\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small \emph{
Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 63, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/63\hfil Dynamics near manifolds of equilibria]
{Dynamics near manifolds of equilibria of codimension one
 and bifurcation without parameters}

\author[S. Liebscher\hfil EJDE-2011/63\hfilneg]
{Stefan Liebscher}

\address{Stefan Liebscher \newline
Free University Berlin,  Institute of Mathematics \\
Arnimallee 3,  D-14195 Berlin, Germany}
\email{stefan.liebscher@fu-berlin.de\;
 http://dynamics.mi.fu-berlin.de}

\thanks{Submitted April 10, 2010. Published May 17, 2011.}
\thanks{Supported by the Deutsche Forschungsgemeinschaft}
\subjclass[2000]{34C23, 34C20, 58K05}
\keywords{Manifolds of equilibria;
bifurcation without parameters; \hfill\break\indent
singularities of vector fields}

\begin{abstract}
 We investigate the breakdown of normal hyperbolicity of a manifold
 of equilibria of a flow. In contrast to classical bifurcation
 theory we assume the absence of any flow-invariant foliation at
 the singularity transverse to the manifold of equilibria. We call
 this setting bifurcation without parameters. We provide a
 description of general systems with a manifold of equilibria of
 codimension one as a first step towards a classification of
 bifurcations without parameters. This is done by relating the
 problem to singularity theory of maps.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction} \label{Lie:secIntroduction}


We study dynamical systems with manifolds of equilibria near points
at which normal hyperbolicity of these manifolds is violated.
Manifolds of equilibria appear frequently in classical bifurcation theory
by continuation of a trivial equilibrium.
Here, however, we are interested in manifolds of equilibria
which are not caused by additional parameters.
In fact we require the absence of any flow-invariant foliation
transverse to the manifold of equilibria at the singularity.
We therefore call the emerging theory \emph{bifurcation without parameters}.


Albeit the apparent degeneracy of our setting
(of infinite codimension in the space of all smooth vectorfields)
there is a surprisingly rich and diverse set of applications ranging from
networks of coupled oscillators \cite{Liebscher97-Diplom},
viscous and inviscid profiles of stiff hyperbolic balance laws \cite{HaerterichLiebscher05-TravWaves},
standing waves in fluids \cite{AfendikovFiedlerLiebscher07-PlaneKolmogorovFlows,
AfendikovFiedlerLiebscher08-PlaneKolmogorovFlows},
binary oscillations in numerical discretizations \cite{FiedlerLiebscherAlexander98-HopfBinOsc},
population dynamics \cite{Farkas84-ZipBifurcation},
cosmological models \cite{Wainwright05-Cosmology},
and many more.
The present paper is a first step towards a classification
of bifurcations without parameters.


Consider a vector field
\begin{equation}\label{Lie:eqGeneralVectorFieldNxM}
\begin{gathered}
 \dot{x} = f(x,y) \quad \text{in }\mathbb{R}^n, \\
 \dot{y} = g(x,y) \quad \text{in }\mathbb{R}^m
\end{gathered}
\end{equation}
with a manifold of equilibria $\{(0,y):y\in\mathbb{R}^m\}$; i.e.,
\begin{equation}\label{Lie:eqGeneralEqManifold}
 f(0,y) \equiv  0, \quad g(0,y) \equiv  0.
\end{equation}
As long as the manifold remains normally hyperbolic;
i.e., the linearization of $f$ on the manifold has no purely
imaginary eigenvalues,
\begin{equation}\label{Lie:eqGeneralHyperbolicity}
 \operatorname{spec} \partial_x f(0,y) \cap  \mathrm{i}\mathbb{R} = \emptyset,
\end{equation}
there exists a local flow-invariant foliation with leaves
homeomorphic to a standard saddle, for example by the theorem of
Shoshitaishvili \cite{Shoshitaishvili75-Bif}. Bifurcations are
characterized by a non-hyperbolic block $A$ of the linearization
\begin{equation}\label{Lie:eqGeneralLinearization}
\begin{pmatrix}A(y)&0\\ B(y)&0\end{pmatrix}
 = \begin{pmatrix} \partial_x f & \partial_y f \\ \partial_x g
& \partial_y g
      \end{pmatrix}(0,y),
\end{equation}
say at the origin; i.e., the spectrum of $A(0)$ intersects the
imaginary axis,
\begin{equation}\label{Lie:eqGeneralNonHyperbolicity}
 \operatorname{spec} A(0) \cap  \mathrm{i}\mathbb{R} \neq  \emptyset.
\end{equation}
Restricting to a center manifold we can assume
\begin{equation}\label{Lie:eqGeneralCenter}
 \operatorname{spec} A(0) \subset \mathrm{i}\mathbb{R}.
\end{equation}
We will always assume that the vector field is smooth enough
to allow suitable expansions.


