\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 83, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/83\hfil [Bifurcation diagram of a cubic system]
{Bifurcation diagram of a cubic three-parameter autonomous
system}

\author[L. Bar\'akov\'a, E. P. Volokitin\hfil EJDE-2005/83\hfilneg]
{Lenka Bar\'akov\'a, Evgenii P. Volokitin}  % in alphabetical order

\address{Lenka Bar\'akov\'a \hfill\break
Mendel University, Dept. of Math., Zemedelska 1,
613 00 Brno, Czech Rep.}
\email{barakova@mendelu.cz}

\address{Evgenii P. Volokitin \hfill\break
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia}
\email{volok@math.nsc.ru}


\date{}
\thanks{Submitted February 9, 2005. Published July 19, 2005.}
\thanks{Supported by the Russian Foundation for Basic Research}
\subjclass[2000]{34C05, 34D45, 34C23}
\keywords{Phase portrait; bifurcation; central manifold;
topological equivalence; \hfill\break\indent
structural stability; bifurcation diagram;
limit cycle}


\begin{abstract}
 In this paper, we study the cubic three-parameter autonomous planar system
 \begin{gather*}
 \dot x_1 = k_1 + k_2x_1 - x_1^3 - x_2,\\
 \dot x_2 = k_3 x_1 - x_2,
 \end{gather*}
 where $k_2, k_3 >0$. Our goal is to obtain a
 bifurcation diagram; i.e., to divide the parameter space into
 regions within which the system has topologically equivalent phase
 portraits and to describe how these portraits are transformed at
 the bifurcation boundaries. Results may be applied to the
 macroeconomical model IS-LM with Kaldor's assumptions. In this
 model existence of a stable limit cycles has already been studied
 (Andronov-Hopf bifurcation). We present the whole bifurcation
 diagram and among others, we prove existence of more difficult
 bifurcations and existence of unstable cycles.
\end{abstract}

\maketitle


\newcommand{\tr}{\mathop{\text{tr}}\nolimits}
\newcommand{\re}{\mathop{\text{Re}}\nolimits}
\newcommand{\im}{\mathop{\text{Im}}\nolimits}
\newcommand{\sgn}{\mathop{\text{sgn}}\nolimits}
\newcommand{\dif}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\diff}[2]{\frac{d #1}{d #2}}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

In the present paper we shall consider
the real dynamical autonomous system
\begin{equation} \label{e1}
\begin{gathered}
\dot x_1 = k_1 + k_2x_1 - x_1^3 - x_2,\\
\dot x_2 = k_3 x_1 - x_2,
\end{gathered}
\end{equation}
where $x_1, x_2 \in \mathbb{R}$ and $K = \{(k_1,k_2,k_3)
\in \mathbb{R}^3 : k_2 > 0, k_3 > 0 \}$ is a parameter space. Note that if
$x_1(t)$, $x_2(t)$ are solutions of \eqref{e1},
$\tilde x_1(t)= - x_1(t)$, $\tilde x_2(t)=-x_2(t)$ are solutions of the
 system
\begin{gather*}
\dot x_1 = -k_1 + k_2x_1 - x_1^3 - x_2,\\
\dot x_2 = k_3 x_1 - x_2.
\end{gather*}
This implies that the bifurcation sets of  \eqref{e1} are
symmetric with respect to the plane $k_1 = 0$, because the phase
portraits of \eqref{e1} with the parameters
$(k_1, k_2, k_3)=(\tilde k_1, \tilde k_2, \tilde k_3)$ and
$(k_1, k_2, k_3)= (-\tilde k_1, \tilde k_2, \tilde k_3)$ are symmetric
about the origin.
We denote
\begin{gather*}
A= \begin{pmatrix}  k_2- 3x_1^2& -1\\
               k_3    & -1 \end{pmatrix},\\
\tr A = k_2 - 3x_1^2 - 1,\\
\det A= 3x_1^2 - k_2 + k_3,\\
p_A(\lambda) = \det(A - \lambda I) = \lambda^2 - \lambda \tr A +
\det A,
\end{gather*}
where $A$ is Jacobi's matrix of the system \eqref{e1}, its trace $\tr A$,
determinant $\det A$ and characteristic polynomial $p_A(\lambda)$
are functions of variable $x_1$.

All equilibrium points $(\xi_1,\xi_2)$ of the system \eqref{e1} have to
be solutions of the  equations
\begin{gather*}
k_1+k_2x_1-x_1^3-x_2=0,\\
k_3x_1-x_2=0,
\end{gather*}
which gives that $\xi_1$ has to satisfy the equality
\begin{equation} \label{e2}
k_1 + (k_2-k_3)\xi_1 -\xi_1^3 = 0
\end{equation}
and $\xi_2 = k_3 \xi_1$. System \eqref{e1} has from one to three
equilibrium points.

