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 computation 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.
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All data and models generated or used during the study appear in the submitted article, and code generated during the study is available from the corresponding author by request.
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This work was supported by the National Natural Science Foundation of China under Grants No. 11775035 and also by the Fundamental Research Funds for the Central Universities with Contract No.2019XD-A10.
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Wang, D., Lan, Y. A reduced variational approach for searching cycles in high-dimensional systems. Nonlinear Dyn 111, 5579–5592 (2023). https://doi.org/10.1007/s11071-022-08130-x
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DOI: https://doi.org/10.1007/s11071-022-08130-x