# Nilpotent center conditions in cubic switching polynomial Liénard systems by
higher-order analysis

Ting Chen<EMAIL_ADDRESS>Feng Li<EMAIL_ADDRESS>Pei Yu<EMAIL_ADDRESS>College of Science, National University of Defense Technology, Changsha,
410073, China  School of Statistics and Mathematics, Guangdong University of
Finance and Economics,
Guangzhou, 510320, China  School of Mathematics and Statistics, Linyi
University, Linyi, 276005, China  Department of Mathematics, Western
University, London, Ontario, N6A 5B7, Canada

###### Abstract

The aim of this paper is to investigate two classical problems related to
nilpotent center conditions and bifurcation of limit cycles in switching
polynomial systems. Due to the difficulty in calculating the Lyapunov
constants of switching polynomial systems at non-elementary singular points,
it is extremely difficult to use the existing Poincaré-Lyapunov method to
study these two problems. In this paper, we develop a higher-order Poincaré-
Lyapunov method to consider the nilpotent center problem in switching
polynomial systems, with particular attention focused on cubic switching
Liénard systems. With proper perturbations, explicit center conditions are
derived for switching Liénard systems at a nilpotent center, which is
characterized as global. Moreover, with Bogdanov-Takens bifurcation theory,
the existence of five limit cycles around the nilpotent center is proved for a
class of switching Liénard systems, which is a new lower bound of cyclicity
for such polynomial systems around a nilpotent center.

###### keywords:

Switching Liénard system; Higher-order Poincaré-Lyapunov method; Nilpotent
center; Global; Limit cycle

###### MSC:

34C07, 34C23

##  1 Introduction

During the past several decades, a large number of works has been focused on
the study of the so-called Liénard equation,

$\ddot{x}+f(x)\dot{x}+g(x)=0,$ (1)

which frequently appears in many disciplines and applications [16]. The dot in
(1) denotes derivative with respect to time $t$. By introducing
$y=F(x)+\dot{x}=\int_{0}^{x}f(x)dx+\dot{x}$, the equation (1) can be brought
to the following planar first-order differential system,

$\begin{array}[]{ll}\dot{x}=y-F(x),\\\ \dot{y}=-\,g(x),\end{array}$ (2)

which is the so-called generalized Liénard system.

For the Liénard system (2), three important problems related to qualitative
dynamical behaviours are classified as the center conditions, the number of
limit cycles and the global phase portraits, see [3, 9, 29, 45]. Christopher
[10] introduced an algebraic approach to classify linear-type centers in
smooth polynomial Liénard systems, which are called elementary singular points
characterized by a pair of purely imaginary eigenvalues. Further, such a
center is considered as global if the whole vector field of the system is
filled with periodic orbits except for this point. Later, Llibre and Valls
[29] studied all types of generalized Liénard systems having a global linear-
type center.

Assume that the polynomials $F(x)$ and $g(x)$ are given in the form of
$F(x)=\sum^{n}_{i=0}a_{i}x^{i}$ and $g(x)=\sum^{m}_{i=0}b_{i}x^{i}$. The
problem of the number of bifurcating limit cycles and their relative positions
for the classical Liénard system (i.e., $g(x)=x$) is the well-known Smale’s
13th problem [35, 2]. The authors of [27] conjectured that the classical
Liénard system can have at most $[\frac{n-1}{2}]$ limit cycles, where
$[\cdot]$ denotes the integer function. Maesschalck and Dumortier [32] proved
that some classical Liénard system can have at least $[\frac{n-1}{2}]+2$ limit
cycles for $n\geq 6$.

It has been noted that much attention for the generalized Liénard system was
paid to consider the maximum number of limit cycles bifurcating from a
monodromic singular point, which is classified as either a center or a focus,
usually called Hopf cyclicity, see for example [11, 38, 22, 23]. For the Hopf
cyclicity at the origin of (2), Han [19] proved it to be $[\frac{n-1}{2}]+2$
when $g(-x)=-g(x)$; Tian and Han [38] showed it to be $[\frac{2n-1}{3}]$ when
$\text{deg}(g)=2$; and Tian et al. [39] proved it to be $[\frac{3n-1}{4}]$
when $\text{deg}(g)=3$. Chen el at. [4, 5] proved the existence of two limit
cycles in two classes of cubic Liénard systems, and analyzed their global
dynamics. However, to the best of our knowledge, the three problems for the
general Liénard system are still open.

