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Lyapunov and reversibility errors for Hamiltonian flows.

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Title: Lyapunov and reversibility errors for Hamiltonian flows.
Authors: Panichi, F.1 federico.panichi@stud.usz.edu.pl, Turchetti, G.2 turchetti@bo.infn.it
Source: Chaos, Solitons & Fractals. Jul2018, Vol. 112, p83-91. 9p.
Subjects: Lyapunov exponents, Differential equations, Numerical analysis, Hamiltonian systems, Chaos theory
Abstract: We discuss the stability of a Hamiltonian system by comparing the standard Lyapunov error (LE) with the forward error (FE) due to a small random perturbation. We introduce also the reversibility error (RE) where the evolution is computed forward up to time t and backwards to t = 0 in presence of noise. This procedure has been investigated in the case of symplectic maps, but it turns out that the results are simpler in the case of a noisy flow, in the limit of zero noise amplitude. Indeed the stochastic processes defined by the displacement of the noisy orbit at time t for FE, or at time 0 for RE after the evolution up to time t , satisfy linear Langevin equations, are Gaussian processes, and the errors are just their root mean square deviations. All the errors are expressed in terms of the fundamental matrix L ( t ) of the tangent flow and can be evaluated numerically using a symplectic integrator. Letting e L ( t ) be the Lyapunov error and e R ( t ) be the reversibility error a very simple relation holds e R 2 ( t ) = ∫ 0 t e L 2 ( s ) d s . The integral relation is quite natural since the local errors due to a random perturbations accumulate during the evolution whereas for the Lyapunov case the error is introduced only at time zero and propagated. The plot of errors for initial conditions in a Poincaré section reflects the phase portrait, whereas in the action plane it allows to single out the resonance strips. We have applied the method to a 3D Hamiltonian model H = H 0 ( J ) + λ V ( Θ ) , where analytic estimates can be obtained for the single resonances from perturbation theory. This allows to inspect the double resonance structure where the single resonance strips intersect. We have also considered the Hénon–Heiles Hamiltonian to show numerically the equivalence of the errors apart from a shift of 1/2 in the power law exponent in the case of regular orbits. The reversibility error method (REM), previously introduced as the error due to round off in the symplectic integration, appears to be comparable with RE also for the models considered here. [ABSTRACT FROM AUTHOR]
Copyright of Chaos, Solitons & Fractals is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Lyapunov and reversibility errors for Hamiltonian flows.
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  Data: <searchLink fieldCode="AR" term="%22Panichi%2C+F%2E%22">Panichi, F.</searchLink><relatesTo>1</relatesTo><i> federico.panichi@stud.usz.edu.pl</i><br /><searchLink fieldCode="AR" term="%22Turchetti%2C+G%2E%22">Turchetti, G.</searchLink><relatesTo>2</relatesTo><i> turchetti@bo.infn.it</i>
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  Data: <searchLink fieldCode="JN" term="%22Chaos%2C+Solitons+%26+Fractals%22">Chaos, Solitons & Fractals</searchLink>. Jul2018, Vol. 112, p83-91. 9p.
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  Data: We discuss the stability of a Hamiltonian system by comparing the standard Lyapunov error (LE) with the forward error (FE) due to a small random perturbation. We introduce also the reversibility error (RE) where the evolution is computed forward up to time t and backwards to t = 0 in presence of noise. This procedure has been investigated in the case of symplectic maps, but it turns out that the results are simpler in the case of a noisy flow, in the limit of zero noise amplitude. Indeed the stochastic processes defined by the displacement of the noisy orbit at time t for FE, or at time 0 for RE after the evolution up to time t , satisfy linear Langevin equations, are Gaussian processes, and the errors are just their root mean square deviations. All the errors are expressed in terms of the fundamental matrix L ( t ) of the tangent flow and can be evaluated numerically using a symplectic integrator. Letting e L ( t ) be the Lyapunov error and e R ( t ) be the reversibility error a very simple relation holds e R 2 ( t ) = ∫ 0 t e L 2 ( s ) d s . The integral relation is quite natural since the local errors due to a random perturbations accumulate during the evolution whereas for the Lyapunov case the error is introduced only at time zero and propagated. The plot of errors for initial conditions in a Poincaré section reflects the phase portrait, whereas in the action plane it allows to single out the resonance strips. We have applied the method to a 3D Hamiltonian model H = H 0 ( J ) + λ V ( Θ ) , where analytic estimates can be obtained for the single resonances from perturbation theory. This allows to inspect the double resonance structure where the single resonance strips intersect. We have also considered the Hénon–Heiles Hamiltonian to show numerically the equivalence of the errors apart from a shift of 1/2 in the power law exponent in the case of regular orbits. The reversibility error method (REM), previously introduced as the error due to round off in the symplectic integration, appears to be comparable with RE also for the models considered here. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Chaos, Solitons & Fractals is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.chaos.2018.03.019
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      – Code: eng
        Text: English
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        PageCount: 9
        StartPage: 83
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      – SubjectFull: Lyapunov exponents
        Type: general
      – SubjectFull: Differential equations
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Hamiltonian systems
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      – SubjectFull: Chaos theory
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      – TitleFull: Lyapunov and reversibility errors for Hamiltonian flows.
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              M: 07
              Text: Jul2018
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              Y: 2018
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