TY - JOUR
T1 - Method for measuring unstable dimension variability from time series
AU - McCullen, N. J.
AU - Moresco, P.
PY - 2006
Y1 - 2006
N2 - Many of the results in the theory of dynamical systems rely on the assumption of hyperbolicity. One of the possible violations of this condition is the presence of unstable dimension variability (UDV), i.e., the existence in a chaotic attractor of sets of unstable periodic orbits, each with a different number of expanding directions. It has been shown that the presence of UDV poses severe limitations to the length of time for which a numerically generated orbit can be assumed to lie close to a true trajectory of such systems (the shadowing time). In this work we propose a method to detect the presence of UDV in real systems from time series measurements. Variations in the number of expanding directions are detected by determining the local topological dimension of the unstable space for points along a trajectory on the attractor. We show for a physical system of coupled electronic oscillators that with this method it is possible to decompose attractors into subsets with different unstable dimension and from this gain insight into the times a typical trajectory spends in each region.
AB - Many of the results in the theory of dynamical systems rely on the assumption of hyperbolicity. One of the possible violations of this condition is the presence of unstable dimension variability (UDV), i.e., the existence in a chaotic attractor of sets of unstable periodic orbits, each with a different number of expanding directions. It has been shown that the presence of UDV poses severe limitations to the length of time for which a numerically generated orbit can be assumed to lie close to a true trajectory of such systems (the shadowing time). In this work we propose a method to detect the presence of UDV in real systems from time series measurements. Variations in the number of expanding directions are detected by determining the local topological dimension of the unstable space for points along a trajectory on the attractor. We show for a physical system of coupled electronic oscillators that with this method it is possible to decompose attractors into subsets with different unstable dimension and from this gain insight into the times a typical trajectory spends in each region.
UR - http://www.scopus.com/inward/record.url?scp=33645765939&partnerID=8YFLogxK
U2 - 10.1103/PhysRevE.73.046203
DO - 10.1103/PhysRevE.73.046203
M3 - Article
AN - SCOPUS:33645765939
SN - 1539-3755
VL - 73
JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
IS - 4
M1 - 046203
ER -