Bifurcation of an equilibrium state with one zero exponent

Bifurcation of an equilibrium state with one zero exponent... 

We can now discuss the bifurcation of an equilibrium state with two zero characteristic exponents. This bifurcation is worth being distinguished because its analysis includes nearly all bifurcations of codimension one. 

The procedure for studying a fc-parameter family is similar to that for the one-parameter case firstly, divide the space of the parameters p into the regions of topologically equivalent behavior of trajectories, and study the system in each of these regions. Secondly, describe the boundaries of these regions (the bifurcation set), and finally study what happens at the bifurcation parameter values. We will see below that in the simplest cases (e.g. an equilibrium state with one zero or a pair of imaginary characteristic exponents, or a periodic orbit with one multiplier equal to 1 or to —1) one can almost always, except for extreme degeneracies, choose a correct bifurcation surface of a suitable codimension and analyze completely the transverse families. Moreover, all of these families turn out to be versal. 

In this section, we will discuss some algorithms for constructing normal forms. Due to the reduction principle, it is sufficient to construct the normal forms for the system on the center manifold only. Therefore, in order to consider bifurcations of an equilibrium state with a single zero characteristic root, we need a one-dimensional normal form. If it has a pair of zero characteristic exponents, one should examine the corresponding family of two-dimensional normal forms, and so on. 

For cases having an extra degeneracy (for example an equilibrium state with zero characteristic exponent and zero first Lyapunov value) the boundary of the stability region may lose smoothness at the point There may also exist situations where the boimdary is smooth but bifurcations in different nearby one-parameter families are different (i.e. there does not exist a versal one-parameter family, for example, such as the case of an equilibrium state with a pair of purely imaginary exponents and zero first Lyapunov value). In such cases the procedure is as follows. Consider a surface 971 of a smaller dimension (less than (p — 1)) which passes through the point and is a part of the stability boundary, selected by some additional conditions in the above examples the condition is that the first Lyapunov value be zero. If (fc — 1) additional conditions are imposed, then the surface 971 will be (P fc)-dimensional and it is defined by a system of the form... 

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