What is symmetry in bifurcation analysis? I did a quick google but I couldn't find much. Could someone please explain when a system has symmetries or link me to some good resources? 
For example, the system 
$x'=\mu x-y+x^3$,
$y'=bx-y$ has certain symmetries which are not present if the $x^3$ is replaced by $x^2$. Why?
Thanks! Any help at all would be great.
 A: The word symmetry in bifurcation analysis can refer to the change of variables
\begin{equation}
x \to -x,
\end{equation}
with $x \in \mathbb{R}^n$ being a system's state vector.
If the system's dynamics are invariant under this change, then the phase portrait for that system will be symmetric under reflection through the origin. 
The system you give,
\begin{equation}
\left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) = 
\left(\begin{array}{c} \mu x - y + x^3 \\ bx - y \end{array}\right)
\end{equation}
exhibits exactly this symmetry. To answer your question, if we replace the $x^3$ term by $x^2$, then this type of symmetry doesn't hold because $x^2$ is even while $x^3$ is odd and we lose a minus sign where we need one. 
One place this type of symmetry is encountered in bifurcation analysis is in dealing with pitchfork bifurcations in the plane. For example, the system
\begin{equation}
\left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) = \left(\begin{array}{c} \mu x - x^3 \\ -y \end{array}\right) 
\end{equation}
will experience a supercritical pitchfork bifurcation as $\mu$ passes through zero. Here, although pitchfork bifurcations are codimension two (meaning they generally require two parameters to be varied), we can get a pitchfork bifurcation by varying only a single parameter because of the system's symmetry.
Another place in which we can use symmetry is with the Lorenz equations,
\begin{equation}
\left(\begin{array}{c} \dot{x} \\ \dot{y} \\ \dot{z} \end{array}\right) = \left(\begin{array}{c} \sigma(y - x) \\ rx - y - xz \\ xy - bz \end{array}\right). 
\end{equation}
These equations are known to have chaotic dynamics for some choices of parameters (e.g., $\sigma = 10$, $b = 8/3$, $r = 28$), though just by observing that the dynamics are invariant under the change of variables $(x, y, z) \to (-x, -y, -z)$, we know immediately that every solution is either itself symmetric or that it can be reflected through the origin to give another solution. The neat thing about this type of symmetry is that it lets us make such conclusions immediately without doing any other work like constructing a phase portrait. 