\begin{lemma} \label{lem1.1}
Let $(\xi_1,\xi_2)$ be an equilibrium point of \eqref{e1}.
Then the set
\begin{equation*}
\{ (x_1,x_2) \in {\mathbb{R}}^2 : k_3 (x_1-\xi_1)^2 + (x_2-k_3
\xi_1)^2 \leq R \},
\end{equation*}
where
$$R = - k_3 \min_{x_1 \in \mathbb{R}} \{ x_1^2 (x_1^2 + 3\xi_1 x_1 - k_2 +
3\xi_1^2-1)  \}$$
is globally attractive.
\end{lemma}

For the proof of the above lemma se  \cite[Theorems 5.1 and 5.2]{barakova} .

\begin{remark} \label{rmk1.1}\rm
A planar dynamical system
\begin{equation} \label{e3}
\begin{gathered}
\dot y = \alpha [i(y,r) - s(y,r)],\\
\dot r = \beta  [l(y,r) - m],
\end{gathered}
\end{equation}
where $\alpha, \beta > 0$, may represent a macroeconomical model
IS-LM (see \cite{barakova}). The variable $y = \ln Y$ is the
logarithm of the product (GNP), $r$ is the interest rate.
Functions $i$ and $s$ are propensities to invest and save, $l$ and
the constant $m$ - demand and supply of money. Using basic
economic properties of the functions $i$, $s$ and $l$ (including
Kaldor's assumptions), we can concretize the system \eqref{e3} to the
most simple one - a cubic system
\begin{equation} \label{e4}
\begin{gathered}
\dot y = \alpha (a_0 + a_1y + br + a_2y^2 + a_3y^3),\\
\dot r = \beta (c_0 + cy + dr),
\end{gathered}
\end{equation}
where $\alpha>0$, $\beta >0$, $b<0$, $a_3<0$, $c>0$, $d<0$ and
the quadratic equation $a_1+2a_2x+3a_3x^2=0$ has two different
real roots. System \eqref{e4} can be replaced by the system \eqref{e1} using
some efficient transformation (see \cite{volokitin}).
\end{remark}

The aim of this paper is to continue in the study of the dynamical
system \eqref{e4} from \cite{barakova}(the system \eqref{e1} respectively) and
to obtain deeper results concerning its stability, topological
properties and types of bifurcations, especially existence and
stability of limit cycles. From the economic point of view stable
limit cycles correspond to business cycles. Economists are used to
presume that economic equilibrium is globally stable always, i.e.
they assume there exists some mechanism of adaptation in economy.
This is true for a linear IS-LM model, with $a_2=0$, $a_3 = 0$. If
the economy satisfies the Kaldor's assumptions, such mechanism
need not exist. This was pointed out already in the original
Kaldor's paper \cite{kaldor}, but dealing with this problem all
authors provided just numerical results or made some other
specific assumptions to the model and to the best of my knowledge
never found any unstable cycle. Although the system \eqref{e4} is
``only'' cubic, we will show that even more than one cycle can
appear and surely it need not be stable. Moreover, the described
cycles are not evoked by external influences, but they are
entirely determined by internal structure of the system, which is
a problem passed by so called ``invisible hand'' that should lead
the economy to the globally stable equilibrium.

\section{Local bifurcations}

\begin{lemma} \label{lem2.1}
Let $(\xi_1, \xi_2)$ be an equilibrium point of  \eqref{e1} and
let
$$ k_2 = k_3 + 3\xi_1^2, \quad k_3 \neq 1.
$$
Then the equilibrium point $(\xi_1,\xi_2)$ is a saddle-node for
$\xi_1 \neq 0$. The origin is topologically equivalent to a node in
the case $\xi_1 = 0$.
\end{lemma}

\begin{proof}
After transformation of the equilibrium point $(\xi_1, \xi_2)$
to the origin by the change of variables
$u_1=x_1-\xi_1$, $u_2 = x_2-\xi_2$ we get the system
\begin{gather*}
\dot u_1 = k_3 u_1 - 3 \xi_1 u_1^2 - u_1^3 - u_2,\\
\dot u_2 = k_3 u_1 - u_2.
\end{gather*}
For $k_3 \neq 1$, the following regular transformation
$$u_1 = y_1 + y_2, \quad u_2 = k_3 y_1 + y_2$$
(the matrix of the trasformation is given by the eigenvectors
corresponding with one zero and one non-zero eigenvalues)
and the time change
$\tau = (k_3-1) t$
give the canonical form of system \eqref{e1}:
\begin{gather*}
\dot y_1 = F(y_1,y_2),\\
\dot y_2 = y_2 - k_3 F(y_1,y_2),
\end{gather*}
where
$$
F(y_1,y_2) = \frac{3 \xi_1}{(k_3-1)^2}(y_1+y_2)^2 + \frac{1}{(k_3-1)
^2}(y_1+y_2)^3.
$$
Let $y_2= \varphi (y_1)$ be a solution of the equation
$$
y_2 - k_3 F(y_1,y_2) = 0
$$
in the neighbourhood of the origin. We approximate this solution
corresponding with the central manifold of the system by a Taylor
expansion
$$
\varphi(y_1) = \sum_{i=0}^\infty a_i y_1^i
$$
in the neighbourhood of the origin and get
$$
\sum_{i=0}^\infty a_i y_1^i = \frac{3 k_3 \xi_1}{(k_3-1)^2}(y_1+
\sum_{i=0}^\infty a_i y_1^i)^2 + \frac{k_3}{(k_3-1)^2}(y_1+
\sum_{i=0}^\infty a_i y_1^i)^3.
$$
We equate coefficients of equal powers of $x$ on the left and the
righthand side and find
$$
a_0 = 0, \quad a_1 = 0, \quad a_2 = \frac{3 k_3 \xi_1}{(k_3-1)^2}
\neq 0.
$$
The equilibrium point $(\xi_1,\xi_2)$ of the system \eqref{e1}
is a saddle-node according to \cite[Theorem 65 (par. 21)]{kolektiv}.

