Near Tangent Dynamics in a Class of Hamiltonian Impact Systems

On the structure of Hamiltonian impact systems

“…By the definition of H tmin,dc (q 2 ), for H < H tmin,dc min , the wall does not intersect the Hill region. Since, by (30), the two global minima of V are within the billiard domain, and thus H tmin,dc min is larger than the global minima, the corresponding leaves are in the billiard by claim 2 of proposition 4.1 and the first item of the proposition follows.…”

“…We demonstrate the application of this construction for finding the IEMBD to the case of the DC Hamiltonian (6) with a slanted wall (q w (q 2 ) = cot α • q 2 , q 2 ) with α ∈ (0, π 2 )). For concreteness, we consider the Hamiltonian (6) when the extrema points of the Duffing-center potential are within the billiard domain: q 1s 1, q 1s − q 2c cot α > 1 ω = 1 (30) and choose parameters in a certain range so that the critical energies obey certain ordering; let:…”

“…Indeed, in the integrable limits, some of the critical energies above which the asymptotic analysis is applicable become infinite; for α → 0, H 1,± → V 2 (0) + H min = − 1 4 + ω 2 2 • (q 2c ) 2 and H 0 → ∞ whereas for α → π 2 , H 1,− → ∞ and H 0 → V 1 (0) = − 1 2 • (q 1s ) 2 + 1 4 • (q 1s ) 4 , see figure 17. See [30] for some detailed calculations of various asymptotic limits.…”