Note the analogy to classical bifurcation theory where $y$ would
be a parameter; i.e., $g \equiv 0$. For references on classical
bifurcation theory see for example
\cite{Arnold83-GeometricalMethods, HaleKocak91-Dynamics,
Kuznetsov95-Bifurcation, Vanderbowhede89-CentManifolds} and the
references there. In the classical case, $g \equiv 0$, however,
the flow invariant transverse foliation $\{ y = \mathrm{const.} \}$ is also
present in a neighborhood of the bifurcation point. This is no
longer true in the general case
\eqref{Lie:eqGeneralVectorFieldNxM},
\eqref{Lie:eqGeneralEqManifold} without parameters. Indeed,
generic functions $g$ of the form \eqref{Lie:eqGeneralEqManifold}
yield a drift in the ``parameter'' direction $y$ which excludes
any flow-invariant foliation transverse to the manifold of
equilibria near a singularity
\eqref{Lie:eqGeneralNonHyperbolicity}. Thus, the resulting
nonlinear local dynamics differ considerably from classical
bifurcation scenarios.


A rigorous analysis of bifurcations without parameters
\eqref{Lie:eqGeneralVectorFieldNxM},
\eqref{Lie:eqGeneralEqManifold},
 \eqref{Lie:eqGeneralCenter}
has been carried out for the following cases:
\begin{itemize}
\item[(i)] A simple eigenvalue zero of $A$ (transcritical point),
$n=m=1$, with linearization at the bifurcation point:
\[
\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},
\]
see \cite{Liebscher97-Diplom}.

\item[(ii)]
A pair of purely imaginary nonzero eigenvalues of $A$ (Andronov-Hopf point),
$n=2$, $m=1$,
with linearization at the bifurcation point:
\[
\begin{pmatrix}
0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0
\end{pmatrix},
\]
see \cite{FiedlerLiebscherAlexander98-HopfTheory}.
A partial description can also be found in \cite{Farkas84-ZipBifurcation}.
\item[(iii)]
An algebraically double and geometrically simple eigenvalue zero
of $A$ (Bogdanov-Takens point), $n=m=2$, with linearization at the
bifurcation point:
\[
\begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\
       0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},
\]
see \cite{FiedlerLiebscher01-TakensBogdanov}.
With additional symmetries,
this case is also studied in
\cite{AfendikovFiedlerLiebscher07-PlaneKolmogorovFlows,
AfendikovFiedlerLiebscher08-PlaneKolmogorovFlows}
\end{itemize}

Note the nonzero blocks $B$ at transcritical and Bogdanov-Takens
points yielding a drift in $y$-direction and excluding any
flow-invariant transverse foliation. At Andronov-Hopf points this
drift is induced by a generic second-order term
$\Pi_y\Delta_x\begin{pmatrix}f\\ g\end{pmatrix}(0) \neq 0$ which is the leading order term of the drift
in $y$-direction averaged over the periodic motion of the
linearization.


Bifurcations without parameters can also appear combined with
additional parameters, for example $g(y) = g(y_1,y_2) =
(g_1(y_1,y_2),0)$. For Bogdanov-Takens points the case of a
generic vector field with two-dimensional equilibrium manifold
turns out to be equivalent to the case of a generic one-parameter
family of vector fields with one-dimensional equilibrium
manifolds, at least up to leading order of the suitably rescaled
normal form; see \cite{FiedlerLiebscher01-TakensBogdanov}. Both
viewpoints are closely related in this case. In the example of a
transcritical point with drift singularity, studied in section
\ref{Lie:secTranscritDriftSingularity}, however, both settings
lead to drastically different dynamical systems, see Remark
\ref{Lie:remTranscritDriftSingularityNoEquilibria}.


In the present paper we analyze the case $x\in\mathbb{R}$,
$y\in\mathbb{R}^m$ of dynamical systems with a codimension-one
manifold of equilibria. As it turns out, the dynamics near
bifurcation points of codimension $m$ on these manifolds can be
related to singularities of smooth maps
$h:\mathbb{R}^{m+1}\to\mathbb{R}$; the Theorem
\ref{Lie:thSingularity}. This correspondence permits the
application of singularity theory and most notably of the
classification of singularities to bifurcations without
parameters. It might serve as a first step towards a
classification of general bifurcations without parameters.


In section \ref{Lie:secTranscritDriftSingularity} we will start
with an example to illustrate the general theorem  formulated and
proved in section \ref{Lie:secGeneralTheorem}. In section
\ref{Lie:secDiscussion} we conclude with a discussion of further
questions, possibilities and problems.


\section{Transcritical points with drift singularity}
\label{Lie:secTranscritDriftSingularity}


Let us first discuss the cases $m=1,2$ of bifurcations
along one- and two-dimensional equilibrium manifolds
as an example to illustrate the general theorem in
section \ref{Lie:secGeneralTheorem}.



