Blue sky catastrophe

A straightforward generalization of two-dimensional bifurcations was developed soon after. So were some natural modifications such as, for instance, the bifurcation of a two-dimensional invariant torus from a periodic orbit. Also it became evident that the bifurcation of a homoclinic loop in high-dimensional space does not always lead to the birth of only a periodic orbit. A question which remained open for a long time was could there be other codimension-one bifurcations of periodic orbits Only one new bifurcation has so far been discovered recently in connection with the so-called blue-sky catastrophe as found in [152]. All these high-dimensional bifurcations are presented in detail in Part II of this book. 

The boundaries of the second type correspond to the merging of a periodic orbit into an equilibrium state (Sec. 11.5) or to a homoclinic loop, or a blue-sky catastrophe (Chaps. 12 and 13). 

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. 

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. 

We see that Medvedev s example describes an extremely degenerate situation. A generic example of the blue sky catastrophe for a stable periodic orbit (when the degree m of the essential map is equal to zero) is given in the next section. 

Fig. 12.4.1. (a) Illustrates the mechanism of a blueunstable manifold returns to the saddle-node from the node region so that the circles of its intersection with the cross-section S tighten with each subsequent iterate, (b) The return map along... 

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... 

Note that the specific topological structure of is not yet sufficient for realizing a blue sky catastrophe there exists also a quantitative condition in Theorem 12.9 which is needed to ensure contraction. If this condition is violated, i.e. if fo 1 at some then infinitely many bifurcations occur in the region /i > 0, just like the cases considered in the preceding sections. Indeed, consider the lift of the map (12.4.1) onto... 

By following this recipe, a family of three-dimensional systems with analytically defined right-hand side has been explicitly designed which realizes the blue sky catastrophe [53]. This family is as follows... 

Fig. 12.4.3. A phemenological scenario of development of the blue sky catastrophe when the saddle-node equilibrium O disappears, the unstable manifold of the saddle-node periodic orbit L has the desired configuration, as the one shown in Fig. 12.4.1(a).

Another kind of examples where our blue sky catastrophe may appear naturally is given by singularly perturbed systems i.e. the systems of the form... 

In fact, the triggering from one stable branch to another is the most typical phenomenon in singularly perturbed systems, so one may encounter for our blue sky catastrophe every time when jumps between the branches of fast periodic orbits and fast equilibrium states are observed. 

The fourth and last situation corresponds to the blue sky catastrophe , i.e. when both period and length of the periodic orbit go to infinity upon approaching the stability boundary. This boundary is distinguished by the existence of a saddle-node periodic orbit under the assumption that all trajectories of the unstable set W ( ) return to as t -> -hoc, where W C ) n — 0. The tra-... 

The computing algorithms of most of these bifurcations have been well developed and can therefore be implemented in software we mention here the packages designed to settle these bifurcation problems LOCBIF [76], AUTO [46] and CONTENT [83]. The exception is the blue sky catastrophe, Despite the fact that it is a codimension-one boundary, this bifurcation has not yet been found in applications of nonlinear dynamics although an explicit mathematical model does exist [53]. 

One more codimension-one boundary of stability of periodic trajectories which corresponds to the blue sky catastrophe [152]. It may occur in n-dimensional systems where n > 3. 

Fig. 14.2.5. A sketch of the blue sky catastrophe the shape of the periodic orbit L(e) looks like a helix condensed near a saddle-node cycle.

C.6. 78. The Medvedev s construction of the blue-sky catastrophe on torus [95] is illustrated by Fig. C.6.11. It is supposed that there exists a pair... 

Sec. 12.4, find the blue-sky catastrophe in the modified Hindmarsh-Rose model of neuronal activity... 

Gavrilov, N. K. and Shilnikov, A. L. [1996] On a blue sky catastrophe model, Proc. Int. Conf. Comtemp. Problems of Dynamical Systems Theory, ed. Lerman, L. (Nizhny Novgorod State University Nizhny Novgorod). [1999] An Example of blue sky catastrophe, in Ams Transl Series II. Methods of qualitative theory of differential equations and related topics. (AMS, Providence, Rhode Island). 

---