Journal
PHYSICAL REVIEW E
Volume 105, Issue 1, Pages -Publisher
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevE.105.014217
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Funding
- Swiss National Science Foundation (SNSF) [200021-160088]
- European Research Council (ERC) under the European Union [865677]
- State Secre-tariat for Education, Research and Innovation SERI via the Swiss Government Excellence Scholarship
- European Research Council (ERC) [865677] Funding Source: European Research Council (ERC)
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This study proposes a matrix-free method for computing periodic orbits in high-dimensional spatiotemporal chaotic systems. The method is globally convergent and unaffected by exponential error amplification. Experimental results demonstrate the robustness and convergence of the method.
Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to highdimensional spatiotemporal systems including fluid turbulence is supported by nonchaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatiotemporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification, or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent, and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatiotemporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.
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