\subsection{Transcritical point}


The case $m=1$ of a line of equilibria in a 2-dimensional center manifold
has already been studied in \cite{Liebscher97-Diplom},
see also \cite{FiedlerLiebscher02-ICM-Beijing}.
In classical bifurcation theory, the only robust bifurcation of
\begin{equation}\label{Lie:eqClassicalTranscrit}
\begin{gathered}
 \dot{x}       = f(x,\lambda) \quad \text{in }\mathbb{R},
 \quad f(0,\lambda) \equiv 0,\\
 \dot{\lambda} = 0   \quad    \text{in }\mathbb{R}
\end{gathered}
\end{equation}
is the transcritical bifurcation with normal form
\begin{equation}\label{Lie:eqClassicalTranscritNF}
 \dot{x} = x(x-\lambda) + \text{high order terms}
\end{equation}
see Figure \ref{Lie:figTranscritical}(a). It is caused by the
nontrivial eigenvalue of the linearization at the equilibrium
$x=0$ crossing zero with nonvanishing speed as the parameter
$\lambda$ increases. Together with the non-degeneracy condition
$\partial_x^2f(0,0) \neq 0$, this implies the above normal form
\eqref{Lie:eqClassicalTranscritNF}.


\begin{figure}
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\caption{\label{Lie:figTranscritical}
Transcritical bifurcation point: classical (a) and without
parameters (b).}
\end{figure}


Without parameters,
\begin{equation}\label{Lie:eqTranscrit}
\begin{gathered}
 \dot{x}  = f(x,y) \quad \text{in }\mathbb{R}, \quad
f(0,y) \equiv 0,\\
 \dot{y}  = g(x,y)  \quad\text{in }\mathbb{R}, \quad
g(0,y) \equiv 0,
\end{gathered}
\end{equation}
the nontrivial eigenvalue $\partial_xf(0,y)$ can change sign
along lines of equilibria $\{y=0\}$.
The generic normal form reads
\begin{equation}\label{Lie:eqTranscritNF}
\begin{gathered}
 \dot{x}      = xy + \text{high order terms},\\
 \dot{y}      = x,
\end{gathered}
\end{equation}
see Figure \ref{Lie:figTranscritical}(b). It requires the same
transversality condition of the nontrivial eigenvalue as the
classical transcritical bifurcation. The non-degeneracy condition,
however, is replaced by $\partial_x g(0,0) \neq 0$ and yields the
two-dimensional Jordan block of the linearization at the
transcritical point. Trajectories form parabolas with tangency to
the line of equilibria at the transcritical point. The flow
direction is reversed on opposite sides of the equilibrium line.


\subsection{Parameter dependent transcritical point with drift
singularity}


Along two-dimensional equilibrium manifolds we expect
transcritical  points to form one-dimensional curves, by the implicit
function theorem. At isolated points one of the non-degeneracy
conditions may fail and codimension-two singularities appear. We
will discuss the case of failing drift condition, first in a
one-parameter family of lines of equilibria and then along a
two-dimensional equilibrium surface.

With one parameter, the setting is as follows. We consider a
system
\begin{equation}\label{Lie:eqSemiclassicalDriftsingularity}
\begin{gathered}
\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix}
 = F(x,y,\lambda)
 = \begin{pmatrix} f(x,y,\lambda) \\ g(x,y,\lambda) \end{pmatrix}
\quad x,y,\lambda \in \mathbb{R}, \\
\dot{\lambda} = 0,
\end{gathered}
\end{equation}
with the following properties:
\begin{itemize}
\item[(i)]
For all parameter values,
there exists a line of equilibria, $F(0,y,\lambda) \equiv 0$,
forming a plane of equilibria in the extended phase space.

\item[(ii)]
For all parameter values, the origin is a transcritical point;
i.e., the origin has an eigenvalue zero in transverse direction to
the equilibrium plane, $\partial_x f(0,0,\lambda) \equiv 0$.

\item[(iii)]
For all parameter values,
this nontrivial eigenvalue crosses zero with nonvanishing speed as $y$ increases,
$\partial_y \partial_x f(0,0,0) > 0$.

\item[(iv)]
At $\lambda=0$ the drift non-degeneracy condition fails, $\partial_x g(0,0,0) = 0$.

\item[(v)]
This drift degeneracy is transverse; i.e., the drift changes
direction with nonvanishing speed, as $\lambda$ increases,
$\partial_\lambda \partial_x g(0,0,0) > 0$.
\end{itemize}
The first condition is our structural assumption, (iii), (v) are
non-degeneracy assumptions fulfilled generically, and (ii), (iv)
describe our bifurcation point. This setup is robust; i.e., under
small perturbations of $F$ respecting (i) there is a point near
the origin satisfying (ii)--(v) for the perturbed system. From the
viewpoint of singularity theory, (ii,iv) define a singularity of
codimension two that is unfolded versally by the coordinate $y$
along the line of trivial equilibria and the parameter $\lambda$.


