The Mathematical Institute, University of Oxford, Eprints Archive

When are projections also embeddings?

Moroz, I. M. and Letellier, C. and Gilmore, R. (2007) When are projections also embeddings? Physical Review E, 75 (4).

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Abstract

We study an autonomous four-dimensional dynamical system used to model certain geophysical processes.This system generates a chaotic attractor that is strongly contracting, with four Lyapunov exponents $\lambda_i$ that satisfy $\lambda_1+ \lambda_2+\lambda_3<0$, so the Lyapunov dimension is $D_L=2+|\lambda_3|/\lambda_1 < 3$ in the range of coupling parameter values studied. As a result, it should be possible to find three-dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior. We study mappings into $R^3$ to determine which can be used as embeddings to reconstruct the dynamics. We find dramatically different behavior in the two simplest mappings: projections from $R^4$ to $R^3$. In one case the one-parameter family of attractors studied remains topologically unchanged for all coupling parameter values. In the other case, during an intermediate range of parameter values the projection undergoes self-intersections, while the embedded attractors at the two ends of this range are topologically mirror images of each other.

Item Type:Article
Subjects:D - G > Geophysics
D - G > Dynamical systems and ergodic theory
O - Z > Ordinary differential equations
D - G > General topology
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:586
Deposited By:Irene Moroz
Deposited On:20 Apr 2007
Last Modified:20 Jul 2009 14:22

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