Nonlinear Theory


In the general introduction we observed that the irruption of the “nonlinear” led to a profound transformation of a great number of scientific fields. The behaviors resulting from the “nonlinear” make it possible to better understand the natural phenomena considered as complex. The “nonlinear” introduced a set of concepts and tools, i.e. analysis and investigation instruments of dynamics generated by the “nonlinear”. We try to gather these investigation tools, knowing that today it is possible to say that there exists a kind of “conceptual unification” through the notions of attractors, period-doublings, subharmonic cascades, bifurcations, Lyapunov exponent, sensitive dependence on initial conditions, etc. Belonging to this set, there are techniques gathered under the name of nonlinear signal processing based mainly on the Takens theorem. The purpose is the reconstruction of the time series based on the topological equivalence concept. While saving an important amount of calculation, it allows the identification of the nonlinear nature1 of the studied time-series. The stakes are important: while working in a reconstructed phase-space of very low dimension, the objective is to reproduce the essential features of the original dynamics, without necessarily knowing the equations of the dynamical system which generated the studied series.

The topological equivalence enables us to study geometrical objects of low dimension which will provide the desired information about the original dynamics. In Economics, we can face this type of problems. Indeed, the number of variables implied in a dynamics is not necessarily known, just as the dimension of the system which can be infinitely large. Moreover, it is known that the attractor of large dynamical systems (even infinite) can have low dimensions. Consequently, with considerably reduced series, the study of low dimension objects can make it possible to extract information which we need to identify dynamics. These concepts of topo-logical equivalence and attractor of low dimensions lead to a major concept which is that of the capacity dimension (non-integer dimension). Indeed, it makes it possible, on the one hand to characterize the attractor which we have to face and, on the other hand, to highlight the difference between deterministic chaos and random walk. It is said for example that a “Brownian motion”2 has a capacity dimension equal to two, which is not necessarily the case of an apparent deterministic chaos. In connection with these (investigation) tools of nonlinear dynamics, it is interesting to highlight the results of recent work completed by Butkovskii, Kravtsov and Brush concerning the predictibility of the final state of a nonlinear model. The model used here is that of the logistic equation with its cascade of subharmonic bifurcations. The observed phenomenon is rather singular and paradoxical in relation to an a priori knowledge that we could have before the experimentation. It results from this work that the predictibility of the final state of the model, after the first bifurcation point, depends on the speed of change of its control parameter and on the background noise. A relation is established between the probability of the final state, the transition speed and the noise level. A critical value of the speed is highlighted by the experimentation. Indeed, when the speed is strictly higher than its critical value, for a given noise level, the probability is close to 1, whereas if the speed is lower than its critical value, the probability of the state is close to 1/2. Such works, which introduce the control speed in nonlinear dynamic models, could find interesting transpositions in economic models. Thus, this type of experimental for-malization introduces a further dimension for the effectiveness and the control of economic policies.


Periodic Orbit Lyapunov Exponent Hopf Bifurcation Unstable Manifold Homoclinic Orbit 
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© Springer-Verlag Berlin Heidelberg 2009

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