Condition (i) allows us to factor out $x$,
\begin{equation}\label{Lie:eqSemiclassicalDriftsingularityFactorX}
F(x,y,\lambda) = x\tilde{F}(x,y,\lambda),
\end{equation}
with smooth $\tilde{F}$.
Conditions (ii-v) yield an expansion
\begin{equation}\label{Lie:eqSemiclassicalDriftsingularityExpansion}
\tilde{F}(x,y,\lambda) = \begin{pmatrix} ax + by \\ cx + dy +
\sigma\lambda \end{pmatrix}
 + \mathcal{O}((|x|+|y|+|\lambda|)^2),
\end{equation}
with coefficients $a,b,c,d,\sigma\in\mathbb{R}$, $b > 0$, $\sigma > 0$.
We assume an additional non-degeneracy condition
\begin{itemize}
\item[(vi)] The matrix
\[
\partial_{(x,y)}\big( \frac{1}{x}F \big)(0,0,0) =
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]
is hyperbolic; i.e., has no purely imaginary eigenvalues.
\end{itemize}
Setting
\begin{equation}\label{Lie:eqSemiclassicalDriftsingularityDetTrace}
\delta := ad-bc, \quad \tau:=a+d,
\end{equation}
for determinant and trace,
we therefore have $\delta \neq 0$, and $\tau\neq0$ if $\delta>0$.


Applying the multiplier $x^{-1}$ to system
\eqref{Lie:eqSemiclassicalDriftsingularityFactorX} preserves
trajectories for $x \neq 0$ but reverses their direction for
$x <0$. After the coordinate transformation $\tilde{x} = x$,
$\tilde{y} = ax+by$, $\tilde{\lambda} = b\sigma\lambda$, we obtain
\begin{equation}\label{Lie:eqSemiclassicalDriftsingularityNF}
\begin{pmatrix} \tilde{x}' \\ \tilde{y}' \end{pmatrix} =
\begin{pmatrix}  \tilde{y}  \\
      -\delta \tilde{x} + \tau \tilde{y}
      + \tilde{\lambda} \end{pmatrix}
 + \mathcal{O}((|x|+|y|+|\lambda|)^2).
\end{equation}
This yields a bifurcating equilibrium at $(\tilde{x},\tilde{y})
\approx (\tilde{\lambda}/\delta,0)$. Transversality of the branch
of equilibria with respect to the trivial line of equilibria as
well as the hyperbolicity of the nontrivial equilibria is ensured
by condition (vi). Therefore, terms of higher order in
\eqref{Lie:eqSemiclassicalDriftsingularityNF} will preserve this
structure. See Figure \ref{Lie:figSemiclassicalDriftsingularity}
for phase portraits in various cases. Note the appearance of the
generic transcritical bifurcation without parameters, Figure
\ref{Lie:figTranscritical}, for $\lambda\neq0$.


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\bigskip

Stable set of the origin in green, unstable set in red.

\caption{\label{Lie:figSemiclassicalDriftsingularity}
Drift singularity along a one-parameter family of transcritical points.}
\end{figure}


\subsection{Transcritical point with drift singularity without
parameters}

Replacing the parameter $\lambda$ discussed above by an additional
direction of a plane of equilibria, the drift along this manifold
of equilibria is now a two-dimensional  vector. If will not vanish
along generic one-dimensional curves. The drift singularity along
curves of transcritical points is therefore not characterized by a
vanishing drift but rather by a drift direction orthogonal to the
curve of transcritical points. (A drift in $\lambda$-direction was
not possible before.)


The correct setup is given by a system
\begin{equation}\label{Lie:eqDriftsingularity}
\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix}
 =  F(x,y)
 = \begin{pmatrix} f(x,y) \\ g(x,y) \end{pmatrix},
\quad x \in \mathbb{R},\quad y \in \mathbb{R}^2,
\end{equation}
$y=(y_1,y_2)$, $g=(g_1,g_2)$, with the following properties:
\begin{itemize}
\item[(i)]
The $y$-plane consists of equilibria, $F(0,y) \equiv 0$.
\item[(ii)]
There is a transcritical point at the origin; i.e., the $y$-plane
loses normal hyperbolicity at this point, $\partial_x f(0,0,0) =
0$.
\item[(iii)]
This loss of normal hyperbolicity is caused by the transverse
eigenvalue crossing zero transversally, $\nabla_{y} \partial_x
f(0,0,0) \neq 0$. Without loss of generality, the gradient points
in $y_1$-direction; i.e., $\partial_{y_1} \partial_x f(0,0,0) >
0$, $\partial_{y_2} \partial_x f(0,0,0) = 0$. By implicit function
theorem, this gives rise to a curve of transcritical points
tangential to the $y_2$-axis.
\item[(iv)]
At the origin, the drift non-degeneracy transverse to the curve of
transcritical points fails,
$\partial_x g_1(0,0,0) = 0$.
\item[(v)]
This drift degeneracy is transverse; i.e., the drift direction
crosses the tangent to the curve of transcritical points with
nonvanishing speed along the curve of transcritical points,
\[
\partial_{y_1} \partial_x f(0,0,0)  \partial_{y_2} \partial_x g_1(0,0,0)
 + \partial_{y_2}^2 \partial_x f(0,0,0)  \partial_x g_2(0,0,0) \neq 0.
\]
\item[(vi)]
The drift does not vanish at the origin; i.e., there is a
component tangential to the curve of transcritical points,
$\partial_x g_2(0,0,0) > 0$.
\end{itemize}
Note that conditions (i)--(v) correspond to the conditions of the
previous section. Again, the degeneracies (ii,iv) are robust under
perturbations satisfying (i), provided the non-degeneracy
conditions (iii), (v), (vi) hold. The signs of $\partial_{y_1}
\partial_x f(0,0,0)$ and $\partial_x g(0,0,0)$ in (iii), (v) can be
inverted by reflecting $y_1 \mapsto -y_1$ and $y_2 \mapsto -y_2$,
respectively.