In recent years, increasing attention has been attracted to the research in
dynamics of non-smooth systems since non-smoothness has been included more and
more in models describing problems arising in engineering [6], epidemiology
[37] and electronics [13], and in particular in the generalized Liénard system
[1, 28, 33, 34]. For the non-smooth Liénard system (2), $F(x)$ and $g(x)$ are
usually assumed to be piecewise smooth. For example, Chen el at. [6] studied
the global dynamics of a mechanical system with dry friction, which can be
transformed to a piecewise smooth Liénard system. However, the center and Hopf
cyclicity problems become extremely complicated in switching systems. To
overcome the difficulty, the authors of [17] developed a useful approach for
computing the Lyapunov constants for the switching polynomial systems with an
elementary singular point. With this method, they solved the linear-type
center problem for a class of switching Liénard systems. In [38, 39], the Hopf
cyclicities are obtained for system (2) when $F(x)$ is a piecewise polynomial
having a switching manifold at $x=0$ with $\text{deg}(g)=2,\,3$.

It should be pointed out that although many research results studies have been
obtained on the center problem and bifurcation of small-amplitude limit cycles
for switching Liénard systems associated with elementary singular points, no
attention has been paid to switching Liénard systems associated with an
isolated nilpotent singular point. By an isolated nilpotent singular point in
planar polynomial systems, it means that the two eigenvalues of the Jacobian
matrix of the system, evaluated at the singular point, are zero but the
Jacobian matrix is not null, more details can be found in [15, 18, 25, 24, 30,
31, 36, 42, 44]. From Theorem 3.5 in [14] we know that if $m$ is the smallest
integer satisfying $b_{m}\neq 0$, the multiplicity of the nilpotent singular
point of system (2) is exactly $m$. Further, let $n$ be the smallest number
for which $a_{n}\neq 0$. Then, the local qualitative properties on the
nilpotent origin $(0,0)$ of (2) are summarized in Table 1.

Table 1: The local qualitative properties on the nilpotent singular point $(0,0)$ of (2). Conditions | Type of the origin
---|---
$a_{n}=0$, $b_{m}\neq 0$ |  $m$ odd, $b_{m}>0$ | a center or a focus
$m$ odd, $b_{m}<0$ | a saddle
$m$ even | a cusp
$\begin{array}[]{l}\\!\\!\\!a_{n}\neq 0,\ b_{m}\neq 0,\\\\[6.02777pt] \end{array}$ | $m=2k\\!+\\!1$ | $b_{m}<0$ | a saddle
$\Delta=-4(n\\!+\\!1)b_{m}\\!+\\!n^{2}a_{n}^{2}$ | $b_{m}>0$ | $k>n-1$, or $k=n-1$ and $\Delta\geq 0$, $n$ even | consisting of one
hyperbolic sector and
one elliptic sector
$k>n-1$, or $k=n-1$ and $\Delta\geq 0$, $n$ odd | a node
$k<n-1$, or $k=n-1$ and $\Delta<0$ | a center or a focus
$m=2k$ | $n<k$ | a saddle-node
$n\geq k$ | a cusp

Regarding Table 1, it is worth to mention that the origin of (2) is a
monodromic singular point and a cusp when the parameters satisfy

$\displaystyle\Xi_{1}=$ $\displaystyle\
\big{\\{}(a_{n},b_{m})\in\mathbb{R}^{2}:\ a_{n}=0,\
b_{2k+1}>0\big{\\}}\bigcup\big{\\{}(a_{n},b_{m})\in\mathbb{R}^{2}:\ a_{n}\neq
0,\ b_{2k+1}>0,\ k>n\big{\\}}$ (3) $\displaystyle\
\bigcup\big{\\{}(a_{n},b_{m})\in\mathbb{R}^{2}:\ a_{n}\neq 0,\ b_{2k+1}>0,\
k=n,\ \Delta<0\big{\\}}$

and

$\displaystyle\Xi_{2}=\big{\\{}(a_{n},b_{m})\in\mathbb{R}^{2}:\,\,a_{n}=0,\,\,b_{2k}\neq
0\big{\\}}\bigcup\big{\\{}(a_{n},b_{m})\in\mathbb{R}^{2}:\,\,a_{n}\neq
0,\,\,b_{2k}\neq 0,\,\,k\leq n\big{\\}},$ (4)

respectively.

