
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Santos MS, Mugnaine M, Szezech JD, Batista AM, Caldas IL, Baptista MS, Viana RL. Recurrence-based analysis of barrier breakup in the standard nontwist map. CHAOS (WOODBURY, N.Y.) 2018; 28:085717. [PMID: 30180633 DOI: 10.1063/1.5021544] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2018] [Accepted: 03/22/2018] [Indexed: 06/08/2023]
Abstract
We study the standard nontwist map that describes the dynamic behaviour of magnetic field lines near a local minimum or maximum of frequency. The standard nontwist map has a shearless invariant curve that acts like a barrier in phase space. Critical parameters for the breakup of the shearless curve have been determined by procedures based on the indicator points and bifurcations of periodical orbits, a methodology that demands high computational cost. To determine the breakup critical parameters, we propose a new simpler and general procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition. We also show that the coexistence of islands and chaotic sea in phase space can be analysed by using the recurrence plot. In particular, the measurement of determinism from the recurrence plot provides us with a simple procedure to distinguish periodic from chaotic structures in the parameter space. We identify an invariant shearless breakup scenario, and we also show that recurrence plots are useful tools to determine the presence of periodic orbit collisions and bifurcation curves.
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Affiliation(s)
- Moises S Santos
- Pós-Graduação em Ciências, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, Paraná, Brazil
| | - Michele Mugnaine
- Pós-Graduação em Ciências, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, Paraná, Brazil
| | - José D Szezech
- Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, Paraná, Brazil
| | - Antonio M Batista
- Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, Paraná, Brazil
| | - Iberê L Caldas
- Instituto de Física, Universidade de São Paulo, 05508-900 São Paulo, São Paulo, Brazil
| | - Murilo S Baptista
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, SUPA, Aberdeen AB24 3UE, United Kingdom
| | - Ricardo L Viana
- Departamento de Física, Universidade Federal do Paraná, 80060-000, Curitiba, Paraná, Brazil
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7
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Mugnaine M, Mathias AC, Santos MS, Batista AM, Szezech JD, Viana RL. Dynamical characterization of transport barriers in nontwist Hamiltonian systems. Phys Rev E 2018; 97:012214. [PMID: 29448491 DOI: 10.1103/physreve.97.012214] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2017] [Indexed: 06/08/2023]
Abstract
The turnstile provides us a useful tool to describe the flux in twist Hamiltonian systems. Thus, its determination allows us to find the areas where the trajectories flux through barriers. We show that the mechanism of the turnstile can increase the flux in nontwist Hamiltonian systems. A model which captures the essence of these systems is the standard nontwist map, introduced by del Castillo-Negrete and Morrison. For selected parameters of this map, we show that chaotic trajectories entering in resonances zones can be explained by turnstiles formed by a set of homoclinic points. We argue that for nontwist systems, if the heteroclinic points are sufficiently close, they can connect twin-islands chains. This provides us a scenario where the trajectories can cross the resonance zones and increase the flux. For these cases the escape basin boundaries are nontrivial, which demands the use of an appropriate characterization. We applied the uncertainty exponent and the entropies of the escape basin boundary in order to quantify the degree of unpredictability of the asymptotic trajectories.
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Affiliation(s)
- M Mugnaine
- Graduate Program in Sciences, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
| | - A C Mathias
- Department of Physics, Federal University of Paraná, Curitiba, Paraná, Brazil
| | - M S Santos
- Graduate Program in Sciences, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
| | - A M Batista
- Graduate Program in Sciences, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
| | - J D Szezech
- Graduate Program in Sciences, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
- Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
| | - R L Viana
- Department of Physics, Federal University of Paraná, Curitiba, Paraná, Brazil
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9
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Datseris G, Diakonos FK, Schmelcher P. Effective intermittency and cross correlations in the standard map. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012914. [PMID: 26274254 DOI: 10.1103/physreve.92.012914] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2015] [Indexed: 06/04/2023]
Abstract
We define auto- and cross-correlation functions capable of capturing dynamical characteristics induced by local phase-space structures in a general dynamical system. These correlation functions are calculated in the standard map for a range of values of the nonlinearity parameter k. Using a model of noninteracting particles, each evolving according to the same standard map dynamics and located initially at specific phase-space regions, we show that for 0.6<k≤1.2 long-range cross correlations emerge. They occur as an ensemble property of particle trajectories by an appropriate choice of the phase-space cells used in the statistical averaging. In this region of k values the single-particle phase space is either dominated by local chaos (k≤k(c) with k(c)≈0.97) or it is characterized by the transition from local to global chaos (k(c)<k≤1.2). Introducing suitable symbolic dynamics we demonstrate that the emergence of long-range cross correlations can be attributed to the existence of an effective intermittent dynamics in specific regions of the phase space. Our findings support the recently established relation of intermittent dynamics and cross correlations [F. K. Diakonos, A. K. Karlis, and P. Schmelcher, Europhys. Lett. 105, 26004 (2014)] in simple one-dimensional intermittent maps, suggesting its validity also for two-dimensional Hamiltonian maps.
