  Standard bifurcation theory is concerned with families of vector fields
$dx/dt = f(x,\lambda)$, $x \in \R^n$, involving one or several constant real
parameters $\lambda$. Viewed as a differential equation for the pair
$(x,\lambda)$, we observe a foliation of the total phase space by constant
$\lambda$. Frequently, the presence of a trivial stationary solution $x=0$ is
also imposed: $0 = f(0,\lambda)$.
  Bifurcation without parameters, in contrast, discards the foliation by a
constant parameter $\lambda$. Instead, we consider systems $dx/dt = f(x,y),
dy/dt = g(x,y)$. Standard bifurcation theory then corresponds to the special
case $y=\lambda, g=0$. To preserve only the trivial solution $x=0$, instead, we
only require $0 = f(0,y) = g(0,y)$ for all $y$. A rich dynamic phenomenology
arises, when normal hyperbolicity of the trivial stationary manifold $x=0$
fails, due to zero or purely imaginary eigenvalues of the Jacobian $f_x(0,y)$.
  Specifically, we address the cases of failure of normal hyperbolicity due to
a simple eigenvalue zero, a simple purely imaginary pair (Hopf bifurcation
without parameters), a double eigenvalue zero (Takens-Bogdanov bifurcation
without parameters), and due to a double eigenvalue zero with additional time
reversal symmetries.
  The results are joint work with Andrei Afendikov, James C. Alexander, and
Stefan Liebscher.