The non-degeneracy condition (vi) indeed yields
\begin{equation}\label{Lie:eqDriftNondegeneracy}
 \frac{\operatorname{d}}{\operatorname{d} y_2}
  \langle
    \nabla_{(y_1,y_2)} \partial_x f, \partial_x g
  \rangle (0,\vartheta(y_2),y_2)
\big|_{y_2=0} \neq 0,
\end{equation}
where $(x,y_1,y_2) = (0,\vartheta(y_2),y_2)$, $\vartheta(0) = 0$,
$\vartheta'(0) = 0$, is the curve $\gamma$ of transcritical
points. Locally, we could reparametrize $y$ to achieve $\vartheta
\equiv 0$. Conditions (ii), (iii), (vi) would then read:
$\partial_x f(0,0,y_2) \equiv 0$, $\partial_{y_1} \partial_x
f(0,0,y_2) > 0$, $\partial_{y_2} \partial_x g_1(0,0,0) \neq 0$.
But let us continue with the general setup.


As in the parameter-dependent case
\eqref{Lie:eqSemiclassicalDriftsingularityFactorX}, we can factor
out $x$ due to condition (i),
\begin{equation}\label{Lie:eqDriftsingularityFactorX}
F(x,y) = x\tilde{F}(x,y)
       = x\begin{pmatrix} \tilde{f}(x,y) \\ \tilde{g}(x,y) \end{pmatrix}.
\end{equation}
However, this time, due to non-degeneracy (vi) no equilibrium remains,
\begin{equation}\label{Lie:eqDriftsingularityRegularOrigin}
\tilde{F}(0,0,0) = (0,0,\partial_x g_2(0,0,0)) \neq  0.
\end{equation}
Applying the flow-box theorem, there exists a local smooth
diffeomorphism
\begin{equation}\label{Lie:eqDriftsingularityFlowboxTransform}
h(z_0,z_1,z_2) = \tilde{\Phi}_{z_2}(z_0,z_1,0),
\end{equation}
where $\tilde{\Phi}_t$ denotes the flow generated by the vector
field $\tilde{F}$. This diffeomorphism fixes the origin and
transforms $\tilde{F}$ into the constant vectorfield,
\begin{equation}\label{Lie:eqDriftsingularityFlowbox}
[\operatorname{D} h(z_0,z_1,z_2)]^{-1} \tilde{F}(h(z_0,z_1,z_2)) =
\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.
\end{equation}
Applying the same transformation to the original vector field $F$,
we obtain
\begin{equation}\label{Lie:eqDriftsingularityTranform}
[\operatorname{D} h(z)]^{-1} F(h(z)) = [\operatorname{D} h(z)]^{-1} h_0(z)
\tilde{F}(h(z)) = \begin{pmatrix} 0 \\ 0 \\ h_0(z)
\end{pmatrix},
\end{equation}
where $h=(h_0,h_1,h_2)$.

