Change of the adiabatic invariant at a separatrix in a volume-preserving 3D system

Neishtadt, A. I.; Vasiliev, A. A.

doi:10.1088/0951-7715/12/2/010

Cited by 48 publications

(96 citation statements)

References 14 publications

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“…The change of the adiabatic invariant due to separatrix crossing is estimated applying Neishtadt's theory [3][4][5], since the IAI (9) tends to the IAI of the interpolating Hamiltonian. Moreover, the evolution of the frozen energy H(ρ, ψ, λ) = E and of the scaled period ǫ T (J, λ) can be described by the dynamics of the interpolating Hamiltonian up to an error O(ǫ 2 ).…”

Section: B Trapping Into Resonance and Change Of Adiabatic Invariantmentioning

confidence: 99%

Analysis of adiabatic trapping for quasi-integrable area-preserving maps

Bazzani¹,

Frye

Giovannozzi

et al. 2014

Phys. Rev. E

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Trapping phenomena involving non-linear resonances have been considered in the past in the framework of adiabatic theory. Several results are known for continuous-time dynamical systems generated by Hamiltonian flows in which the combined effect of non-linear resonances and slow timevariation of some system parameters is considered. The focus of this paper is on discrete-time dynamical systems generated by two-dimensional symplectic maps. The possibility of extending the results of neo-adiabatic theory to quasi-integrable area-preserving maps is discussed. Scaling laws are derived, which describe the adiabatic transport as a function of the system parameters using a probabilistic point of view. These laws can be particularly relevant for physical applications. The outcome of extensive numerical simulations showing the excellent agreement with the analytical estimates and scaling laws is presented and discussed in detail. Trapping phenomena involving non-linear resonances have been considered in the past in the framework of adiabatic theory. Several results are known for continuous-time dynamical systems generated by Hamiltonian flows in which the combined effect of non-linear resonances and slow time-variation of some system parameters is considered. The focus of this paper is on discrete-time dynamical systems generated by two-dimensional symplectic maps. The possibility of extending the results of neo-adiabatic theory to quasi-integrable area-preserving maps is discussed. Scaling laws are derived, which describe the adiabatic transport as a function of the system parameters using a probabilistic point of view. These laws can be particularly relevant for physical applications. The outcome of extensive numerical simulations showing the excellent agreement with the analytical estimates and scaling laws is presented and discussed in detail.

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Section: B Trapping Into Resonance and Change Of Adiabatic Invariantmentioning

confidence: 99%

Analysis of adiabatic trapping for quasi-integrable area-preserving maps

Bazzani¹,

Frye

Giovannozzi

et al. 2014

Phys. Rev. E

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“…The asymptotic formulas for the jump of adiabatic invariant on a separatrix were obtained by Timofeev (Timofeev, 1978) for particular case of a pendulum in a slowly varying gravity field, by Neishtadt (Neishtadt, 1986) and Cary and co-workers (Cary et al, 1986) for systems with one degree of freedom plus slowly varying parameter and by Neishtadt (Neishtadt, 1987) for systems with two degrees of freedom, one corresponding to the fast motion and the other corresponding to the slow motion. This theory was first applied to magnetospheric problems in (Büchner and Zelenyi, 1989).…”

Section: Separatrix Crossingsmentioning

confidence: 99%

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

Vainchtein

Rovinsky

Zelenyǐ

et al. 2004

Journal of Nonlinear Science

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In the present paper we investigate the resonant interaction between monochromatic electromagnetic waves and charged particles in configurations with magnetic field reversals (e.g. in the earth magnetotail). The smallness of certain physical parameters allows us to solve this problem using perturbation theory reducing the problem of resonant wave-particle interaction to the analysis of slow passages of a particle through a resonance.We discuss in details two of the most important resonant phenomena: capture into resonance and scattering on resonance. We show that these processes result in destruction of the adiabatic invariants, chaotization of particles, may lead to significant (almost free) acceleration of particles and govern transport in the phase space.We calculate the characteristic times of mixing due to resonant effects and separatrix crossings and discuss the relative importance of these phenomena.

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“…The variables of the classical Hamiltonian are defined as (17) into the time-dependent GP equation, one gets [38] i dψ 1 …”

Section: Nonlinear Two-mode Model For Two Coupled Becmentioning

confidence: 99%

“…A conceptual phenomenon of classical adiabatic theory is destruction of adiabatic invariance at separatrix crossings which is encountered, in particular, in plasma physics and hydrodynamics, classical and celestial mechanics [10,11,12,13,14,15,16,17,18,19,20,21,22]. The phenomenon is very important for BEC physics: we consider here nonlinear two-mode models related to tunnelling between coupled BEC in a double well [23], nonlinear Landau-Zener tunnelling [24,25], Feshbach resonance passage in atom-molecule systems [26,27,28].…”

Section: Introductionmentioning

confidence: 99%

“…The asymptotic formula for this jump was obtained in [11,12,13]. Later, the general theory of adiabatic separatrix crossings was also developed for slow-fast Hamiltonian systems [10,17], and was applied to certain physical problems (see, for example, [16,18,19,20]). It was also noticed that nonlinear Landau-Zener (NLZ) tunnelling models constitute a particular case for which the general theory can be applied [27,28].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Universality in nonadiabatic behavior of classical actions in nonlinear models of Bose-Einstein condensates

Itin

Watanabe

2007

Phys. Rev. E

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We discuss dynamics of approximate adiabatic invariants in several nonlinear models being related to physics of Bose-Einstein condensates (BEC). We show that nonadiabatic dynamics in Feshbach resonance passage, nonlinear Landau-Zener (NLZ) tunnelling, and BEC tunnelling oscillations in a double-well can be considered within a unifying approach based on the theory of separatrix crossings. The separatrix crossing theory was applied previously to some problems of classical mechanics, plasma physics and hydrodynamics, but has not been used in the rapidly growing BEC-related field yet. We derive explicit formulas for the change in the action in several models. Extensive numerical calculations support the theory and demonstrate its universal character. We also discovered a qualitatively new nonlinear phenomenon in a NLZ model which we propose to call separated adiabatic tunnelling (AT).

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Change of the adiabatic invariant at a separatrix in a volume-preserving 3D system

Cited by 48 publications

References 14 publications

Analysis of adiabatic trapping for quasi-integrable area-preserving maps

Analysis of adiabatic trapping for quasi-integrable area-preserving maps

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

Universality in nonadiabatic behavior of classical actions in nonlinear models of Bose-Einstein condensates

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