Guidelines

The goal of this package is to find solutions $x$ to nonlinear equations

\[F(x,p) = 0 \quad\tag{E}\]

in large dimensions as function of a real parameter $p$. We want to be able to do so on GPU, distributed systems...

The core of the package is built around a Newton-Krylov solver (see newton) which allows to solve equations of the form $F(x)=0$, or find a solution $x_0$ to (E) for a particular $p_0$.

Once such a solution (or several) $(x_0,p_0)$ is known, we can continue it by computing a 1d curve of solutions $\gamma = (x(s),p(s))_{s\in I}$ passing through $(x_0,p_0)$ (see continuation and Deflated Continuation).

In practice, you don't need to know exactly $(x_0,p_0)$ to compute $\gamma$. Indeed, continuation will call newton to refine any initial guess that you pass.

Bifurcation analysis of Equilibria

We can detect if the curve of solutions $\gamma$ crosses another curve of solutions $\gamma^{bif}$ without knowing $\gamma^{bif}$! The intersection point $(x^b,p^b)\in\gamma$ is called a bifurcation point and is such that $\partial_xF(x^b,p^b)$ is non invertible. When calling γ = continuation(...) with the option detect_bifurcation > 1 inside ContinuationPar, the bifurcation points are automatically detected.

Branch switching

In the simple case where $dim\ker \partial_xF(x^b,p^b) = 1$, we can compute automatically the bifurcated branch $\gamma^{bif}$ by calling continuation and passing $\gamma$. This is explained in Branch switching from simple branch point to equilibria.

Recursively, we can compute the curves of solutions which are connected to $(x_0,p_0)$, this is called a bifurcation diagram. This bifurcation diagram can be automatically computed using the function bifurcationdiagram. More information is provided in Automatic Bifurcation diagram computation and a simple example is provided.

When $d\equiv dim\ker \partial_xF(x^b,p^b) > 1$, you can still compute automatically the bifurcated branches $\gamma^{bif}$s by calling continuation. It is based on a reduction of (E) to a small system of $d$ dimensional multivariate polynomial equations in $d$.

In the case $d=1$, the reduced equation can be further simplified into a normal form. This is also automatically computed by the package.

Bifurcation analysis of Cauchy problems

The goal of this part is to study the dynamics of the Cauchy problem

\[\frac{d}{dt}x - F(x,p) = 0. \quad\tag{C}\]

The equilibria are time independent solutions of (C) hence satisfying (E). The previous part can be applied to compute curves of equilibria. However, we can do more and look for time dependent solutions as well.

We can detect the existence of periodic solutions close to $\gamma$. This is done automatically and those bifurcation points are called Hopf bifurcation points.

Branch switching at Hopf points

We focus on computing the branch of periodic solutions branching from a Hopf point. This is done automatically by calling continuation, passing $\gamma$ and choosing a time discretization algorithm (see Periodic orbits computation). Some details about this branch switching are given in From Hopf point to periodic orbits.

Branch switching at bifurcation points of periodic orbits

Let us consider the case where a branch of periodic orbits $\gamma^{po}$ have been computed (see for example previous section) and several bifurcation points have been detected (branch point, period doubling and Neimark Sacker). We can compute the bifurcated branches from $\gamma^{po}$ automatically.

We do not provide an automatic branch switching for those points and for all methods (Shooting, Finite differences). However, for branch points of periodic orbits, you can call continuation by passing $\gamma^{po}$ and some simple arguments (amplitude of the periodic orbits) to perform branch switching in a semi-automatic way. For the case of Periodic orbits based on Trapezoidal rule, see From Branch / Period-doubling point of curve of periodic orbits.

References