A.4 Continuation techniques
A.4.2 Fold bifurcation
At a quadratic fold the eigenvalue of the Jacobian changes sign. The eigenvalue equation near the fold point is
Dxf(x(s), λ(s))q(s) =κ(s)q(s),
where κ is the eigenvalue and q is the eigenvector. Dierentiating the eigenvalue equation and evaluating at s0 = 0, where the fold bifurcation takes place, yields
Dx2f(x(0), λ(0))q(0)q(0) +Dxf(x(0), λ(0))q0(0) =κ0(0)q(0).
Also, multiply the above equation by the left eigenvector ξ∗ to nd κ0(0) = ξ∗D2xf(x(0), λ(0))q(0)q(0)
ξ∗q(0) .
As it can be seen from formulae of the κ0(0) andλ00(0)the stability changes if and only if the fold is quadratic.
Fold bifurcation can be continued in a similar fashion as a xed point, the only dierence is in the dening equation. The zero eigenvalue of the Jacobian Dxf(x, λ, ν) can be exploited to dene bifurcation points as
f(x, λ, ν) = 0,
Dxf(x, λ, ν)q= 0, (A.15)
q∗0q−1 = 0.
A.4 Continuation techniques 90 Here q0, a previously computed eigenvector is used to determine the length of the actual eigenvector. Because we have 2n+ 1 equations we introduced the second parameter ν, what can be used in continuation.
The Jacobian of (A.15) is
Dxf(x0, λ0, ν0) 0 Dλf(x0, λ0, ν0) D2xf(x0, λ0, ν0)q0 Dxf(x0, λ0, ν0)DλDxf(x0, λ0, ν0)
0∗ q∗0 0
,
which is non-singular at a simple quadratic fold. This can be proved in a similar way as the bordering lemma (see [24]).
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