Reconstruction of Phase Portraits

One practical issue that arises immediately when experimental data are analyzed is the necessity of constructing a phase space to generate a portrait of the attractor. Sometimes it is not feasible or desirable to measure, simultaneously, more than one of the dynamical variables. Recall that simultaneous measurements are necessary for the construction of attractors, and the full phase space for any given system will consist of ail the dynamical variables and their velocities or derivatives. At least two such variables are needed to look at projections of the attractor in a lower dimensional phase space, so the measurement of only one dynamical variable does not seem, at first glance, to be enough for this task. 

However, a very useful technique for reconstructing the attractor in Ac absence of a multiple number of measured variables exists. An adequate representation of the attractor can be created by measuring just a single dynamical variable and creating an -dimensional phase space using Ae method of time delays. This method rests on the topological equivalence of portraits constructed from time-delayed readings to those constructed with derivatives. 

An illustration of this method is found in an investigation of the Be-lousov-Zhabotinskii (BZ) reaction carried out by Hudson and Mankin. ° The BZ reaction involves the bromination of an organic acid and proceeds through 

Br electrode reading, Pt electrode reading, time derivative of Pt electrode or 

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In conclusion of our short excursion into the qualitative theory of differential equations, we shall discussed the often-used term "bifurcation . It is ascribed to the systems depending on some parameter and is applied to point to a fundamental reconstruction of phase portrait when a given parameter attains its critical value. The simplest examples of bifurcation are the appearance of a new singular point in the phase plane, its loss of stability, the appearance (birth) of a limit cycle, etc. Typical cases on the plane have been discussed in detail in refs. [11, 12, and 14]. For higher dimensions, no such studies have been carried out (and we doubt the possibility of this). 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

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