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UPOsHam is a Python package for computing unstable periodic orbits in two degrees of freedom Hamiltonian systems
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This Python package is a collection of three methods for computing unstable periodic orbits in Hamiltonian systems that model a diverse array of problems in physical sciences and engineering. 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) (Wiggins 2016). The three methods (Pollak, Child, and Pechukas 1980,Koon et al. (2011),Naik and Wiggins (2019)) described below have been implemented for three Hamiltonian systems of the form kinetic plus potential energy and a brief description can be found in the paper.pdf. The scripts are written as demonstration of how to modify and adapt the code for a problem of interest.


Clone/download the git repository using

$ git clone
$ cd UPOsHam
$ pip install -r requirements.txt

and check the modules shown in requirements.txt are installed using conda/pip. Specific problems can be imported as modules for use in further analysis.


Comparison of the three methods for the coupled quartic Hamiltonian Comparison of the results (data is located here) obtained using the three methods for the coupled quartic Hamiltonian can be done using

>> run

and the generated figure is located here

To obtain the unstable periodic orbits for a specific model Hamiltonian using a specific method, one uses

>> run

Other tests can be performed by running the cells in the run_alltests.ipynb

Copyright and License

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.


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.

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). American Physical Society: 022204. doi:10.1103/PhysRevE.100.022204.

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. doi:10.1063/1.439276.

Wiggins, Stephen. 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. doi:10.1134/S1560354716060034.

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