DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING

作者：Coy Benjamin^*

来源：International Journal of Bifurcation and Chaos, 2012, 22(1): 1230001.

DOI：10.1142/S0218127412300017

• Unstable periodic orbit
• locally linear embedding
• linking number

An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

• 出版日期2012-1