C. Letellier, G. Gouesbet & N. Rulkov
Topological analysis of chaos in equivariant electronic circuits,
International Journal of Bifurcation & Chaos, 6 (12B), 2531-2555, 1996. Online _
Abstract: Chaotic oscillations in an electronic circuit are studied by recording simultaneously two time series. The chaotic dynamics is characterized by using the topological analysis. A comparison with two models is also discussed. Some prescriptions are given in order to take into account the symmetry properties of the experimental system to perform the topological analysis.
C. Letellier & G. Gouesbet
Topological characterization of reconstructed attractors modding out symmetries,
Journal de Physique II, 6, 1615-1638, 1996. Online
Abstract: Topological characterization is important in understanding the subtleties of chaotic behaviour. Unfortunately it is based on the knot theory which is only efficiently developed in 3D spaces (namely R3 or in its one-point compactification S3). Consequently, to achieve topological characterization, phase portraits must be embedded in 3D spaces, i.e. in a lower dimension than the one prescribed by Takens’ theorem. Investigating embedding in low-dimensional spaces is, therefore, particularly meaningful. This paper is devoted to tridimensional systems which are reconstructed in a state space whose dimension is also 3. In particular, an important case is when the system studied exhibits symmetry properties, because topological properties of the attractor reconstructed from a scalar time series may then crucially depend on the variable used. Consequently, special attention is paid to systems with symmetry properties in which specific procedures for topological characterization are developed. In these procedures, all the dynamics are projected onto a so-called fundamental domain, leading us to the introduction of the concept of restricted topological equivalence, i.e. two attractors are topologically equivalent in the restricted sense, if the topological properties of their fundamental domains are the same. In other words, the symmetries are moded out by projecting the whole phase space onto a fundamental domain.
C. Letellier, G. Gouesbet, F. Soufi, J. R. Buchler & Z. Kollath
Chaos in variable stars : topological analysis of W Vir model pulsations,
Chaos, 6 (3), 466-476, 1996. Online
The topological characterization of chaos is applied to the irregular pulsations of a model for a star of the W Virginis type, computed with a state‐of‐the‐art numerical hydrodynamical code. The banded W Vir attractor is found to possess an additional twist when compared to the Rössler band. It is shown that the stellar light‐curve contains the same dynamical information about the attractor as the stellar radius or as the radial velocity variations.
C. Letellier, P. Dutertre, J. Reizner & G. Gouesbet
Evolution of multimodal map induced by an equivariant vector field,
Journal of Physics A, 29, 5359-5373, 1996. Online
It has been shown that the topological characterization of an equivariant system should preferably be achieved by working in a fundamental domain generated by the symmetry properties appearing in the phase space. In this paper, we discuss the case when the equivariance of the studied system is taken into account to study the evolution of the population of periodic orbits when a control parameter is varied. The Burke - Shaw system is considered here as an example. It is shown that the equivariance of this system may be used to reduce the multimodal first-return map in a Poincaré section to a unimodal map. A relationship between four-symbol sequences and two-symbol sequences is given. The non-trivial evolution of the orbit spectrum of a multimodal map is then predicted from the much simpler unimodal map to which the multimodal map reduces.
L. Le Sceller, C. Letellier & G. Gouesbet
Global vector field reconstruction taking into account a control parameter evolution,
Physics Letters A, 211 (4), 211-216, 1996. Online
A global vector field reconstruction method including a control parameter dependence is derived and tested with the Rössler model. The reconstructed model is checked by comparing its bifurcation diagram with the one of the original system.