Quadratic tangency

This result is due to Palis, who had fotmd that two-dimensional diffeomor-phisms with a heteroclinic orbit at whose points an unstable manifold of one saddle fixed point has a quadratic tangency with a stable manifold of another saddle fixed point can be topologically conjugated locally only if the values of some continuous invariants coincide. These continuous invariants are called moduli. Some other non-rough examples where moduli of topological conju-gacy arise are presented in Sec. 8.3. 

However, a similar classification of two-dimensional diffeomorphisms, or of three-dimensional fiows, is not that trivial. Let us illustrate this with an example. Consider a diffeomorphism T which has two saddle fixed points 0 and O2 with the characteristic roots )Ai) < 1 and i > 1 at (z = 1,2). Suppose that Wq and have a quadratic tangency along a heteroclinic orbit as shown in Fig. 8.3.1. The quadratic tangency condition implies that all similar diffeomorphisms form a surface of codimension-one in the space of all diffeomorphisms with a C -norm. 

The value 9 is also a modulus of topological equivalence in the case of a three-dimensional fiow which has two saddle periodic orbits such that an unstable manifold of one periodic orbit has a quadratic tangency with a stable manifold of another orbit along a heteroclinic trajectory. 

The non-smooth case appears, for example, when Wq touches the strong-stable manifold Wq, as shown in Fig. 12.2.2. The latter, in turn, may be detected via a small time-periodic perturbation of a system with an on-edge homoclinic loop to a saddle-node (see the previous section). Generically, the non-transversality of with respect to is also preserved under small smooth perturbations (say, if the tangency between and the corresponding leaf of is quadratic). 

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