Jupo

A julia package for numerically finding Unstable Periodic Orbits (UPOs) in dynamical systems.

This library accompanies the study currently under review:

Hamilton, O., Demaeyer, J., Crucifix, M. & Vannitsem, S. (2025) Using Unstable Periodic Orbits to Understand Blocking Behaviour in a Low Order Land-Atmosphere Model [under review] https://arxiv.org/abs/2503.02808

It is planned to improve this library and then add to ChaosTools.jl, see issue 347.

Examples

Here are some of the UPOs of the Lorenz 63 system:

Usage

To find the UPOs of a given dynamical system, define a System, as the collection of the dynamical rule, the jacobian, and some other information about the system:

lorenz = System(
        name="Lorenz", 
        f=lorenz!, 
        jac=lorenz_j!, 
        ndim=3, 
        p=[10., 28., 8/3],
        p_names=["sig", "r", "b"]
        )
end

this is passed to the method of choice for finding the UPOs, along with a Vector containing the initial conditions and the initial period: ic = [x_0; T].

upo = find_upo_nm(lorenz, ic)

This will produce a UPO_sol struct, which contains:

• success (whether the algorithm ran successfully)
• ic (initial conditions)
• period
• system (summary of system parameters)
• floquet_multipliers

Currently only the newton method finder is working. In the future the stabilising transforms method will be added.