Dynamical Systems
Now suppose a (complex) dynamical system is simply defined by the equation
where is the (-dimensional) curve parametrization of an orbit on the Riemann surface of the dynamical system, whereas is an complex matrix whose elements are complex functions of a -dimensional parameter . Even if (i.e. continuously depends on the parameter ) the Jordan normal form of the matrix is continuously deformed almost everywhere on but, in general, not everywhere: there is some critical submanifold of which the Jordan form abruptly changes its structure whenever the parameter crosses or simply “travels” around it (monodromy). Such changes substantially mean that several Jordan blocks (either belonging to different eigenvalues or not) join together to a unique Jordan block, or vice versa (i.e. one Jordan block splits in two or more different ones). Many aspects of Bifurcation theory for both continuous and discrete dynamical systems can be interpreted with the analysis of functional Jordan matrices. From the tangent space dynamics this means that the orthogonal decomposition of the dynamical systems' phase space changes and, for example, different orbits gain periodicity, or lose it, or shift from a certain kind of periodicity to another (such as period-doubling, cfr. Logistic map). In just one sentence, the qualitative behaviour of such a dynamical system may substantially change as the versal deformation of the Jordan normal form of .
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