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more informationMore information: Multifario | Fields Institute lecture (2001) on Multiparameter Continuation
For smooth nonlinear systems that depend on one or more parameters ( F(u,λ)=0 ), the "usual" situation is that u is a smooth function u(λ) which is "isolated' (locally unique). This is the Implicit Function Theorem, which holds when the Jacobian ∂F/∂u is non-singular (there are other conditions). Bifurcation Theory deals with the case when the Jacobian is singular.
When the Jacobian is singular, and the null space is finite dimensional, Lyapunov-Schmidt reduction can be used to reduce the system to a finite set of nonlinear equations -- the Bifurcation Equations (whose Jacobian is zero). For linear systems any multiple of the null space can be added to u. For nonlinear systems the bifurcation equations restrict the allowed points in the null space.
The final step in the analysis is to construct a non-singular system for solutions of the bifurcation equations (the Algebraic Bifurcation Equations). This produces an algebraic system of equations, and each isolated solution of this new system corresponds to a family of solutions of the bifurcation equations.
This field is closely related to Nonlinear Functional Analysis, Germ Theory, and Catastrophe Theory. Applications include chemical reactors, where ignition and extinction of the reaction are bifurcations, and mechanical systems, where resonances and buckling are bifurcations
Unlike linear systems, finding all solutions of a nonlinear system is very difficult. In low dimensions the solution space can be searched, and there are ways of testing if a solution lies in a sub-region, but for large systems these are not practical. One use of bifurcation theory is to trace all of the solution connected to a known solution (continuation methods).
The algebraic bifurcation equations are algebraic, and so the number of complex roots doesn't depend on the parameters. My work on complex bifurcation involved switching to complex solutions at bifurcation points in the hope that disconnected solution branches might be connected through complex solutions. This led, for example, to silly things like computing flow fields of the complex Navier-Stokes equations. While it is easy to find model systems where this is true (u*u+μu-λ*λ = 0) algebraic approximations of differential equations have so many complex solutions that this seems to not happen very often.
The simplest continuation method (numerical continuation/path following) is to change a parameter a little and solve the system using the solution at the previous parameter value as an initial guess.
There are basically two generalizations of this, depending on the size of the system. Simplicial Continuation or Piecewise linear Continuation puts a mesh on the solution space and traces paths from mesh cell to mesh cell. The dimension of the solution space is the dimension of u plus the number of parameters. When solution space (u,λ) is three this is called Isosurface Extraction or Marching Cubes. The other approach is to put a mesh on the solution branch (the dimension is now only the number of parameters). Psuedo-arclength Continuation method uses a local arclength parameterization to avoid those problems.
It is natural to want to apply the same approach when there are more parameters. Simplical Continuation was extended by Allgower and Schmidt. My algorithm extends pseudo-arclength continuation. An open-source implementation is available on SourceForge