In a suitable neighborhood of the origin, the vectorfield $F$ is
flow-equivalent to a vectorfield
\begin{equation}\label{Lie:eqDriftsingularity1dVectorfield}
 \dot{z_2} = h_0(z_0,z_1,z_2)
\end{equation}
on the real line depending on two (classical) parameters
$(z_0,z_1)$. Expansion of $h_0$ using
\eqref{Lie:eqDriftsingularityFlowboxTransform} and conditions
(ii)-(vi) yields
\begin{equation}\label{Lie:eqDriftsingularity1dExpansion}
 \dot{z_2} = a z_2^3
  + (c_0 z_0 + c_1 z_1) z_2^2
  + (b z_1 + c_2 z_0 + c_3 z_0^2 + c_4 z_0z_1 + c_5 z_1^2) z_2
  + z_0 + \mathcal{O}(|z|^4)
\end{equation}
with
\begin{equation}\label{Lie:eqDriftsingularity1dCoeffs}
\begin{gathered}
a = \left(
 \partial_{y_1} \partial_x f(0) \partial_{y_2}\partial_x g_1(0)
        + \partial_{y_2}^2 \partial_x f(0) \partial_x g_2(0)
      \right) \partial_x g_2(0)
      \neq  0,
\\
b = \partial_{y_1}\partial_x f(0) \neq  0.
\end{gathered}
\end{equation}
In particular $h_0(0,0,z_2) = a z_2^3 + \mathcal{O}(|z_2|^4)$. This is a
cusp singularity. See
\cite{GolubitskyGuillemin73-SingularityTheory,
Gibson79-SingularityTheory, Arnold94-CatastropheTheory,
ArnoldGusejnZadeVarchenko85-Singularities, Murdock03-Unfoldings}
for a background on singularity theory and its connection to
dynamical systems. In fact, non-degeneracies
\eqref{Lie:eqDriftsingularity1dCoeffs} allow to diffeomorphically
transform \eqref{Lie:eqDriftsingularity1dExpansion} into the
normal form
\begin{equation}\label{Lie:eqDriftsingularity1dNF}
 \dot{z_2} = \pm z_2^3 + z_1 z_2 + z_0 + \mathcal{O}(z_2^N),
\end{equation}
for arbitrary normal-form order $N$, see for example
\cite{BruceGiblin92-Singularities}, proposition 6.10.
This is a minimal versal unfolding of the cusp singularity.
See Figure \ref{Lie:figCusp}.


\begin{figure}
\centering
\setlength{\unitlength}{0.48\textwidth}
\begin{picture}(1.0,1.0)(0.0,0.0)
\put(0,0){\makebox(1,1){%
  \includegraphics[width=\unitlength]{fig2} %{liebscher-cusp}
}}
\put(0.99,0.38){\makebox(0,0)[rt]{$z_0$}}
\put(0.01,0.39){\makebox(0,0)[lt]{$z_1$}}
\put(0.49,0.98){\makebox(0,0)[lt]{$z_2$}}
\put(0.28,0.43){\makebox(0,0)[t]{$\gamma$}}
\put(0.28,0.87){\makebox(0,0)[b]{$\sigma$}}
\put(0.60,0.10){\rotatebox{36}{\makebox(0,0)[bl]{equilibria}}}
\put(0,0){\framebox(1,1){}}
%\put(0.98,0.02){\makebox(0,0)[br]{(a)}}
\end{picture}
\caption{\label{Lie:figCusp}
Cusp singularity $\dot{z_2} = a z_2^3 + z_1 z_2 + z_0$
with $a=-1$. Reverse direction of trajectories and signs of $z_0$, $z_1$ for $a=+1$.
The fold line $\gamma$ is connected by heteroclinic orbits to the
curve $\sigma$,
both curves have a common tangent at the origin.}
\end{figure}


Reverting the flow-box transformation,
the cusp singularity yields a description of the local dynamics near
a transcritical point with drift singularity on a two-dimensional
manifold of equilibria. Note in particular the cusp-shaped fold line
\[
\gamma: \quad z_1^3 = \mp \frac{27}{4} z_0^2 + \mathcal{O}(z_0^{N/3}),
\quad z_2^3 = \pm \frac{1}{2} z_0 + \mathcal{O}(z_0^{N/3})
%(z_0,z_1) = (\pm 2 z_2^3, \mp 3 z_2^2)
\]
of the manifold of equilibria that is connected by heteroclinic
orbits to the curve
\[
\sigma:\quad z_1^3 = \mp \frac{27}{4} z_0^2 + \mathcal{O}(z_0^{N/3}),
\quad z_2^3 = \mp 4 z_0 + \mathcal{O}(z_0^{N/3}).
\]


\begin{proposition} \label{prop2.1}
Under condition {\rm (i)-(vi)} the vector field
\eqref{Lie:eqDriftsingularity} in a local neighborhood $U$ of the
origin is flow-equivalent to the cusp singularity
\eqref{Lie:eqDriftsingularity1dNF}. Depending on the sign of the
cubic term, that is the sign of $a = \operatorname{sign} (
\partial_{y_1} \partial_x f(0)  \partial_{y_2} \partial_x g_1(0)
 + \partial_{y_2}^2 \partial_x f(0)  \partial_x g_2(0))$,
all trajectories in $U$ converge to an equilibrium $(0,y)$ in
forward time ($a=-1$) or backward time ($a=+1$).

