Various Closeness Results in Discretized Bifurcations

Discretizing the transcritical and pitchfork bifurcations – conjugacy results

“…(In fact, [10, Lemma 2.4.3] is more general, and that estimate has been formulated in the context of the fold bifurcation. A similar version with proof is found in [11,Lemma 3.9]. See also Lemma 4.14 in the present work.)…”

“…There is a rich literature on connections between discretizations and bifurcations. The interested reader will find a selection of about 50 references on this topic in our recent work [11]. Let us add that the above brief summary in the previous paragraphs is also put into a wider context in [11], including the notion of structural and numerical structural stability with earlier important results under various hyperbolicity conditions; references on the properties of numerical methods near bifurcation points of ODEs together with convergence questions; earlier proofs on the discretized fold bifurcation; finally, several theorems on other types of discretized bifurcations of codimension 1, 2 or higher, in 1 or N dimensions, and comparisons between the corresponding bifurcation diagrams, critical eigenvectors or normal form coefficients.…”

“…We omit the technical proofs of the above two results; they are given in [10]. The idea of their proofs is to follow the corresponding sections of [8] with h suitably built into the computations: the smoothness assumptions allow multivariate Taylor polynomials of Φ and ϕ with integral remainders to be constructed explicitly, then qualitative and quantitative inverse and implicit function theorems (see, e.g., [11,Theorem 2.2]) are applied. We remark that the value of s = ±1 is the same in Lemma 2.4 and Theorem 2.5, and this sign is only responsible for the orientation of the bifurcation diagrams.…”

“…In addition, we proved optimal O(h p )-closeness results in the α ≤ 0 region where the two branches of fixed points are located and merge at (α, x) = (0, 0); in the fixed-point-free α > 0 region however we only managed to get either a singular O(h p | ln α|) or a O(h) estimate between the conjugacy map J and the identity id, depending on the construction of J. A detailed proof of the α ≤ 0 case is given in [5], while a proof of the singular estimate is published in [11]. Therefore, although numerical structural stability has been established, the problem of optimal conjugacy estimates in the fold bifurcation case is still open.…”

Detection of high codimensional bifurcations in variational PDEs

“…We can impose v, α E = Ω ∇v, ∇α dx = 0 in(19) or v, w E = 0 in(20) as discussed in Remark 2.11. The uniqueness condition allows us to interpret v as F (0) and w as F (0).Singularity A n , n ≥ 4 The equation to determine F (n−2) (0) : Ω → R reads (0) + B n−2 (α, F (0), .…”