Boundary Layer Problems Exhibiting Resonance

Overstability and resonance

“…In our context, a solution bounded on an interval I c [a, b] is a function defined and bounded on a rectangle of the form I x ]0, Eo], Eo > 0 such that for every ~, x ~--~ u(x, E) is a solution of (1) for that value E of the parameter. Note that no regularity condition with respect to c is required.…”

“…We call slow curve a solution uo of the reduced equation 0 = u, 0). We suppose that the interval ]a, b[ contains 0 and that equation (1) admits a slow curve that is attractive for x 0 and repulsive for x > 0. We will show that canard solutions [4] having bounded derivatives of any order Keywords : Resonance -Canard solution -Overstability -Singular perturbation.…”

Uniform Solution of Boundary Layer Problems Exhibiting Resonance

“…Introduction. The problem discussed in this paper is one whose anomalies became well known with a paper by Ackerberg and O'Malley [1]. Let a(e) [-1, b].…”

The Sufficiency of the Matkowsky Condition in the Problem of Resonance

“…Indeed, Ackerberg and O'Mally [2] found that unless / = -g(0,0)//x(0,0) is a nonnegative integer, the solution y(x, e) of (1.1) and (1.2) converges to zero as e tends to zero, and the term " resonance" is applied to those cases when the limit of y(x, e) as e tends to zero is a nontrivial solution of (1.3). Watts [16] considered the problem with/(x, e) = -x and g(x, e) -I + x, where / is a nonnegative integer, and showed that the above condition is not sufficient for resonance.…”

“…Note that if a > a2 then (1.14) exhibits resonance on [-a, a]. where/» is a positive odd integer, Sx U S2 U • • • U Sv = {e: 0 <| e | < p0}, and (i) 8j(e) is holomorphic in S-= (e: a} < arg e < fy, 0 < | e | < p0}; (ü) Sj(e) is asymptotically zero as e tends to zero in Sf, (2) it follows from the connection formulas of Weber's equation ( In this paper we consider the general case where D is a domain in the x-plane containing the real interval [-a, b]. The main problem is to remove from Sibuya's result the assumption that D is a disc.…”