In $U$, the transcritical points on the manifold of equilibria form a curve $\gamma$
through the origin.
The unstable (for $a=-1$) and stable (for $a=+1$) sets, respectively,
of the two components $\gamma_1,\gamma_2$ of $\gamma\setminus\{0\}$ form manifolds of
heteroclinic orbits on opposite sides of the manifold of equilibria.
Their targets in forward time ($a=-1$) or backward time ($a=+1$)
again form curves $\sigma_{1,2}$ on the manifold of equilibria with
$\sigma_1\cup\{0\}\cup\sigma_2$ being a tangential curve to $\gamma$.
See Figure \ref{Lie:figDriftSingularity}.
\end{proposition}


\begin{remark}\label{Lie:remTranscritDriftSingularityNoEquilibria}
\rm In contrast to the parameter-dependent drift singularity no
equilibria bifurcate. In fact, the drift non-degeneracy excludes
any kind of recurrent or stationary orbits except the primary
manifold of equilibria.
\end{remark}


\begin{figure}
\setlength{\unitlength}{0.48\textwidth}
\begin{picture}(1.0,1.0)(0.0,0.0)
\put(0,0){\makebox(1,1){%
  \includegraphics[width=\unitlength]{fig3} %{liebscher-driftsingbottom}
}}
\put(0,0){\framebox(1,1){}}
\put(0.02,0.50){\rotatebox{25}{\makebox(0,0)[bl]{equilibria}}}
\put(0.70,0.56){\makebox(0,0)[tl]{$\gamma$}}
\put(0.35,0.47){\makebox(0,0)[br]{$\sigma$}}
%\put(0.98,0.02){\makebox(0,0)[br]{(a)}}
\end{picture}
\hfill
\begin{picture}(1.0,1.0)(0.0,0.0)
\put(0,0){\makebox(1,1){%
  \includegraphics[width=\unitlength]{fig4} %{liebscher-driftsingtop}
}}
\put(0,0){\framebox(1,1){}}
\put(0.35,0.57){\rotatebox{12}{\makebox(0,0)[bl]{equilibria}}}
\put(0.73,0.52){\makebox(0,0)[tl]{$\gamma$}}
\put(0.62,0.57){\makebox(0,0)[tr]{$\sigma$}}
%\put(0.98,0.02){\makebox(0,0)[br]{(b)}}
\end{picture}
\caption{\label{Lie:figDriftSingularity}
  Transcritical point with drift singularity on a plane of equilibria.
  Stable set of the line $\gamma$ of transcritical points in green, unstable set in red,
  selected trajectories in blue. Two different views for $a=-1$.
  Reverse direction of trajectories and switch colors of manifolds for $a=+1$.}
\end{figure}


\section{General Bifurcation at codimension-one manifolds}
\label{Lie:secGeneralTheorem}

We consider the general case of a manifold of equilibria of
codimension one,
\begin{equation}\label{Lie:eqGeneralSingularity}
\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix}
 = F(x,y) =
 \begin{pmatrix} f(x,y) \\ g(x,y) \end{pmatrix},
\quad
x \in \mathbb{R},\; y \in \mathbb{R}^m.
\end{equation}
Typically, such a system will arise as a reduced system on a
center manifold of finite smoothness.
Following the discussion in the previous section we obtain
the following theorem.


\begin{theorem}\label{Lie:thSingularity}
The exists a generic subset of the class of all smooth vector
fields \eqref{Lie:eqGeneralSingularity} with an equilibrium
manifold $\{x=0\}$ of codimension one. For every vector field in
that class the following holds true:

At every point $(x=0,y)$ the vector field is locally flow
equivalent to a $m$-parameter family
\begin{equation}\label{Lie:eqodimEllSingularity}
\dot{z}_m = \pm z_m^{\ell+1} + \sum_{k=0}^{\ell-1} z_{k} z_m^k
+ \mathcal{O}(z_m^N),
\end{equation}
$0 \leq \ell \leq m$, of vector fields on the real line. Here $N$
is the arbitrary but finite normal-form order bounded by the
smoothness of the initial vector field
\eqref{Lie:eqGeneralSingularity}, $f,g\in\mathcal{C}^M$, $N \le
M$, $N < \infty$. This is a versal unfolding of the singularity
$\dot{z}_m = \pm z_m^{\ell+1}$ at the origin.

In particular, near bifurcation points of codimension $m$,
that appear robustly at isolated points on the equilibrium manifold,
the vector field is locally flow equivalent to
\begin{equation}\label{Lie:eqodimMSingularity}
\dot{z}_m = \pm z_m^{m+1} + \sum_{k=0}^{m-1} z_k z_m^k +
\mathcal{O}(z_m^N);
\end{equation}
i.e., a universal unfolding of the singularity $\dot{z}_m = \pm
z_m^{m+1}$ at the origin.
\end{theorem}


\begin{proof}
The equilibrium condition $f(0,y)=g(0,y)=0$ for all $y\in\mathbb{R}^m$
allows us to factor out $x$.
\begin{equation}\label{Lie:eqGeneralSingularityFactorX}
F(x,y) = x\tilde{F}(x,y)
       = x\begin{pmatrix} \tilde{f}(x,y) \\ \tilde{g}(x,y) \end{pmatrix}.
\end{equation}
The resulting vector field
$\tilde{F}:\mathbb{R}^{m+1}\to\mathbb{R}^{m+1}$ does not vanish
on the $m$-dimensional submanifold $\{x=0\}$, for generic $F$.
Without loss of generality, consider a neighborhood $U\subset\mathbb{R}^{m+1}$ of the origin.

