Saddle-node periodic orbit

The general setting of the problem of global bifurcations on the disappearance of a saddle-node periodic orbit is as follows. Assume that there exists a saddle-node periodic orbit and that all trajectories which tend to this periodic orbit as i — 00 also tend to it as -f-oo along some center manifold. In other words, assume that the unstable manifold of the saddle-node returns to the saddle-node orbit from the side of the node region. In this case, either ... 

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. 

Note that in the n-dimensional case, where n > 4, other topological configurations of may be realized. Such saddle-node bifurcations will definitely lead the system out of the class of systems with simple dynamics. For example, it is shown in [139, 152] that a hyperbolic attractor of the Smale-Williams type may appear just after the disappearance of a saddle-node periodic orbit. ... 

Fig. 10.2.4. Saddle-node periodic orbits in (a) the cycle L is stable in the interior region and unstable in the exterior region. When hp < 0, it is attractive for the point inside it, and repelling for outer trajectories (b).

When the map (11.3.1) is a Poincare map near a periodic orbit of some system of ODE s, the fixed point under consideration corresponds to a saddle-node periodic orbit (at /x = 0). The phase portraits for this case are shown in Figs. 11.3.5-11.3.7. 

Fig. 11.3.7. Scenario of a saddle-node bifurcation of periodic orbits in The stable periodic orbit and the saddle periodic orbit in (a) coalesce at /i = 0 in (b) into a saddle-node periodic orbit, and then vanishes in (c). The unstable manifold of the saddle-node orbit in (c) is homeomorphic to a semi-cylinder. A trajectory following the path in (b) slows down along a transverse direction near the virtual saddle-node periodic orbit (i.e., the ghost of the saddle-node orbit in (b)) so that its local segment is similar to a compressed spring.

As /i increases within a resonant zone other periodic orbits with the same rotation number M/N may appear. In some cases, the boundary of the resonant zone can lose its smoothness at some points, like in the example shown in Fig. 11.7.4 here, the resonant zone consists of the union of two regions D and Z>2 corresponding to the existence of, respectively, one and two pairs of periodic orbits on the torus. The points C and C2 in Fig. 11.7.4 correspond to a cusp-bifurcation. At the point S corresponding to the existence of a pair of saddle-node periodic orbits the boundary of the resonant zone is non-smooth. 

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. 

Let us consider a one-parameter family of n-dimensional C -smooth (r > 2) systems having a saddle-node periodic orbit L at /i = 0. We assume that /jL is the governing parameter for local bifurcations. Thus (recall Fig. 11.3.7), for /i < 0, there exist stable and saddle periodic orbits which collapse into one orbit L at /X = 0. The local imstable set is homeomorphic to a half-cylinder... 

Fig. 12.2.3. The structure of intersection of the unstable manifold of a saddle-node periodic orbit L with a solid-torus-like cross-section Sq in the case m = 2. A trace of the intersection is a doubly-twisted curve 1. Consequently, it has at least two intersections with each level

Let us consider next the bifurcation of the saddle-node periodic orbit L in the case where the unstable manifold is a Klein bottle, as depicted in Fig. 12.3.1, i.e. when the essential map has degree m = -1. By virtue of Theorem 12.3, if is smooth, then a smooth invariant attracting Klein bottle persists when L disappears. In its intersection with a cross-section So, the flow on the Klein bottle defines a Poincare map of the form (see (12.2.26))... 

Fig. 12.3.1. A saddle-node periodic orbit on the Klein bottle.

Let us now consider the case where the global unstable set of the saddle-node periodic orbit L is not a manifold, but has the structure like shown in Fig. 12.4.1. This means that the integer m which determines the homotopy class of the curve fl jSq in the cross-section 5q x = —d is... 

Theorem 12.9. Consider a one-parameter family of dynamical systems which has a saddle-node periodic orbit L at = 0 such that all orbits in the global unstable set tend to L as t -foo, but do not lie in W[. Let the essential map satisfy m = 0 and fo (p) < 1 for all (p. Then after disappearance of the saddle-node for /i > 0, the system has a stable periodic orbit non-homotopic to L in U) which is the only attractor for all trajectories in U. 

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... 

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).

This section addresses the question on the local behavior of the flow near a saddle-node periodic orbit. Since the d3mamics in the directions transverse to the center manifold is trivial (it is a strong contraction), we restrict our consideration to the system on the center manifold ... 

A saddle-node periodic orbit (fold bifurcation) = 0, l2 e) 0. 

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-... 

Verify that for ci = C3 = 1/6 it vanishes at the point (a 1.72886, b 1.816786), labeled by Li = 0 on the Andronov-Hopf curve in Fig. C.2.1. This is the point of codimension two from which a curve of saddle-node periodic orbit originates. ... 

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