A reduced variational approach for searching cycles in high-dimensional systems

TL;DR

This paper introduces an improved variational method to efficiently find periodic orbits in high-dimensional chaotic systems by leveraging inertial manifolds, reducing computational costs while maintaining exponential convergence.

Contribution

An enhanced variational approach that accelerates the search for cycles in high-dimensional systems using a loop evolution equation and local coordinate modifications.

Findings

Effective reduction in storage and computation time.

Successful demonstration on well-known high-dimensional systems.

Maintains exponential convergence and stability.

Abstract

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.