Path-following strategy with consistent Jacobian for periodic solutions in multi-DOF nonlinear dynamic systems

We propose an enhanced pseudo-arclength path-following technique for recovering periodic solutions in high-dimensional nonlinear dynamic systems using the Poincaré map method. The key innovation is the direct computation of the Jacobian matrix within the time-marching algorithm used to obtain periodic orbits, including both the monodromy matrix and derivatives with respect to the continuation parameter. For smooth problems, the resulting Jacobian matrix is algorithmically exact: while the equations of motion are approximated using a user-selected time-integration scheme, the differentiation of the computed solution is performed exactly. This approach eliminates the need for numerical differentiation, significantly improving both the efficiency and robustness of the path-following process. Although the theoretical framework assumes differentiability, the method effectively handles piecewise smooth problems as well. Numerical tests demonstrate the superior performance of the proposed approach compared to traditional techniques that rely on numerical differentiation. To further validate its effectiveness and versatility, we present numerical examples involving the Finite Element discretization of three-dimensional problems, including shell structures.