The analysis in this question is concerned with 2-D piece-wise linear maps that are noninvertible on one side. It was shown in (Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P., Piecewise-smooth Dynamical Systems, Applied Mathematical Sciences, Springer-Verlag 2008) that under some conditions that for the map (3.60) there exists a transverse homoclinic intersection between the stable and the unstable manifolds of the point b defined in the book page 160.
the stable manifold is:
S_{b}: x₂=-τ₂x₁+((τ₂)/(1-τ₂))μ
and the unstable manifold is given by:
U_{b}: x₂=((-δ₁(τ₂+1))/(τ₂τ₁+τ₁))x₁+((δ₁(τ₂+1))/(τ₂τ₁+τ₁))μ
and the intersection between them is:
S_{b}∩U_{b}={X_{Cr}=( ((δ₁-τ₁τ₂-τ₂δ₁)/((τ₂-1)(-δ₁+τ₁τ₂)))μ, ((τ₂²δ₁)/((τ₂-1)(-δ₁+τ₁τ₂)))μ)}
where all the parameters above are related to the map (3.60). All the detaills are given in the attached PDF file.
By the Smale–Birkhoff homoclinic theory there exists infinitely many
transverse homoclinic intersections between the stable and the unstable manifolds of b.
My **question** is: Is it possible for the map (3.60) to have infinitely many intersections in the line x₂=x₁
I am not sure that this fact is possible for this case.
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