ABSTRACT
An extra advantage of problem (132) is that the homotopic connections of basic patterns can be revealed more easily. Namely, as above, we consider ODE (132) in (−R,R), with a sufficiently large R > 0 to see the first l patterns, which are still exponentially small for y ≈ ±R. Consider the following homotopic path with the operators:
Aε(F ) ≡ F (4) + εF − F 3 = 0. (190) Consider the corresponding linearized operator:
A′ε(0) = D 4 y + εI. (191)
Let σ(D4y) = {λl > 0, l ≥ 0} be the discrete spectrum of simple eigenvalues of D4y > 0 in L
2((−R,R)) with the Dirichlet boundary conditions. The orthonormal eigenfunctions {ψl} satisfy Sturm’s zero property; we again refer to [108] for most general results. By classic bifurcation theory [252, p. 391], for such variational problems,
εl = −λl < 0 for l = 0, 1, 2, ... are bifurcation points, so there exists a countable number of branches emanating from these points (but we take into account the first ones). In order to identify the type of bifurcations, in a standard manner, setting ε = εl + s for |s| > 0 small and
F = Cψl + w, where w⊥ψl, substituting into (190) and multiplying by ψl yields
s = C2(s) ∫ ψ4l + ... > 0 =⇒ s > 0 and C(s) = ±
+ ... .