In the case that $\xi_1 = 0$, the system \eqref{e1} has a unique
equilibrium point $(0,0)$. We analogically aproximate the central
manifold by the Taylor expansion with zero coefficients up to the
second order (including) and get
$$
a_3 = \frac{k_3}{(k_3-1)^2} > 0.
$$
Consequently, the origin is topologically equivalent to a node
according to \cite[Theorem 65 (par. 21)]{kolektiv}.
\end{proof}

\begin{theorem} \label{thm2.1}
The subset $M_T$ of the parameter space $K$,
$$
M_T=\{ (k_1,k_2,k_3) \in K : k_1 = - 2 \xi_1^3, k_2 = k_3 + 3\xi_1^2,
k_3 \neq 1, \xi_1 \in \mathbb{R} - \{0\} \},
$$
is a bifurcation set of codimension 1 - double equilibrium (also
called ``saddle-node bifurcation'').
The double equilibrium point $(\xi_1,k_3 \xi_1)$ is a saddle-node.
\end{theorem}

\begin{proof}
Let $(\xi_1,\xi_2)$ be an equilibrium point of the system \eqref{e1}. The
bifurcation ``double equilibrium'' occurres in the case that the
parameters $k_1$, $k_2$, $k_3$ satisfy the following condition
\begin{equation} \label{e5}
3 \xi_1^2 - k_2 + k_3 = 0.
\end{equation}
In this case two equilibrium points coincide to one. So called
non-degeneracy condition is $\xi_1 \neq 0$, because the
equilibrium point is triple for $\xi_1 = 0$. Conditions \eqref{e2} and
(5) together with the non-degeneracy condition define the subset
of $K$, where the system \eqref{e1} has exactly two equilibrium points:
the double equilibrium point $(\xi_1,k_3 \xi_1)$ and the single
equilibrium point $(-2\xi_1,-2k_3 \xi_1)$. In the case that $k_3 =
1$, the double equilibrium point has two zero eigenvalues and
bifurcation of codimension 2 takes place (this case is studied in
Theorem \ref{thm2.4}).

The set $M_T$ consists of two components $M_{Tl}$ and $M_{Tr}$.
They correspond with the case $\xi_1 < 0$ (the double equilibrium
point lies left of the single one) and $\xi_1>0$ (the double equilibrium
point lies right of the single one). These sets are symmetrical according
to the axis $k_1 = 0$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}\\
\end{center}
\caption{The set $M_T$. \label{fig1}}
\end{figure}


The closure of the set $M_T$ divides the parameter space $K$ into
two sets $M_1$, $M_3$
\begin{gather*}
M_1=\{ (k_1,k_2,k_3) \in K : k_1 = - 2 \xi_1^3, k_2 < k_3 + 3\xi_1^2,
\xi_1 \in \mathbb{R} \},\\
M_3=\{ (k_1,k_2,k_3) \in K : k_1 = - 2 \xi_1^3, k_2 > k_3 + 3\xi_1^2,
\xi_1 \in \mathbb{R} \}.
\end{gather*}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}\\
\end{center}
\caption{The section of $M_T$ and the parameter space in $k_3$.
\label{fig2}}
\end{figure}

The set $M_1$ contsists of all the parameters from $K$, for which
the system \eqref{e1} has a unique equilibrium point (non-saddle), the
set $M_3$ consists of those, for which the system \eqref{e1} has 3
equilibrium points (non-saddle, saddle, non-saddle). While
crossing the boundary $M_T$ from the set $M_3$ to $M_1$, two
equilibrium points coincide and disappear then. According to
Lemma \ref{lem2.1}, the double equilibrium point is a saddle-node.
A qualitative local change of the phase portraits occurres, a
local bifurcation of codimension 1 - ``saddle-node''.
\end{proof}

\begin{theorem} \label{thm2.2}
The subset $M_H$ of the parameter space $K$,
$$
M_H=\{ (k_1,k_2,k_3) \in K : k_1 = \xi_1(k_3 - 1 -2\xi_1^2), k_2 = 1
+ 3\xi_1^2, k_3 > 1, \xi_1 \in \mathbb{R} \},
$$
is a bifurcation set corresponding with Andronov-Hopf bifurcation.
The equilibrium point $(\xi_1,k_3 \xi_1)$ is a multiple focus.
\end{theorem}

\begin{proof}
Let $(\xi_1,\xi_2)$ be an equilibrium point of \eqref{e1}.
The trace $\tr A=0$ and the determinant $\det A > 0$
if and only if  the Jacobi's matrix $A$ has two purely imaginary eigenvalues.
We get the following conditions
\begin{gather*}
k_1 + (k_2-k_3) \xi_1 - \xi_1^3 = 0,\\
k_2 - 3 \xi_1^2 -1 = 0,\\
3 \xi_1^2 -k_2 + k_3 > 0.
\end{gather*}
These three conditions define the set $M_H$ (see fig. 3).


