Real equations of the form g(x,λ) = 0 are shown to have a complex extension G(u,λ) = 0, defined on the complex Banach space 𝔹 ⊕ i𝔹. At a singular point of the real equation this extension has solution branches corresponding to both the real and imaginary roots of the Algebraic Bifurcation Equations (ABE's).
We solve the ABE's at simple quadratic folds, quadratic bifurcation points, and cubic bifurcation points, and show that these are complex bifurcation points. We also show that at a Hopf bifurcation point of the real equation there are two families of complex periodic orbits, parametrized by three real parameters.
By taking sections of solutions of complex equations with two real parameters, we show that complex branches may connect disjoint solution branches of the real equation. These complex branches provide a simple and practical means of locating disjoint branches of real solutions.
Finally, we show how algorithms for computing real solutions may be modified to compute complex solutions. We use such an algorithm to find solutions of several example problems, and locate two sets of disjoint real branches.
