We explore the possibility of extending the stabilizing transformations approach [ J. J. Crofts and R. L. Davidchack, SIAM J. Sci. Comput. (USA) 28, 1275 (2006) ]. to the problem of locating large numbers of unstable periodic orbits in high-dimensional flows, in particular those that result from spatial discretization of partial differential equations. The approach has been shown to be highly efficient when detecting large sets of periodic orbits in low-dimensional maps. Extension to low-dimensional flows has been achieved by the use of an appropriate Poincaré surface of section [ D. Pingel, P. Schmelcher, and F. K. Diakonos, Phys. Rep. 400, 67 (2004) ]. For the case of high-dimensional flows, we show that it is more efficient to apply stabilizing transformations directly to the flows without the use of the Poincaré surface of section. We use the proposed approach to find many unstable periodic orbits in the model example of a chaotic spatially extended system-the Kuramoto-Sivashinsky equation. The performance of the proposed method is compared against other methods such as Newton-Armijo and Levenberg-Marquardt algorithms. In the latter case, we also argue that the Levenberg-Marquardt algorithm, or any other optimization-based approach, is more efficient and simpler in implementation when applied directly to the detection of periodic orbits in high-dimensional flows without the use of the Poincaré surface of section or other additional constraints.
- nonlinear dynamical systems
- partial differential equations
Crofts, J. J., & Davidchack, R. L. (2009). On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows. Chaos, Solitons and Fractals, 19(3). https://doi.org/10.1063/1.3222860