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3}\\
\end{center}
\caption{The set $M_H$. \label{fig3}}
\end{figure}


The eigenvalues are purely imaginary on $M_H$,
$$
\lambda_{1,2} = \pm i \omega, \quad \omega = \sqrt{\det A(\xi_1)}=
\sqrt{k_3 - 1},
$$
and the equilibrium point $(\xi_1, \xi_2)$ is a multiple focus.
While crossing the bound $M_H$, the equilibrium point may change its
stability. We will compute the value of $\diff{\re \lambda_{1,2}}{k_2}$
to describe the change of stability.
Since
$$
\diff{p_A}{\lambda} = 2 \lambda - \tr A,$$
we have
\begin{equation} \label{e6}
\diff{p_A}{\lambda} \Big|_{M_H} = \pm i  2 \sqrt{k_3-1} \neq 0
\end{equation}
on the set $M_H$ and we can apply the implicit function theorem and
get
\begin{equation} \label{e7}
\diff{\lambda}{k_2} \bigg |_{M_H} = -\frac{\diff{p_A}{k_2}}
{\diff{p_A}{\lambda}}\bigg |_{M_H}.
\end{equation}
The coordinates of the equilibrium point depend on the parameters.
Let us denote $\xi_1 = \varphi(k_1,k_2,k_3)$. Then we get
\begin{equation} \label{e8}
\diff{p_A}{k_2} = - (\lambda + 1) \bigl (1 - 6 \varphi \dif{\varphi}{k_2}
\bigr).
\end{equation}
Since the equality \eqref{e2} gives
$$
\varphi + (k_2 -k_3)\dif{\varphi}{k_2} - 3 \varphi^2 \dif{\varphi}
{k_2} = 0,
$$
we can express the partial derivative $\dif{\varphi}{k_2}$
on the set $M_H$ as
$$ \dif{\varphi}{k_2} = \frac{\xi_1}{k_3 - 1}.
$$
Using this expression, equalities \eqref{e6} and \eqref{e8} in \eqref{e7}, we get
\begin{equation*}
\diff{\lambda}{k_2} \bigg |_{M_H} = \frac{ \bigl (1 - 6\frac{\xi_1^2}
{k_3 -1} \bigr )(\pm i\, \sqrt{k_3-1}+1)}{\pm i\, 2\sqrt{k_3-1}}.
\end{equation*}
That yields
\begin{equation}
\diff{\re \lambda}{k_2}\bigg |_{M_H} = \frac{ k_3 -1 - 6\xi_1^2}{ 2
(k_3-1)} = \frac{k_3 + 1 -2k_2}{2(k_3-1)}.
\end{equation}
Taking $M_H$ as a parametric function of $\xi_1$, we have
$$
\diff{k_1}{\xi_1}=k_3-1-6\xi_1^2.
$$
The derivative $\diff{\re \lambda}{k_2}$ is zero if and only if
$\diff{k_1}{\xi_1}=0$, that is in the case that the tangent to
$M_H$ is parallel to the axis $k_2$. In this situation, there is
no crossing of $M_H$ (just a contact) and there is also no change
in stability of the focus. In the case that $k_3+1 > 2k_2$, a
stable focus changes to an unstable focus, while crossing $M_H$ in
the direction of the axis $k_2$. In the opposite case, an unstable
focus changes to a stable focus. (These results correspond to
 Theorem \ref{thm2.3} on subcritical and supercritical bifurcation.)

While crossing the bifurcation bound $M_H$, the focus changes its
stability and a limit cycle arises in its neighbourhood. There
occurres a local qualitative change of the phase portraits called
Andronov-Hopf bifurcation.

The set $M_H$ is divided by the set $M_T$  into three parts
$M_{Hr}$, $M_{Hl}$ and $M_{Hu}$ (see fig. 4).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4}\\
\end{center}
\caption{The section of $M_T$ and $M_H$ in $k_3>1$. \label{fig4}}
\end{figure}


These sets correspond with Andronov-Hopf bifurcation of the right,
left (in the case of three equilibrium points) and unique equilibrium
point.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
Stability of the limit cycle depends on stability of the multiple
focus and is determined by the sign of the first Lyapunov number
of this multiple focus. The cycle is stable for $l_1<0$ and unstable
for $l_1 > 0$. Parameters corresponding with zero values of
the first Lyapunov number $l_1$ determine a subset of codimension 2
of $M_H$ - degenerate Andronov-Hopf bifurcation.
\end{remark}

\begin{theorem} \label{thm2.3}
The subset $M_{DH}$ of the parameter space $K$
$$
M_{DH}=\{ (k_1,k_2,k_3) \in K :  k_1 = 4 \xi_1^3, k_2 = 1 +
3\xi_1^2, k_3 = 1+6 \xi_1^2, \xi_1 \in \mathbb{R} - \{0\} \}
$$
is a bifurcation set of codimension 2 corresponding
with degenerate Andronov-Hopf bifurcation.
\end{theorem}