We can apply the flow-box theorem to $\tilde{F}$:
Take a local smooth section
\begin{equation}\label{Lie:eqGeneralTransverseSection}
\Sigma: \mathbb{R}^m \supset V \longrightarrow U,
\end{equation}
through the origin, $\Sigma(0)=0$, transverse to the vector field
$\tilde{F}$ in $U$. Let $\tilde{\Phi}_t$ be the flow generated by
$\tilde{F}$. Then the flow-box transformation
\begin{equation}\label{Lie:eqGeneralFlowboxTransform}
h(z_0,\dots ,z_m) = \tilde{\Phi}_{z_m}(\Sigma(z_0,\dots ,z_{m-1}))
\end{equation}
transforms $\tilde{F}$ into the constant vector field
$[\operatorname{D} h]^{-1}(\tilde{F} \circ h) = (0,\ldots,0,1)$.
Again, $\tilde{\Phi}_t$ denotes the flow to the vector field $\tilde{F}$.
Applying the transformation $h$ to the vector field $F|_U$,
we obtain a $m$-parameter family
$[\operatorname{D} h]^{-1}(F \circ h) = (0,\ldots,0,\pi_x h)$
of vector fields on the real line in a neighborhood $V$ of the origin.

Classification of germs of vector fields and their versal unfoldings is the
topic of singularity or catastrophe theory.

Singularities on the real line have the form $\dot{z}_m = \pm
z_m^{\ell+1}$. In generic $m$-parameter families at most $m+1$
leading coefficients of the Tailor expansion vanish; i.e., $\ell
\leq m$ and
\[
\dot{z}_m = \pm z_m^{\ell+1} + \sum_{k=0}^{\ell-1}
\zeta_k(z_0,\dots ,z_{m-1}) z_m^k + \mathcal{O}(z_m^{\ell+2}).
\]
The coefficient $\zeta_\ell$ vanishes by linear transformation of
$z_m$ and the map
\[
(z_0,\dots ,z_{m-1}) \mapsto (\zeta_0,\dots,\zeta_{\ell-1})
\]
has full rank, generically. Remainder terms,
$\mathcal{O}(z_m^{\ell+2})$, can be pushed to any finite normal-form
order, by a suitable coordinate change. This yields system
\eqref{Lie:eqodimEllSingularity}. See also \cite[Chapter 6]
{BruceGiblin92-Singularities}.

Genericity conditions amount to algebraic conditions of the
coefficients of the Taylor expansion at the origin. These
conditions translate via \eqref{Lie:eqGeneralFlowboxTransform} to
generic conditions on $F$.

The versal unfolding \eqref{Lie:eqodimEllSingularity}, one the
other hand, is a system of the form
\eqref{Lie:eqGeneralSingularity}. Therefore, it represents the
versal unfolding of a generic singularity along $m$-dimensional
manifolds of equilibria in $(m+1)$-dimensional phase space.
\end{proof}


\section{Discussion} \label{Lie:secDiscussion}

The present result is a first step towards a more systematic
treatment of bifurcations without parameters than done by case
studies in \cite{FiedlerLiebscherAlexander98-HopfTheory,
FiedlerLiebscher01-TakensBogdanov,
AfendikovFiedlerLiebscher07-PlaneKolmogorovFlows}.

The removal of the manifold of equilibria by a scalar, albeit
singular, multiplier greatly facilitates the analysis but
restricts it to the case of manifolds of codimension one in the
phase space, see \eqref{Lie:eqDriftsingularityFactorX} and
\eqref{Lie:eqGeneralSingularityFactorX}. Hopf points and
Bogdanov-Takens points require manifolds of equilibria of at least
codimension two. Their analysis in
\cite{FiedlerLiebscherAlexander98-HopfTheory,
FiedlerLiebscher01-TakensBogdanov} uses a blow-up or rescaling
procedure reminiscent of the scalar multiplier used here. It seems
worthwile to closer connect these bifurcations without parameters
to singularity theory. This might provide a suitable setting to
also include singularities of the set of equilibria and generalize
the manifold to varieties.

Contrary to classical bifurcation theory, no recurrent dynamics has been found so far
near bifurcation points without parameters.
For the codimension-one manifolds of equilibria discussed here,
the drift nondegeneracy yielding the flow-box transformation prevents any recurrent dynamics.
Similar drift conditions hold true at generic Hopf and Bogdanov-Takens points.
In fact this is the drift which distinguishes bifurcations without parameters
from classical bifurcations
by preventing any flow-invariant transverse foliation.
Recurrent dynamics should be possible at bifurcations points of higher codimension
as the drift condition gets less restrictive.

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\end{document}