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Affiliation(s)
- G Datseris
- Department of Physics, University of Athens, GR-15771 Athens, Greece
| | - F K Diakonos
- Department of Physics, University of Athens, GR-15771 Athens, Greece
| | - P Schmelcher
- Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany and The Hamburg Centre for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
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10
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Kroetz T, Livorati ALP, Leonel ED, Caldas IL. Global ballistic acceleration in a bouncing-ball model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012905. [PMID: 26274245 DOI: 10.1103/physreve.92.012905] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2015] [Indexed: 06/04/2023]
Abstract
The ballistic increase for the velocity of a particle in a bouncing-ball model was investigated. The phenomenon is caused by accelerating structures in phase space known as accelerator modes. They lead to a regular and monotonic increase of the velocity. Here, both regular and ballistic Fermi acceleration coexist in the dynamics, leading the dynamics to two different growth regimes. We characterized deaccelerator modes in the dynamics, corresponding to unstable points in the antisymmetric position of the accelerator modes. In control parameter space, parameter sets for which these accelerations and deaccelerations constitute structures were obtained analytically. Since the mapping is not symplectic, we found fractal basins of influence for acceleration and deacceleration bounded by the stable and unstable manifolds, where the basins affect globally the average velocity of the system.
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Affiliation(s)
- Tiago Kroetz
- Universidade Tecnológica Federal do Paraná, Pato Branco, Paraná, Brazil
| | - André L P Livorati
- Instituto de Física, Universidade de São Paulo, São Paulo, São Paulo, Brazil
- Departamento de Física, Universidade Estadual Paulista, Rio Claro, São Paulo, Brazil
| | - Edson D Leonel
- Departamento de Física, Universidade Estadual Paulista, Rio Claro, São Paulo, Brazil
- Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
| | - Iberê L Caldas
- Instituto de Física, Universidade de São Paulo, São Paulo, São Paulo, Brazil
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11
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Corrêa da Silva TM, Pakter R, Rizzato FB, de Sousa MC, Caldas IL, Steffens FM. Chaotic particle heating due to an obliquely propagating wave in a magnetized plasma. Phys Rev E 2013; 88:013101. [PMID: 23944562 DOI: 10.1103/physreve.88.013101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2013] [Revised: 05/29/2013] [Indexed: 11/07/2022]
Abstract
We study the dynamics of a relativistic charged particle in the presence of a uniform magnetic field and a stationary electrostatic wave that propagates at an arbitrary angle. The wave is considered as a series of periodic pulses which allows us to derive an exact map for the system. In particular, we investigate the heating process of an initially low-energy particle. It is found that abrupt changes in the maximum energy attained by the particle may occur as the angle between the wave propagation and the magnetic field varies. To determine what is the mechanism behind this phenomenon a reduced Hamiltonian that retains the important dynamical features is obtained. Using both Poincaré plots and perturbation theory, we identify that a separatrix reconnection is the key mechanism for the abrupt change in particle response.
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Affiliation(s)
- T M Corrêa da Silva
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil.
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12
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Vieira Abud C, Caldas IL. Secondary nontwist phenomena in area-preserving maps. CHAOS (WOODBURY, N.Y.) 2012; 22:033142. [PMID: 23020481 DOI: 10.1063/1.4750040] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Phenomena as reconnection scenarios, periodic-orbit collisions, and primary shearless tori have been recognized as features of nontwist maps. Recently, these phenomena and secondary shearless tori were analytically predicted for generic maps in the neighborhood of the tripling bifurcation of an elliptic fixed point. In this paper, we apply a numerical procedure to find internal rotation number profiles that highlight the creation of periodic orbits within islands of stability by a saddle-center bifurcation that emerges out a secondary shearless torus. In addition to the analytical predictions, our numerical procedure applied to the twist and nontwist standard maps reveals that the atypical secondary shearless torus occurs not only near a tripling bifurcation of the fixed point but also near a quadrupling bifurcation.
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Affiliation(s)
- C Vieira Abud
- Instituto de Física, Universidade de São Paulo, São Paulo, 05315-970 São Paulo, Brazil.