\begin{proof}
Let $(\xi_1,\xi_2)$ be an equilibrium point of the system \eqref{e1}. We
transform the system \eqref{e1} by a substitution
$u = x- \xi_1$,  $v = k_1 +k_2x-x^3-y$
to an equivalent system of Lienard's type
\begin{equation} \label{e10}
\begin{gathered}
\dot u = v,\\
\dot v = p(u)+q(u)v \equiv p_0 +p_1u+p_2u^2+ p_3u^3 + (q_0 + q_1u +
q_2u^2) v,
\end{gathered}
\end{equation}
where
\begin{equation} \label{e11}
\begin{gathered}
p_0=k_1+(k_2-k_3)\xi_1-\xi_1^3, \quad p_1=k_2-k_3-3\xi_1^2, \quad p_2
= -3\xi_1,\quad  p_3=-1, \\
q_0=-1 + k_2-3\xi_1^2, \quad q_1=-6 \xi_1, \quad q_2=-3.
\end{gathered}
\end{equation}
Since  \eqref{e2} holds for the equilibrium point
$(\xi_1,\xi_2)$, we have $p_0 = 0$, system (10) has an
equilibrium point at the origin. The origin is a multiple focus if
and only if $p_1<0$ and $q_0=0$. According to \cite{volokitin} or
\cite{bautin}, we can express the first and the second Lyapunov
numbers as
\begin{equation*}
l_1=p_2q_1-p_1q_2,\quad l_2=-p_3q_2.
\end{equation*}
Consequently from \eqref{e11}
\begin{equation*}
l_1=3(k_2-k_3+3\xi_1^2), \quad l_2=-3.
\end{equation*}
Since $\tr A = 0$ on $M_H$, we get
\begin{equation*}
l_1=3(1-k_3+6\xi_1^2)=3(2k_2-k_3-1), \quad l_2=-3 \neq 0.
\end{equation*}
The condition $l_1=0$ determines the subset $M_{DH}$ on $M_H$
(see fig. 5) that corresponds with the degenerate Andronov-Hopf bifurcation
of codimension 2 (since $l_2 \neq 0$). The curve $M_{DH}$ divides
the surface $M_H$ into parts $M_{DH-}$ corresponding with the
supercritical bifurcation ($l_1 < 0$, a stable limit cycle
occurres) and $M_{DH+}$ corresponding with the subcritical
bifurcation ($l_1 > 0$, an unstable limit cycle occurres).
\end{proof}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig5}\\
\end{center}
\caption{The set $M_{DH}$. \label{fig5}}
\end{figure}


\begin{remark} \label{rmk2.2} \rm
The set $M_{DH-}$ is entirely contained in the set $M_{Hu}$, which
imply that the stable limit cycle (caused by Andronov-Hopf bifurcation)
may occur only in the case of the unique equilibrium point.
\end{remark}

\begin{theorem} \label{thm2.4}
The subset $M_{BT}$ of the parameter space $K$,
$$
M_{BT}=\{ (k_1,k_2,k_3) \in K :  k_1 = - 2 \xi_1^3, k_2 = 1 +
3\xi_1^2, k_3 = 1, \xi_1 \in \mathbb{R} - \{0\} \},
$$
is a bifurcation set of codimension 2 corresponding with
Bogdanov-Takens bifurcation.
\end{theorem}

\begin{proof}
Let $(\xi_1,\xi_2)$ be an equilibrium point of the system \eqref{e1}. The
bifurcation set of codimension 2 corresponding with
Bogdanov-Takens bifurcation includes such parameters from $K$ that
both eigenvalues of Jacobi's matrix $A$ are zero. The set $M_{BT}$
is determined by two conditions $\det A = 0$ and $\tr A = 0$. The
set $M_{BT}$ lies in the intersection of the closure of $M_H$ and
the set $M_T$. In the case $\xi_1 = 0$, that is for $k_1 = 0, k_2
= k_3 = 1$, bifurcation of higher codimension occurres. Further
analysis of this bifurcation is presented in  Theorem \ref{thm3.1}.
\end{proof}

\begin{theorem} \label{thm2.5}
The subset $M_C$ of the parameter space $K$,
$$
M_C=\{ (k_1,k_2,k_3) \in K :  k_1 = 0, k_2 = k_3, k_3 \neq 1\},
$$
is a bifurcation set of codimension 2 - triple equilibrium point.
The unique equilibrium point $(0,0)$ of  \eqref{e1} is
topologically equivalent to a stable node for $k_3<1$, or an
unstable node surrounded by a stable limit cycle for $k_3 > 1$.
\end{theorem}

\begin{proof}
The Jacobi's matrix on $M_C$ is
$$
A= \begin{pmatrix} k_2 & -1\\
 k_2 & -1 \end{pmatrix}
$$
and its eigenvalues are $\lambda_1=0$ and $\lambda_2=k_2-1$. The
origin is the unique equilibrium point of \eqref{e1} and it is
stable for $k_3 < 1$, unstable for $k_3 > 1$. The unstable unique
equilibrium is surrounded by a stable limit cycle according to
Lemma \ref{lem1.1} on existence of a globally attractive set and the
Poincar\'e's theorem. The origin is topologically equivalent to a
node according to  Lemma \ref{lem2.1}.
\end{proof}

\section{Non-local bifurcations}

In contradiction to local bifurcations, where the bifurcation sets
could be expressed explicitly, bifurcation sets corresponding with
non-local bifurcations can only be studied numerically or can be
approximated with accuracy to a particular order in the
neighbourhood of some important bifurcation points.

