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Proof of the transversality for the standard map. (English) Zbl 1485.37038

Summary: We consider the standard map. The stable and unstable manifolds of the saddle fixed point are proved to intersect transversely at the primary homoclinic point \(u\) for any parameter value. For the proof, we use the particular objects called the dominant axis (DA) and subdominant axis (SD), and symmetric periodic orbits that have orbital points on these axes. The periodic orbit named \(1/q\)-BE has the orbital point \(z_k\) at the intersection point of DA and SD. Let \(\xi_k\) be the slope of SD at \(z_k\). Take a sequence of \(z_k\) accumulating at \(u\) as \(k \rightarrow \infty \). We prove that the slope \(\xi_{} k\) monotonically decreases to the slope \(\xi_{} u(u)\) of the unstable manifold at \(u\) (the monotone inclination property). Using Ushiki’s theorem, the hyperbolic region (HR) is constructed. It is proved that the orbital point \(z_k\) in HR is a saddle point with reflection. Using the monotone inclination property and the properties of \(z_k\) in HR, the transversality at \(u\) for any value of \(a (> 0)\) is proved.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C27 Periodic orbits of vector fields and flows
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37C75 Stability theory for smooth dynamical systems
37D10 Invariant manifold theory for dynamical systems
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References:

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