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13
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Szezech JD, Caldas IL, Lopes SR, Morrison PJ, Viana RL. Effective transport barriers in nontwist systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:036206. [PMID: 23030996 DOI: 10.1103/physreve.86.036206] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2012] [Revised: 08/09/2012] [Indexed: 06/01/2023]
Abstract
In fluids and plasmas with zonal flow reversed shear, a peculiar kind of transport barrier appears in the shearless region, one that is associated with a proper route of transition to chaos. These barriers have been identified in symplectic nontwist maps that model such zonal flows. We use the so-called standard nontwist map, a paradigmatic example of nontwist systems, to analyze the parameter dependence of the transport through a broken shearless barrier. On varying a proper control parameter, we identify the onset of structures with high stickiness that give rise to an effective barrier near the broken shearless curve. Moreover, we show how these stickiness structures, and the concomitant transport reduction in the shearless region, are determined by a homoclinic tangle of the remaining dominant twin island chains. We use the finite-time rotation number, a recently proposed diagnostic, to identify transport barriers that separate different regions of stickiness. The identified barriers are comparable to those obtained by using finite-time Lyapunov exponents.
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Affiliation(s)
- J D Szezech
- Instituto de Física, Universidade de São Paulo, 5315-970, São Paulo, São Paulo, Brazil
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14
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Livorati ALP, Kroetz T, Dettmann CP, Caldas IL, Leonel ED. Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:036203. [PMID: 23030993 DOI: 10.1103/physreve.86.036203] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2012] [Indexed: 06/01/2023]
Abstract
Some phase space transport properties for a conservative bouncer model are studied. The dynamics of the model is described by using a two-dimensional measure preserving mapping for the variables' velocity and time. The system is characterized by a control parameter ε and experiences a transition from integrable (ε=0) to nonintegrable (ε≠0). For small values of ε, the phase space shows a mixed structure where periodic islands, chaotic seas, and invariant tori coexist. As the parameter ε increases and reaches a critical value εc, all invariant tori are destroyed and the chaotic sea spreads over the phase space, leading the particle to diffuse in velocity and experience Fermi acceleration (unlimited energy growth). During the dynamics the particle can be temporarily trapped near periodic and stable regions. We use the finite time Lyapunov exponent to visualize this effect. The survival probability was used to obtain some of the transport properties in the phase space. For large ε, the survival probability decays exponentially when it turns into a slower decay as the control parameter ε is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity.
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Affiliation(s)
- André L P Livorati
- Instituto de Física, IFUSP, Universidade de São Paulo, USP Rua do Matão, Tr. R 187, Cidade Universitária, 05314-970, São Paulo, SP, Brazil
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15
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Rack M, Spatschek KH, Wingen A. Diffusion in a collisional standard map. CHAOS (WOODBURY, N.Y.) 2012; 22:023114. [PMID: 22757521 DOI: 10.1063/1.4704386] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Test particle evaluation of the diffusion coefficient in the presence of magnetic field fluctuations and binary collisions is presented. Chaotic magnetic field lines originate from resonant magnetic perturbations (RMPs). To lowest order, charged particles follow magnetic field lines. Drifts and interaction (collisions) with other particles decorrelate particles from the magnetic field lines. We model the binary collision process by a constant collision frequency. The magnetic field configuration including perturbations on the integrable Hamiltonian part is such that the single particle motion can be followed by a collisional version of a Chirikov-Taylor (standard) map. Frequent collisions are allowed for. Scaling of the diffusion beyond the quasilinear and subdiffusive behaviour is investigated in dependence on the strength of the magnetic perturbations and the collision frequency. The appearance of the so called Rechester-Rosenbluth regime is verified. It is further shown that the so called Kadomtsev-Pogutse diffusion coefficient is the strong collisional limit of the Rechester-Rosenbluth formula. The theoretical estimates are supplemented by numerical simulations.
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Affiliation(s)
- M Rack
- Institut für Theoretische Physik, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
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16
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de Sousa MC, Steffens FM, Pakter R, Rizzato FB. Standard map in magnetized relativistic systems: fixed points and regular acceleration. Phys Rev E 2010; 82:026402. [PMID: 20866921 DOI: 10.1103/physreve.82.026402] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2010] [Indexed: 11/07/2022]
Abstract
We investigate the concept of a standard map for the interaction of relativistic particles and electrostatic waves of arbitrary amplitudes, under the action of external magnetic fields. The map is adequate for physical settings where waves and particles interact impulsively, and allows for a series of analytical result to be exactly obtained. Unlike the traditional form of the standard map, the present map is nonlinear in the wave amplitude and displays a series of peculiar properties. Among these properties we discuss the relation involving fixed points of the maps and accelerator regimes.
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Affiliation(s)
- M C de Sousa
- Departamento de Física, Universidade Prebisteriana Mackenzie, 01302-906 São Paulo, SP, Brazil.
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17
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Martins CGL, Carvalho RED, Caldas IL, Roberto M. Effect of robust torus on the dynamical transport. ACTA ACUST UNITED AC 2010. [DOI: 10.1088/1742-6596/246/1/012005] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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