\subsection*{Non-local bifurcation of codimension 1 - multiple cycle}

The curve $M_{DH}$ is a boundary of a surface $M_D$ corresponding
with non-local bifurcation of codimension 1 - multiple cycle.
While crossing the set $M_D$, two limit cycles (stable and
unstable) merge into one semi-stable cycle that disappears then.
Closures of sets $M_D$ and $M_H$ are tangent to each other in each
point of the curve $M_{DH}$. The following schematic figure 6
shows the lay-out of the sets $M_H$, $M_T$ and $M_D$ only. They
are figured by their intersections with the plane
$k_3 = \mbox{const.} >1$. The numerical computations shows,
that these sets lie closely
to each other and there are technical problems with their
rendering on the same scale.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig6}\\
\end{center}
\caption{The section of $M_{DH}$ in $k_3>1$ . \label{fig6}}
\end{figure}


\subsection*{Non-local bifurcation of codimension 1 - separatrix loop}

The curve $M_{BT}$ is a boundary of the surface $M_L$
corresponding with non-local bifurcation of codimension 1 -
separatrix loop. The surface $M_L$ is tangent to $M_T$ and $M_H$
at each point of $M_{BT}$. The set $M_L$ is contained in the
half-space $k_3>1$ and consists of two components $M_{Lr}$ and
$M_{Ll}$ corresponding with existence of the separatrix loop
surrounding the right or the left equilibrium point respectively.
While crossing the bound $M_L$, the unstable limit cycle
(originated near $M_H$ in consequence of the subcritical
Andronov-Hopf bifurcation) merge into the separatrix loop and
splits.

Let $(\xi_1, \xi_2)$ be the right double equilibrium point of the
system \eqref{e1}. Then the parameters of the system \eqref{e1} lie in the set
$M_{BT}$ (Bogdanov-Takens bifurcation) and the coordinates of the
double equilibrium point satisfy
$$
\xi_1 = \sqrt{\frac{k_2-1}{3}}, \quad \xi_2 = k_3 \sqrt{\frac{k_2-1}{3}}
$$
according to Theorem \ref{thm2.4}. Using the following substitution
$$
x = x_1 - \sqrt{\frac{k_2-1}{3}}, \quad y = x_2 -
k_3 \sqrt{\frac{k_2-1}{3}},
$$
we transform the system \eqref{e1} into a system
\begin{equation} \label{e12}
\begin{gathered}
\dot x = k_1 + \sqrt{\frac{k_2-1}{3}} \left (k_2-k_3-\frac{k_2-1}{3}
\right ) + x - \sqrt{3(k_2-1)} x^2 -x^3 - y,\\
\dot y = k_3 x - y.
\end{gathered}
\end{equation}
The origin is a double equilibrium point of the system (12) with
two zero eigenvalues for parameters from $M_{BT}$.

System \eqref{e12} can be transformed by the linear transformation
$x_1 = y$,  $x_2 = k_3x -y$
into the  system
\begin{equation} \label{e13}
\begin{gathered}
\dot x_1 = x_2,\\
\dot x_2 = h_{00}+h_{10}x_1+\frac{1}{2}h_{20}x_1^2 + h_{11}x_1x_2 +
\frac{1}{2} h_{02} x_2^2 + R(x_1,x_2,k_1,k_2,k_3),
\end{gathered}
\end{equation}
where
\begin{gather*}
h_{00} = k_3 \Big( k_1 + \sqrt{\frac{k_2-1}{3}} \big(k_2-k_3 -
\frac{k_2-1}{3} \big) \Big),\quad
h_{10} = 1-k_3,\\
h_{20} = - \frac{2}{k_3} \sqrt{3(k_2-1)},\quad
h_{11} = - \frac{2}{k_3} \sqrt{3(k_2-1)},\\
h_{02} = - \frac{2}{k_3} \sqrt{3(k_2-1)},\quad
R(x_1,x_2,k_1,k_2,k_3) = - \frac{(x_1+x_2)^3}{k_3^2}.
\end{gather*}
This transformation keeps the equilibrium point at the origin as
well as its zero eigenvalues. In the further analysis, we will
study system \eqref{e13} instead of the equivalent system \eqref{e1}.

