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Geometric Parametrisation of Lagrangian Descriptors for 1 Degree-of-Freedom Systems

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Nonlinear Dynamics and Complexity

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 36))

Abstract

Lagrangian Descriptors (LDs) are scalar quantities able to reveal separatrices, manifolds of hyperbolic saddles, and chaotic seas of dynamical systems. A popular version of the LDs consists in computing the arc length of trajectories over a calibrated time window. Herein we introduce and exploit an intrinsic geometrical parametrisation of LDs, free of the time variable, for 1 degree-of-freedom Hamiltonian systems. The parametrisation depends solely on the energy of the system and on the geometry of the associated level curve. We discuss applications of this framework on classical problems on the plane and cylinder, including the cat’s eye, 8-shaped, and fish-tail separatrices. The developed apparatus allows to characterise semi-analytically the rate at which the derivatives of the geometrical LDs become singular when approaching the separatrix. For the problems considered, the same power laws of divergence are found irrespective from the dynamical system. Some of our results are connected with existing estimates obtained with the temporal LDs under approximations. The geometrical formalism provides alternative insights of the mechanisms driving this dynamical indicator.

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Notes

  1. 1.

    Let us remark that we referred to the quantity ℓ(E) as ‘geometrical Lagrangian Descriptor’ even if ℓ(E) has no Lagrangian nature as it does not rely on the flow.

  2. 2.

    Note that B(r, θ) is computed by using the same mesh of initial conditions that has been used to compute ℓ(E) on each point (r, θ) of the grid. In particular, to compute the norm of the directional derivatives, we do not resample a new and more resolved mesh of initial conditions. It is thus implicitly assumed that the resolution of the mesh is fine enough.

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Acknowledgements

J. D. acknowledges funding from the “Fonds de la Recherche Scientifique”. The authors acknowledge discussions with Ana Maria Mancho, Makrina Agaoglou and Guillermo Garcia-Sanchez that have ensued the 2nd Online Conference on Nonlinear Dynamics and Complexity, October 2021.

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Authors and Affiliations

  1. ENSTA Paris, Institut Polytechnique de Paris, Palaiseau, France

    Rémi Pédenon-Orlanducci

  2. naXys, Namur Institute for Complex Systems, University of Namur, Namur, Belgium

    Timoteo Carletti, Anne Lemaitre & Jérôme Daquin

Authors
  1. Rémi Pédenon-Orlanducci
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  2. Timoteo Carletti
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  3. Anne Lemaitre
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  4. Jérôme Daquin
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Corresponding author

Correspondence to Jérôme Daquin .

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Editors and Affiliations

  1. School of Engineering, Polytechnic Institute of Porto, Porto, Portugal

    Carla M.A. Pinto

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Pédenon-Orlanducci, R., Carletti, T., Lemaitre, A., Daquin, J. (2022). Geometric Parametrisation of Lagrangian Descriptors for 1 Degree-of-Freedom Systems. In: Pinto, C.M. (eds) Nonlinear Dynamics and Complexity. Nonlinear Systems and Complexity, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-031-06632-0_11

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