Catastrophe saddle node bifurcation

Virtual bifurcations of such kind were named the blue sky catastrophes by R. Abraham. The first example of a blue sky catastrophe was constructed by Medvedev [95] for the saddle-node bifurcation on a Klein bottle. The most important feature of Medvedev s example is that the periodic orbit whose length and period are constantly increasing as /i -hO remains stable and does not undergo any bifurcation for all small /x > 0. Theorem 12.8 shows that this is only possible in the case fo (p) = 0, which means that all points (except for the two fixed points) of the essential map are of period two. 

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

The saddle node catastrophe and the Hopf bifurcation may be shown to be structurally stable. Certain additional conditions (see Sections 5.5.2.2, 5.5.2.3) are imposed on the transcritical bifurcation and the pitchfork bifurcation. The system is structurally stable under perturbations not disturbing these additional conditions on the other hand, when arbitrary... 

A totally diflFerent situation becomes possible in the case where the system does not have a global cross-section, and when is not a manifold. In this case (Sec. 12.4), the disappearance of the saddle-node periodic orbit may, under some additional conditions, give birth to another (unique and stable) periodic orbit. When this periodic orbit approaches the stability boundary, both its length and period increases to infinity. This phenomenon is called a hlue-sky catastrophe. Since no physical model is presently known for which this bifurcation occurs, we illustrate it by a number of natural examples. 

To study such bifurcations one should understand the structure of the limit set into which the periodic orbit transforms when the stability boundary is approached. In particular, such a limit set may be a homoclinic loop to a saddle or to a saddle-node equilibrium state. In another bifurcation scenario (called the blue sky catastrophe ) the periodic orbit approaches a set composed of homoclinic orbits to a saddle-node periodic orbit. In this chapter we consider homoclinic bifurcations associated with the disappearance of the saddle-node equilibrium states and periodic orbits. Note that we do not restrict our attention to the problem on the stability boundaries of periodic orbits but consider also the creation of invariant two-dimensional tori and Klein bottles and discuss briefly their routes to chaos. 

We will study the case m = 0 in Sec. 12.4 in connection with the problem of the blue sky catastrophe . In the case m > 2, infinitely many saddle periodic orbits are born (see Theorem 12.5) when the saddle-node disappears moreover, even hyperbolic attractors may arise here (see [139]). We do not discuss such kind of bifurcations in this book. 

Since the return time from/to the cross-section S (i.e. the period of L j) grows proportionally to cj(/i), it must tend to infinity as /i —H-oo (see Sec. 12.2 if L is a simple saddle-node, then the period grows as tt/V/IZ ). Since the vector field vanishes nowhere in 17, it follows that the length of must tend to infinity also. Since L, does not bifurcate when /x > 0, we have an example of the blue sky catastrophe [152]. 

If the saddle-node L is simple, then all neighboring systems having a saddle-node periodic orbit close to L constitute a codimension-one bifurcational surface. By construction (Sec. 12.2), the function /o depends continuously on the system on this bifurcational surface. Thus, if the conditions of Theorem 12.9 are satisfied by a certain system with a simple saddle-node, they are also satisfied by all nearby systems on the bifurcational surface. This implies that Theorem 12.9 is valid for any one-parameter family which intersects the surface transversely. In other words, our blue sky catastrophe occurs generically... 

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