Poincare-Andronov—Hopf bifurcation

An interesting case is the so-called Poincare-Andronov-Hopf bifurcation, which implies that for a single unstable steady state in a two-dimensional case when the variables (concentrations of intermediates) cannot be infinite or negative, the only possible attractor is a Hmit cycle. Such limit cycle is then manifested in selfroscillations of concentrations. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

Eventually, aU of them are based on the methods of general qualitative theory of differential equations developed by Poincare more than a century ago [47]. This theory was essentially developed by Andronov in 1930s [48] and, finally, after Hopf s theorem on bifurcation appeared in 1942 [49] it became a self-consistent branch of mathematics. This subject is currently known luider several names Poincare-Andronov s general theory of dynamic systems theory of non-linear systems theory of bifurcation in dynamic systems. Although the first notion is, in our opinion, the most exact one, we will use the term bifurcation theory , or BT, for the sake of brevity. 

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See also in sourсe #XX -- [ ]

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