Searching for Periodic Orbits

It ha been found that the long scattering orbits (large scattering times) are those that come in the neighborhood of periodic orbits. These scattering trajectories spend time close to unstable periodic orbits before going away since these periodic orbits are unstable.

Finding periodic orbits in open systems (chaotic scattering) or closed systems is notoriously difficult. This is particularly the case for unstable periodic orbits. Tracking these orbits is difficult since any slight deviation from the true orbit will take the trajectory away from the true one by definition of an unstable orbit.

The search for periodic orbits in a chaotic region can proceed via various ways.

• Visually look for region where a chaotic trajectory seems to stay in the neighborhood of a periodic orbit. This may point out to a set of initial conditions from which to start a brute force search.

• Perform a brute force search. This is done via launching numerous trajectories from a grid of initial conditions and recognizing which of these trajectories close on themselves. This is difficult and time consuming in high dimensional systems.

• Recently (2000), Henry, Watt and Wearne proposed a lattice refinement technique to find periodic orbits and demonstrated the usefulness of the method for Maps and Flows. The original paper or a local copy are available in pdf format.

The C code search_for_orbits.c implements the lattice refinement search algorithm for the Three Hills Potential used as scatterer above.

The salient features of the code are

• It is based on a RK4 solution of the scattering problem - the calculation of scattering trajectories is the same as the calculation of a periodic orbit

• The solution of the ODEs is in a separate C function (ODE_solve) for clarity

• ODE_solve assumes initial conditions of the type ( $x, y, v_x, v_y$) = ( $x_o, 0, 0, {v_y}_o$)

• ODE_solve computes the trajectory until a point $y = 0$ is reached with the constraint that $v_y$ be of the same sign as the initial ${v_y}_o$ for the first time or for hits times. This defines a Poincare Surface of Section (SoS).

• This choice for the SoS exploits +y / -y symmetry of the potential function. Other choices ae possible.

• ODE_solve sharpens the coordinates of the point of intersection of the trajectory with the $y = 0$ plane via a linear interpolation.

• The function search implements the search algorithm from a grid of initial conditions covering ( $x, y, v_x, v_y$) = ( $x_0, 0, 0, {v_y}_0$)

• This search function is recursively called to refine the search when any of the 4 projected crossings from the vertices of each cell on the search lattice fall within the same cell. These projections are calculated via affine projections as proposed by Henry et Al..

The output of the code are suggestions for sets of initial conditions that should result in periodic orbits.