Integrable approximation of regular islands: The iterative canonical transformation method

Löbner, Clemens; Löck, Steffen; Bäcker, Arnd; Ketzmerick, Roland

doi:10.1103/physreve.88.062901

Cited by 7 publications

(18 citation statements)

References 32 publications

(63 reference statements)

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“…In order to find H r:s (q, p), we generalize the iterative canonical transformation method of Ref. [31] to include the considered nonlinear resonance chain. The iterative canonical transformation method is based on the idea, that the tori of the regular region and their dynamics can be decomposed into the properties (i) action and frequency as well as (ii) shape.…”

Section: Iterative Canonical Transformation Methods With a Resonancementioning

confidence: 99%

Integrable approximation of regular regions with a nonlinear resonance chain

et al. 2014

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Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the non-integrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.

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Section: Iterative Canonical Transformation Methods With a Resonancementioning

confidence: 99%

Integrable approximation of regular regions with a nonlinear resonance chain

et al. 2014

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“…Expectedly, this feature is completely missing in the dynamics of the effective integrable Hamiltonian, where the phase-space always remains regular and thereby cannot represent the actual time-dependent system for large parameter values. We note that there are studies indicating the disappearance of island like features in certain time-independent nonintegrable systems when they are approximated by integrable models [40].…”

Section: B the Classical Limitmentioning

confidence: 82%

Effective time-independent analysis for quantum kicked systems

Bandyopadhyay

Sarkar

2015

Phys. Rev. E

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We present a mapping of potentially chaotic time-dependent quantum kicked systems to an equivalent approximate effective time-independent scenario, whereby the system is rendered integrable. The time-evolution is factorized into an initial kick, followed by an evolution dictated by a timeindependent Hamiltonian and a final kick. This method is applied to the kicked top model. The effective time-independent Hamiltonian thus obtained, does not suffer from spurious divergences encountered if the traditional Baker-Cambell-Hausdorff treatment is used. The quasienergy spectrum of the Floquet operator is found to be in excellent agreement with the energy levels of the effective Hamiltonian for a wide range of system parameters. The density of states for the effective system exhibits sharp peak-like features, pointing towards quantum criticality. The dynamics in the classical limit of the integrable effective Hamiltonian shows remarkable agreement with the non-integrable map corresponding to the actual time-dependent system in the non-chaotic regime. This suggests that the effective Hamiltonian serves as a substitute for the actual system in the non-chaotic regime at both the quantum and classical level.

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“…In Sec. IV B, we set up an integrable approximation including the nonlinear resonance chain using the iterative canonical transformation method [43,68] as presented in Ref. [43].…”

Section: Perturbation-free Prediction Of Tunneling In the Standarmentioning

confidence: 99%

Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems

et al. 2016

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For generic Hamiltonian systems we derive predictions for dynamical tunneling from regular to chaotic phase-space regions. In contrast to previous approaches, we account for the resonanceassisted enhancement of regular-to-chaotic tunneling in a non-perturbative way. This provides the foundation for future semiclassical complex-path evaluations of resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new class of integrable approximations which mimic the regular phase-space region and its dominant nonlinear resonance chain in a mixed regular-chaotic system. We illustrate the method for the standard map.

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Integrable approximation of regular islands: The iterative canonical transformation method

Cited by 7 publications

References 32 publications

Integrable approximation of regular regions with a nonlinear resonance chain

Integrable approximation of regular regions with a nonlinear resonance chain

Effective time-independent analysis for quantum kicked systems

Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems

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