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From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential

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Abstract

In this paper we compare the method of Lagrangian descriptors with the classical method of Poincaré maps for revealing the phase space structure of two-degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two-degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincaré maps are compared for their ability to determine the phase space structures that govern this dynamical process.

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Funding

The authors would like to acknowledge the financial support provided by the EPSRC Grant No. EP/P021123/1.

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Correspondence to Rebecca Crossley, Makrina Agaoglou, Matthaios Katsanikas or Stephen Wiggins.

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The authors declare that they have no conflicts of interest.

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MSC2010

37N99, 70K44, 70H05, 70H07, 34C45, 34C37

APPENDIX

APPENDIX

1.1 Lagrangian Descriptors

The method of Lagrangian descriptors (LDs) is a trajectory-based scalar diagnostic that has the capability of revealing the geometrical structures of the phase space. In this paper we have used the p-norm definition of the LDs as it is presented in [20]. In particular in this work we have used the value \(p=1/2\). Consider the following dynamical system with time dependence:

$$\dfrac{d\mathbf{x}}{dt}=\mathbf{v}(\mathbf{x},t),\quad\mathbf{x}\in\mathbb{R}^{n},t\in\mathbb{R}$$
(A.1)

where \(\mathbf{v}(\mathbf{x},t)\in C^{r}(r\geqslant 1)\) in \(\mathbf{x}\) and it is continuous in time. Given an initial condition \(x_{0}\) at time \(t_{0}\), take a fixed integration time \(\tau>0\) and \(p\in(0,1]\). The method of LDs is as follows:

$$M_{p}(\mathbf{x}_{0},t_{0},\tau)=\sum_{k=1}^{n}\bigg{[}\int^{t_{0}+\tau}_{t_{0}-\tau}|v_{k}(\mathbf{x}(t;\mathbf{x}_{0}),t)|^{p}dt\bigg{]}=M_{p}^{(b)}(\mathbf{x}_{0},t_{0},\tau)+M_{p}^{(f)}(\mathbf{x}_{0},t_{0},\tau),$$
(A.2)

where \(M_{p}^{(b)}\) and \(M_{p}^{(f)}\) its backward and forward integration parts:

$$M_{p}^{(b)}(\mathbf{x}_{0},t_{0},\tau) = \sum_{k=1}^{n}\bigg{[}\int^{t_{0}}_{t_{0}-\tau}|v_{k}(\mathbf{x}(t;\mathbf{x}_{0}),t)|^{p}dt\bigg{]},\\ M_{p}^{(f)}(\mathbf{x}_{0},t_{0},\tau) = \sum_{k=1}^{n}\bigg{[}\int^{t_{0}+\tau}_{t_{0}}|v_{k}(\mathbf{x}(t;\mathbf{x}_{0}),t)|^{p}dt\bigg{]},$$
(A.3)

The forward integration reveals the stable manifolds of our dynamical system. The backward integration reveals the unstable manifolds of our dynamical system. Finally by combining both reveals all the invariant manifolds. Both the stable (blue) and unstable (red) manifolds, as have been presented in the rest of the paper, have been extracted from the gradient of the scalar field generated by LDs. In this appendix we illustrate how the methods of LDs reveals the geometry of invariant manifolds with increasing complexity as the integration time parameter \(\tau\) is increased. To show this we have calculate the LDs on the Section 3.1 we present the LDs backward in time, in Fig. 18, forward in time and finally in Fig. 19 backward and forward in time.

Fig. 17
figure 17

LDs for \(\tau=2,4,6,8\) respectively for energy \(H=-0.15\), backward in time.

Fig. 18
figure 18

LDs for \(\tau=2,4,6,8\) respectively for energy \(H=-0.15\), forward in time.

Fig. 19
figure 19

LDs for \(\tau=2,4,6,8\) respectively for energy \(H=-0.15\) backward and forward in time.

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Crossley, R., Agaoglou, M., Katsanikas, M. et al. From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential. Regul. Chaot. Dyn. 26, 147–164 (2021). https://doi.org/10.1134/S1560354721020040

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