Bifurcation analysis of 1D ODEs
We consider a single differential equation with a parameter
\begin{equation}
\dot x = f(x, a),\qquad x(0) = x_0,
\label{eq-aut-ivp-with-parameter}
\end{equation}
where $a$ is a parameter (i.e. a constant that appears
in the differential equation).
We assume that the function $f$ is a continuously differentiable
function of $(x, a)$.
Bifurcation diagrams
The fixed points or stationary solutions of the differential equation
\eqref{eq-aut-ivp-with-parameter} are the solutions of
\[
f(x, a) =0.
\]
We can graphically represent the fixed points by drawing the zero set
of the function $f(x, a)$ in the $(x, a)$ plane. On any vertical line
in this diagram the intersections with the zero set are the fixed
points for the particular parameter value. In between the fixed
points the differential equation $\dot x = f(x, a)$ dictates if $x$
increases or decreases along a solution. We can indicate this in the
drawing by including up– or downward pointing arrows.
Regular Fixed Points
Definition. A fixed point $(x_0, a_0)$ is
called regular if
\[
\frac{\pd f} {\pd x}(x_0,a_0)\neq 0.
\]
A fixed point is singular if it is not regular.
If $(x_0, a_0)$ is a regular fixed point then we can apply
the Implicit Function Theorem and conclude that
there is a $\delta\gt0$ such that for each parameter value $a$ with
$|a-a_0|\lt \delta$ there is exactly one equilibrium $x=x(a)$
with $|x-x_0|\lt \delta$. This equilibrium $x(a)$ is a differentiable
function of $a$.
Thus the regular fixed points lie on “branches” which are graphs
of functions $x=x(a)$. These branches meet at the singular points.
Finding the singular points
The singular fixed points are the solutions of the following two equations
\begin{equation}
f(x, a) = 0,\qquad f_x(x, a)=0.
\label{eq-sing-fxd-pt}
\end{equation}
Note that we have two equations for two unknowns ($x$ and $a$), so that the
typical result of solving these equations is a finite list of solutions, or at
least a discrete list of solutions.
Singular fixed points
If $(x_0,a_0)$ is a singular fixed point, so $(x_0, a_0)$ satisfies
\eqref{eq-sing-fxd-pt}, then we call $(x_0,a_0)$ an ordinary singular fixed
point if
\[
\frac{\pd f}{\pd a}(x_0, a_0) \neq0.
\]
If $(x_0,a_0)$ is such a fixed point then we can still apply the Implicit Function Theorem and conclude that near
$(x_0,a_0)$ the zeroset of $f$ is the graph of some function $a=a(x)$ (rather
than $x$ being a function of $a$, as is the case near regular fixed points).
Smoothness of the zeroset
So far we have only assumed that the function $f$ has continuous derivatives of
first order. From here on we assume that $f$ actually has as many derivatives
of higher order as we shall use. It follows from the IFT
that the function $a=a(x)$ that describes the zeroset of $f$ near our singular
point $(x_0, a_0)$ also has derivatives of any order.
Shape of the bifurcation diagram near an ordinary singular point
By applying implicit differentiation we can compute the first few derivatives
at $x_0$ of the function $a(x)$ in terms of the partial derivatives of $f(x,a)$
at $(x_0, a_0)$. The computation goes like this: start with
\[
f(x, a(x)) = 0 \qquad x_0-\delta \lt x\lt x_0+\delta,
\]
and differentiate with respect to $x$, using the chain rule,
\[
f_x(x, a(x)) + f_a(x, a(x)) \, a'(x) = 0,
\]
which we abbreviate to
\[
f_x + f_a \, a_x = 0.
\]
This is also true for all $x\in (x_0- \delta, x_0 + \delta)$, so we can
differentiate again, to get
\begin{equation}
f_{xx} + 2f_{xa}\,a_x + f_{aa} \, (a_x)^2 + f_a \, a_{xx} = 0.
\label{eq-diffd-once}
\end{equation}
We now evaluate these last two equations at $x=x_0$. There we have $a=a(x_0) =
a_0$, and also
\begin{equation}
f_x(x_0, a_0) =0, \qquad f_a(x_0, a_0)\neq 0.
\label{eq-diffd-twice}
\end{equation}
Thus we get from \eqref{eq-diffd-once} that
\[
a'(x_0) = - \frac{f_x(x_0, a_0)}{f_a(x_0, a_0)} = 0.
\]
This simplifies \eqref{eq-diffd-twice}, from which we then conclude that
\[
a''(x_0) = - \frac{f_{xx}(x_0, a_0)}{f_a(x_0,a_0)}.
\]
Based on the sign of $a''(x_0)$ we can now distinguish between three cases
- $a''(x_0)\gt0$: in this case $a(x)$ has a local minimum at $x=x_0$, so
that for $a$ slightly less than $a_0$ there are no fixed points near $x_0$,
while for $a$ slightly larger than $a_0$ there are two fixed points near
$x_0$.
- $a''(x_0)\lt0$: this case is very similar to the previous case.
- $a''(x_0)=0$. In this case we cannot say much about the shape of the
zeroset of $f$ near $(x_0,a_0)$ without computing further derivatives of
$a$.
If $a''(x_0)\neq0$ then the singular point is called a fold point.
In the figure $A$ is a fold point with $a''(x_0)\lt0$, while $B$ is a
fold point with $a' '(x_0)\gt0$.