This Python package is a collection of three methods for computing unstable periodic orbits in two degrees of freedom Hamiltonian systems that model a diverse array of problems in physical sciences and engineering (Parker and Chua 1989; Wiggins 2003). The unstable periodic orbits exist in the bottleneck of the equipotential line V(x, y) = E and project as lines on the configuration space (x, y) for the Hamiltonian system of the form kinetic plus potential energy (Wiggins 2016). The three methods implemented here (and available under src directory) has been used in (Pollak, Child, and Pechukas 1980; De Leon and Berne 1981; Koon et al. 2000, 2011; Naik and Ross 2017; Ross et al. 2018; Naik and Wiggins 2019) for transition dynamics in chemical reactions, celestial mechanics, and ship capsize. We have chosen three Hamiltonian systems that have two wells connected by a bottleneck where the unstable periodic orbits exist for energy above the energy of the saddle equilibrium point. A brief description of these systems can be found in the paper.
Clone the git repository and install uposham as a module using
$ git clone git@github.com:WyLyu/UPOsHam.git
$ cd UPOsHam
$ python setup.py installThe setup.py should also install the modules listed in
requirements.txt
and which are typically installed using
$ pip install -r requirements.txt (or pip3 install -r requirements.txt)Now uposham module is available for import along with the methods and
example systems as submodules, for example
import uposham.differential_correction as diffcorr
import uposham.deleonberne_hamiltonian as deleonberneTo obtain the unstable periodic orbits for a two degree-of-freedom
Hamiltonian system, one needs the expressions described in the examples
section of the
documentation.
These expression can then be implemented as functions as shown in the
script and the function names of
the expressions can then be passed to a method’s function as in the
script. The example
script that computes the unstable periodic orbits (UPOs) for a given
Hamiltonian system is in the examples directory with names
<method>_<system>.py. The example scripts show how the method specific
functions take the system specific functions as input and are meant as
demonstration of how to set-up computation for a new system using the
available methods.
The user will need to write the system specific functions in a script, import this as module, import one of the method as module, and pass the system specific functions to the method specific functions.
Solving the De Leon-Berne Hamiltonian system using differential correction method
For example, to test the computation of unstable periodic orbits (the default setting computes the UPOs for 2 values of the total energy) for the De Leon-Berne Hamiltonian (1981) using the differential correction method, the demonstration script can be called from the root directory at the command-line using
$ python examples/differential_correction_deleonberne.pyThis will save data files in the root directory and save the plot in the ./tests/plots directory.
Fig. 1. Unstable periodic orbits for De Leon-Berne Hamiltonian
Solving the coupled quartic Hamiltonian system using differential correction method
To perform these computations using ipython, for example to obtain the unstable periodic orbits for the coupled quartic Hamiltonian using the differential correction, the user can execute:
$ ipython
>>> run ./examples/differential_correction_coupled.pyThis will save data files in the root directory and save the plot in the ./tests/plots directory.
Fig. 2. Unstable periodic orbits for the coupled quartic Hamiltonian
The uncoupled quartic Hamiltonian is an example of a system where the unstable periodic orbits can be obtained in explicit analytical form. This system is used for testing the numerical solutions obtained using the methods implemented in this package.
Unit test
Passing a test constitutes the following:
-
The Hausdorff distance between a given numerical and analytical orbits is within 10 − 8. This is implemented using Scipy’s spatial module.
-
The element wise comparison test of the orbits agree upto 6 decimal places. This is performed using Numpy’s testing module.
Other nonlinear examples systems can not be solved in this explicit analytical form and these are tested by performing plotting comparison of the orbits obtained using the methods. These are available in the Jupyter notebook.
Comparing the methods for the coupled quartic Hamiltonian
We demonstrate how the UPOs computed using the different methods compare with each other for a specific system: coupled quartic Hamiltonian.
Comparison of the results (data is located in the data directory) obtained using the three methods for the coupled quartic Hamiltonian can be done using
$ ipython
>>> run ./tests/compare_methods_coupled.pyand the generated figure is located in the tests/plots directory.
Guidelines on how to contribute to this package can be found here along with the code of conduct here for engaging with the fellow contributors.
We acknowledge the support of EPSRC Grant No. EP/P021123/1 and Office of Naval Research (Grant No. N00014-01-1-0769). The authors would like to acknowledge the London Mathematical Society and School of Mathematics at the University of Bristol for supporting the undergraduate research bursary 2019. We acknowledge contributions from Shane Ross for writing the early MATLAB version of the differential correction and numerical continuation code.
Copyright 2019 Wenyang Lyu, Shibabrat Naik, Stephen Wiggins.
All content is under Creative Commons Attribution CC-BY 4.0 and all the python scripts are under BSD-3 clause.
De Leon, N., and B. J. Berne. 1981. “Intramolecular Rate Process: Isomerization Dynamics and the Transition to Chaos.” The Journal of Chemical Physics 75 (7): 3495–3510. https://doi.org/10.1063/1.442459.
Koon, Wang Sang, Martin W. Lo, Jerrold E. Marsden, and Shane D. Ross. 2000. “Heteroclinic Connections Between Periodic Orbits and Resonance Transitions in Celestial Mechanics.” Chaos: An Interdisciplinary Journal of Nonlinear Science 10 (2): 427–69. https://doi.org/10.1063/1.166509.
Koon, W. S., M. W. Lo, J. E. Marsden, and S. D. Ross. 2011. Dynamical systems, the three-body problem and space mission design. Marsden books. https://doi.org/10.1142/9789812792617_0222.
Naik, Shibabrat, and Shane D. Ross. 2017. “Geometry of Escaping Dynamics in Nonlinear Ship Motion.” Communications in Nonlinear Science and Numerical Simulation 47 (June): 48–70. https://doi.org/10.1016/j.cnsns.2016.10.021.
Naik, Shibabrat, and Stephen Wiggins. 2019. “Finding Normally Hyperbolic Invariant Manifolds in Two and Three Degrees of Freedom with Hénon-Heiles-Type Potential.” Phys. Rev. E 100 (2): 022204. https://doi.org/10.1103/PhysRevE.100.022204.
Parker, T. S., and L. O. Chua. 1989. Practical Numerical Algorithms for Chaotic Systems. New York, NY, USA: Springer-Verlag New York, Inc. https://doi.org/10.1007/978-1-4612-3486-9.
Pollak, Eli, Mark S. Child, and Philip Pechukas. 1980. “Classical Transition State Theory: A Lower Bound to the Reaction Probability.” The Journal of Chemical Physics 72 (3): 1669–78. https://doi.org/10.1063/1.439276.
Ross, Shane D., Amir E. BozorgMagham, Shibabrat Naik, and Lawrence N. Virgin. 2018. “Experimental Validation of Phase Space Conduits of Transition Between Potential Wells.” Phys. Rev. E 98 (5): 052214. https://doi.org/10.1103/PhysRevE.98.052214.
Wiggins, Stephen. 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos. 2nd ed. Texts in Applied Mathematics 2. New York: Springer. [ISBN: 978-0-387-00177-7](ISBN: 978-0-387-00177-7).
———. 2016. “The Role of Normally Hyperbolic Invariant Manifolds (NHIMs) in the Context of the Phase Space Setting for Chemical Reaction Dynamics.” Regular and Chaotic Dynamics 21 (6): 621–38. https://doi.org/10.1134/S1560354716060034.