The CAPD library has been used in several articles in which chaotic dynamics, bifurcations, heteroclinic/homoclinic solutions and periodic orbits were studied.

The existence of simple choreographies

A simple choreography is defined as a collision free solution of the n-body problem in which all the bodies move on the same closed curve with the same time shift.

Using validated solvers from the CAPD for ODE's and first order variational equations we were able to prove the existence of some choreographies for various numbers of bodies. Also the linear stability of the well known figure eight choreography was verified.

  • T. Kapela, P. Zgliczyński, The existence of simple choreographies for the N-body problem - a computer assisted proof, Nonlinearity, vol. 16(2003), 1899-1918.
  • T. Kapela, C. Simo, Computer assisted proofs for non-symmetric planar choreographies and for stability of the Eight, Nonlinearity 20 (2007) 1241-1255.
All the choreographies presented below are proved to exist.

Figure eight
The figure eight choreography. We proved its existence and linear stability - all the eigenvalues of a return map are on the unit circle.

Gerver orbit
The Gerver orbit.

8 body orbit
Symmetric orbit with 8 bodies.

Nonsymetric 6 bodies
Nonsymmetric choreography with 6 bodies.

Nonsymetric 7 bodies
Nonsymmetric choreography with 7 bodies.