\begin{remark} \label{rmk3.1} \rm
For $(k_1,k_2,k_3) \in M_{BT}$, the following statements hold
\begin{equation*}
h_{00} = 0,\quad
h_{10} = 0,\quad
h_{11} = h_{20} = h_{02} \neq 0.
\end{equation*}
\end{remark}

\begin{theorem} \label{thm3.1}
The system \eqref{e13} can be transformed by a smooth non-degenerate
change of parameters to the Bogdanov-Takens normal canonical form
\begin{equation} \label{e14}
\begin{gathered}
\dot x_1 = x_2,\\
\dot x_2 = \beta_1+\beta_2x_1+x_1^2 + x_1x_2 + O(\|x\|^3),
\end{gathered}
\end{equation}
where
\begin{equation} \label{e15}
\begin{gathered}
\beta_1 = \frac{h_{11}}{(-h_{10}+\frac{1}{4} h_{02}h_{00} +
\frac{1}{2})^3} h_{00},\\
\beta_2 = \frac{1}{(-h_{10}+\frac{1}{4} h_{02}h_{00} +
\frac{1}{2})^2}(h_{10}-h_{00}h_{02}).
\end{gathered}
\end{equation}
In the neighbourhood of the  Bogdanov-Takens curve $M_{BT}$
corresponding with the right double equilibrium point, the set
$M_{Lr}$ can be expressed at the form
\begin{equation}
M_{Lr} = \big\{ (k_1, k_2, k_3) \in \mathbb{R}^3 : \beta_2 < 0, \,
\beta_1 = -\frac{6}{25} \beta_2^2 + o(\beta_2^2)
\big\}.
\end{equation}
The set $M_{Ll}$ is symmetrical to $M_{Lr}$ according to the plane
$k_1=0$.
\end{theorem}

\begin{proof}
The change of time
$dt = (1 - \frac{h_{02}}{2} x_1 ) d\tau$
and the substitution
\begin{equation*}
u_1 = x_1,\quad
u_2 = x_2 - \frac{h_{02}}{2} x_1x_2
\end{equation*}
eliminates the term with $x_2^2$. We get a system of the form
\begin{gather*}
\dot u_1 = u_2,\\
\dot u_2 = \nu_1+\nu_2u_1+C_1u_1^2 + C_2u_1u_2 + O(\|u\|^3),
\end{gather*}
where
\begin{gather*}
\nu_1 = h_{00}, \quad \nu_2 = h_{10}-h_{00}h_{02},\quad
C_1 = -h_{02}h_{10} + \frac{1}{4} h_{02}^2h_{00} + \frac{1}{2}
h_{20}, \quad C_2 = h_{11}.
\end{gather*}
Note that $C_1 = \frac{1}{2}h_{20} \neq 0$ on $M_{BT}$ according to
Remark \ref{rmk3.1}. Introducing a new time (denoted again with $t$)
$$
t= \big | \frac{C_2}{C_1} \big | \tau
$$
and new variables (denoted again with $x_1$ and $x_2$)
$$
x_1 = \frac{C_2^2}{C_1} u_1, \quad x_2 = \sgn \Bigl (\frac{C_2}{C_1}
\Bigr) \frac{C_2^3}{C_1^2}u_2,
$$
we get the Bogdanov-Takens
normal canonical form \eqref{e14}, where
\begin{gather*}
\beta_1 = \frac{h_{11}^4}{(-h_{02}h_{10}+\frac{1}{4} h_{02}^2h_{00} +
\frac{1}{2} h_{20})^3} h_{00},\\
\beta_2 = \frac{h_{11}^2}{(-h_{02}h_{10}+\frac{1}{4} h_{02}^2h_{00} +
\frac{1}{2} h_{20})^2}(h_{10}-h_{00}h_{02}).
\end{gather*}
With respect to the fact that $h_{20} = h_{11} = h_{02}$, we get
the expressions \eqref{e15}.

The coefficient of the term with $x_1 x_2$ corresponds to
$$
s = \sgn \Big(\frac{C_2}{C_1} \Big) \big |_{M_{BT}}
= \sgn \Big(\frac{h_{11}}{-h_{02}h_{10} + \frac{1}{4} h_{02}^2h_{00} +
\frac{1}{2} h_{20}} \Big) \big |_{M_{BT}}.
$$
According to Remark \ref{rmk3.1}, we have
$s = \sgn 2 = 1$.
The Bogdanov-Takens bifurcation is non-degenerate, since
$$
h_{11} = - 2 \sqrt{3(k_2-1)} = -6 \xi_1\neq 0
$$
and $h_{20} \neq 0$ on $M_{BT}$.
The change of parameters is invertible in the neighbourhood of the
origin. It can be verified by a direct computation of the following
determinants and finding
$$
\vmatrix \dif{\beta_1}{k_1} & \dif{\beta_1}{k_2} \\
\dif{\beta_2}{k_1} & \dif{\beta_2}{k_2}\endvmatrix \neq 0, \quad
\vmatrix \dif{\beta_1}{k_2} & \dif{\beta_1}{k_3} \\
\dif{\beta_2}{k_2} & \dif{\beta_2}{k_3}\endvmatrix \neq 0, \quad
\vmatrix \dif{\beta_1}{k_3} & \dif{\beta_1}{k_1} \\
\dif{\beta_2}{k_3} & \dif{\beta_2}{k_1}\endvmatrix \neq 0.
$$
This fact implies that the change of parameters cause no degeneration of
the bifurcation manifold according to the parameter space. (In the
bifurcation theory this regularity of the parameter transformation
is called the transversality condition.)

