  Searching recurrent patterns in complex systems with high-dimensional phase
spaces is an important task in diverse fields. In the current work, an improved
scheme is proposed to accelerate the recently designed variational approach for
finding periodic orbits in systems with chaotic dynamics based on the existence
of inertial manifold widely observed in various spatially extended systems,
especially those with high dimensions. On the premise of keeping exponential
convergence of the variational method, an effective loop evolution equation is
derived to greatly reduce the storage and computing time. With repeated
modification of local coordinates and evolution of the guess loop being carried
out alternately, the rapid convergence and the stability of the reduction
scheme are effectively achieved. The dimension of local coordinate subspaces is
generally larger than the number of nonnegative Lyapunov exponents to ensure
the exponential convergence. The proposed scheme is successfully demonstrated
on several well-known examples and expected to supply a powerful tool in the
exploration of high-dimensional nonlinear systems.
