Big lobe condition

Definition 12.1, The set satisfies the refined) big lobe condition if each leaf of intersects Wf at least twice in the node region Fig. 12.2.4). 

Fig. 12.2.4. Geometrical interpretation of the big lobe condition — each leaf of the strong stable foliation F must cut through not fewer than two points.

Theorem 12.5. If the big lobe condition is satisfied then the system has infinitely many saddle periodic orbits for all small fi > 0. 

Note that this theorem holds independently of the type of the topological structure of (i.e. independently of the degree m of the essential map). Obviously, the big lobe condition is always satisfied when m > 2, so the disappearance of the saddle-node always implies chaos in this case. Therefore, for the rest of this chapter we will be focusing on the cases m = 1,0, —1. 

Note the difference between the transition to chaos under the big lobe condition and without it in the second case the intervals Ai of chaotic dynamics may, in principle, interchange with the intervals where the system has only finitely many saddle and stable periodic orbits [151]. According to the reduction principle (Theorem 12.4), this occurs if, within some interval of u, the essential map... 

So, the results of Theorems 12.3, 12.5 and 12.7 are summarized as follows IfW is a smooth toruSy then a smooth attracting invariant torus persists after the disappearance of the saddle-node L. If is homeomorphic to a torus but it is non-smoothy then chaotic dynamics appears after the disappearance of L, Herey either the torus is destroyed and chaos exists for all small /i > 0 the big lobe condition is sufficient for that)y or chaotic zones on the parameter axis alternate with regions of simple dynamics. 

Moreover, it follows from Theorems 12.5 and 12.6 that chaotic behavior may also be possible if the condition /o(< ) < 1 of Theorem 12.9 is not met. In particular, the big lobe condition (Sec. 12.2) is here equivalent to the existence of a leaf of the strong-stable foliation which intersects at least two connected components of the intersection of with the local cross-section S (p = constant (see Fig. 12.4.2). In terms of the essential map this condition is written as... 

Fig. 12.4.2. The option of chaotic behavior resulted from the disappearance of a saddle-node fixed point of the corresponding Poincare map, assuming the contraction condition is not satisfied but the big lobe condition holds each leaf of the foliation must intersect at least two of the connected components of n S.

Let us now consider the question concerning what happens when is non-smooth. For the first time, this question was studied in [3] where it was discovered that the possibility of the breakdown of the invariant manifold causes an onset of chaos at such bifurcations. In particular, sufficient conditions (the so-called big lobe and small lobe conditions) were given in [3] for the creation of infinitely many saddle periodic orbits upon the disappearance of a saddle-node in the non-smooth case. Subsequent studies have shown that these conditions may be further refined so we may reformulate them as follows. 

Let us now consider briefly the question on what may happen if is a non-smooth Klein bottle. Since Theorems 12.5 and 12.6 are applicable in this case, it follows that when /x > 0, chaos may appear when the big lobe or the small lobe condition is satisfied. However, a direct analogue of Theorem 12.7 does not exist here because of the following possibility ... 

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