The expression for the set $M_L$ can be found in
\cite[Theorem 8.5, Appendix]{kuznecov} or in \cite{bogdanov}.
The set $M_{Ll}$ has to be symmetric to $M_{Lr}$ about to the
plane $k_1=0$.
\end{proof}

\subsection*{Non-local bifurcation of codimension 2 - two separatrix loops}

The curve $M_{LL}$, which is an intersection of the sets $M_{Lr}$
and $M_{Ll}$ and lies in the plane $k_1=0$ (because of the
symmetry of the parameter portrait) corresponds with the non-local
bifurcation of codimension 2 - two separatrix loops. Two
separatrix loops surround both the right and the left equilibrium
points (see fig. 7).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig7}\\
\end{center}
\caption{Structurally unstable two separatrix loops. \label{fig7}}
\end{figure}

\subsection*{Non-local bifurcation of codimension 1 - ``big separatrix loop``}

According to \cite{turaev}, the curve $M_{LL}$ is a boundary of a
bifurcation set $M_{BL}$ corresponding with non-local bifurcation
of codimension 1 - ,,big separatrix loop``. While crossing the set
$M_{BL}$, separatrix loop surrounding both equilibrium points
appears and consequently gives to arise to an unstable limit cycle
containing the saddle and both remaining equilibrium points in its
interior (see fig. 8).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig8}\\
\end{center}
\caption{A structurally unstable big separatrix loop.
\label{fig8}}
\end{figure}

Figure 9 presents the lay-out of the sets $M_T$, $M_H$,
$M_L$ and $M_{BL}$, showing the section of the parameter space $K$
by the plane $k_3 = const. > 1$, near 1.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig9}\\
\end{center}
\caption{The section of $M_{L}$ and $M_{BL}$ in $k_3>1$.
\label{fig9}}
\end{figure}

\section{Global bifurcation diagram}

The bifurcation sets described above divide the parameter space
$K$ into parts, where the phase portraits of system \eqref{e1} are
topologically equivalent and structurally stable. The bifurcation
sets contain those parameters, for which the phase portraits are
structurally unstable.

Figure 10 shows a section of the global bifurcation diagram by
the plane $k_3=$const. for $k_3 \in (0,1]$, and figure 11 this
section for $k_3>1$.
Figure 12 shows the structurally stable
phase portraits corresponding to the marked regions for $k_1 < 0$.
The half-space $k_1>0$ is symmetrical to the opposite one and the
phase portraits are symmetrical according to the origin.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig10}\\
\end{center}
\caption{The section of the bifurcation diagram in $k_3 \in
(0,1]$. \label{fig10}}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig11}\\
\end{center}
\caption{The section of the bifurcation diagram in $k_3 > 1 $.
\label{fig11}}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig12}\\
\end{center}
\caption{Phase portraits corresponding to the marked regions.
\label{fig12}}
\end{figure}

\begin{thebibliography}{00}

\bibitem{kolektiv} Andronov A.A., Leontovich E.A., Gordon I.I., Mayer
A.G.: {\it Kachestvennaya teoriya dinamicheskikh sistem vtorovo
poryadka}, Nauka, Moskva, 1966.

\bibitem{barakova} Bar\'akov\'a L.: {\it Asymptotic Behaviour and
Existence of a Limit Cycle in Cubic Autonomous Systems},
Demonstratio Mathematica Vol. 34 No. 3 (2001), 559-576.

\bibitem{bautin} Bautin N.N.: {\it Povedeniye dinamicheskikh sistem
vblizi granic oblasti ustoychivosti}, Ogiz Gostexizdat,
Leningrad-Moskva, 1949.

\bibitem{bogdanov} Bogdanov R.I.: {\it Versalnaya deformatsiya
osoboy tochki vektornovo polya na ploskosti v sluchae nulevikh
sobstvennikh chisel}, Tr. sem. im. Petrovskovo, No. 2 (1976),
37-65.

\bibitem{kaldor} Kaldor N.: {\it A Model of the Trade
Cycle}, Econ. Jour. 50 (1940), 78-92.

\bibitem{kuznecov} Kuznetsov Y.A., {\it Elements of Applied Bifucation
Theory}, Second Edition, Applied Mathematical Sciences 112, Berlin,
Heidelgerg, New York, Springer-Verlag, 1995, 1998.

\bibitem{turaev} Turaev D.V.: {\it ,,Bifurkatsii dvumernikh
dinamicheskikh sistem, blizkich k sisteme s dvumya petlyami separatris``}, Uspechi
mat. nauk Vol. 40, No. 6 (1985), 203-204.

\bibitem{volokitin} Volokitin E.P., Treskov S.A.: {\it Bifurkatsionnaya
diagramma kubicheskoy sistemi Lienardovskovo tipa}, Sibirskij zhurnal
industrialnoy matematiki Vol. 5, No. 3(11) (2002), 67-75.

\end{thebibliography}
\end{document}