J. Godelle & C. Letellier
Symbolic sequence statistical analysis for free liquid jets,
Physical Review E, 62 (6), 7973-7981, 2000. Online
Free liquid jets are investigated here as nonlinear dynamical systems. A scalar time series corresponding to the time evolution of the jet diameter is then used to investigate the underlying dynamics in terms of reconstructed phase portraits, Poincaré sections, and first-return maps. Particular attention is paid to characterizing the behavior using symbolic sequence statistics that enable different atomization regimes to be distinguished. Such statistics are first applied on theoretical maps to support the results obtained on the jet dynamics.
O. Ménard, C. Letellier, J. Maquet & G. Gouesbet
Map modeling by using rational functions,
Physical Review E, 62 (5), 6325-6331, 2000. Online
Rational functions are not very useful for obtaining global differential models because they involve poles that may eject the trajectory to infinity. In contrast, it is here shown that they allow one to significantly improve the quality of models for maps. In such a case, the presence of poles does not involve any numerical difficulty when the models are iterated. The models then take advantage of the ability of rational functions to capture complicated structures that may be generated by maps. The method is applied to experimental data from copper electrodissolution.
O. Ménard, C. Letellier, J. Maquet, L. Le Sceller & G. Gouesbet
Analysis of a non synchronized sinusoidally driven dynamical system,
International Journal of Bifurcation & Chaos, 10 (7), 1759-1772, 2000. Online
A nonautonomous system, i.e. a system driven by an external force, is usually considered as being phase synchronized with this force. In such a case, the dynamical behavior is conveniently studied in an extended phase space Rm which is the product of the phase space Rm of the undriven system by an extra dimension associated with the external force. The analysis is then performed by taking advantage of the known period of the external force to defi-ne a Poincaré section relying on a stroboscopic sampling. Nevertheless, it may so happen that the phase synchronization does not occur. It is then more convenient to consider the nonautonomous system as an autonomous system incorporating the subsystem generating the driving force. In the case of a sinusoidal driving force, the phase space is Rm+2 instead of the usual extended phase space Rm x S1. It is also demonstrated that a global model may then be obtained by using m+ 2 dynamical variables with two variables associated with the driving force. The obtained model characterizes an autonomous system in contrast with a classical input/output model obtained when the driving force is considered as an input.
C. Letellier, S. Meunier-Guttin-Cluzel & G. Gouesbet
Topological invariants in period-doubling cascades,
Journal of Physics A, 33, 1809-1825, 2000. Online
Topological characterization is a useful description of dynamical behaviours as exemplified by templates which synthesize the topological properties of very dissipative chaotic attractors embedded in tri-dimensional phase spaces. Such a description relies on topological invariants such as linking numbers between two periodic orbits which may be viewed as knots. These invariants may, therefore, be used to understand the structure of dynamical behaviours. Nevertheless, as an example, the celebrated period-doubling cascade is usually investigated by using total twists which are not topological invariants. Instead, we introduce linking numbers between an orbit, viewed as the core of a small ribbon, and the edges of the ribbon. Such a linking number (which is in fact the Calugareanu invariant) is related to the total twist number and the number of writhes of the ribbon. A second topological invariant, called the effective twist number, is also introduced and is useful for investigating period-doubling cascades. In the case of a trivial suspension of a horseshoe map, this topological invariant may be predicted from a symbolic dynamics with the aid of framed braid representations.
C. Lainscsek, C. Letellier, J. Kadtke, G. Gouesbet & F. Schurrer
Equivariance identification using delay differential equations,
Physics Letters A, 265 (4), 264-273, 2000. Online
The construction of delay differential equations (DDE). from recorded data has been shown to be relevant to time series analysis. In particular, it allows one to detect the presence of a deterministic component in the signal. Also, it provides an opportunity to generate classification schemes in which a given signal may be mapped onto an equivalence class. In this Letter, this technique is applied to the important problem of symmetry detection in time series. Testing 24 numerical and 2 experimental time series, it is demonstrated that DDEs allow one to quantify distinction between equivariant and invariant time series.