In this paper, we will develop a higher-order Poincaré-Lyapunov method to
determine the nilpotent center conditions and bifurcation of small-amplitude
limit cycles in switching polynomial systems. It is a more challenging and
interesting compared to the works for switching systems with elementary
singular points. We will apply this method to investigate a class of cubic
switching polynomial Liénard systems with a nilpotent singular point. Without
loss of generality, the cubic switching Liénard systems can be written in the
form of the differential equations,

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}y-(a_{0}^{+}+a_{1}^{+}x+a_{2}^{+}x^{2}+a_{3}^{+}x^{3})\\\\[2.15277pt]
-(b_{0}^{+}+b_{1}^{+}x+b_{2}^{+}x^{2}+b_{3}^{+}x^{3})\end{array}\right),&\text{if}\
\ x\geq 0,\\\
&\left(\begin{array}[]{c}y-(a_{0}^{-}+a_{1}^{-}x+a_{2}^{-}x^{2}+a_{3}^{-}x^{3})\\\\[2.15277pt]
-(b_{0}^{-}+b_{1}^{-}x+b_{2}^{-}x^{2}+b_{3}^{-}x^{3})\end{array}\right),&\text{if}\
\ x<0,\end{aligned}\right.$ (5)

where $x=0$ is the unique switching manifold, and
$\lambda\\!=\\!(a_{i}^{\pm},b_{j}^{\pm})\\!\in\\!\mathbf{R}^{16}$,
$i=0,1,2,3$, represents the parameter vector. For the convenience in the
following analysis, we call the system with “$+$” sign “the first system” and
the system with “$-$” sign “the second system”.

Assume that $(0,0)$ is a singular point of (5). Then, we have
$a_{0}^{\pm}=b_{0}^{\pm}=0$. Thus, the Jacobian matrices of the first and the
second systems of (5) evaluated at the origin are given by

$J^{\pm}=\left[\begin{array}[]{cc}-a_{1}^{\pm}&1\\\
-b_{1}^{\pm}&0\end{array}\right].$ (6)

The necessary and sufficient conditions for the origin of (5) to be an
isolated nilpotent singular point are ${\rm Tr}(J^{\pm})=\det(J^{\pm})=0$,
with $J^{\pm}$ not being identically zero. It is easy to obtain that
$a_{1}^{\pm}=b_{1}^{\pm}=0$. Then, (5) is reduced to

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}y-(a_{2}^{+}x^{2}+a_{3}^{+}x^{3})\\\\[2.15277pt]
-(b_{2}^{+}x^{2}+b_{3}^{+}x^{3})\end{array}\right),&\text{if}~{}~{}x\geq 0,\\\
&\left(\begin{array}[]{c}y-(a_{2}^{-}x^{2}+a_{3}^{-}x^{3})\\\\[2.15277pt]
-(b_{2}^{-}x^{2}+b_{3}^{-}x^{3})\end{array}\right),&\text{if}~{}~{}x<0.\end{aligned}\right.$
(7)

It follows from Table 1 that different types of the nilpotent singular point
can generate much more rich dynamics than that of the elementary one, leading
to that the determination of the center conditions of (7) becomes more
involved. In fact, the center of system (7) on the switching manifold can be
classified as two monodromic singular points, or a cusp and a monodromic
singular point, or two cusps. Due to this complexity it is extremely difficult
to consider the nilpotent center problem in the switching polynomial Liénard
systems of general degree $n$. Here, we derive the necessary and sufficient
conditions for the center problem associated with the nilpotent origin of the
cubic switching polynomial Liénard system (7). We have the following result.

######  Theorem 1.1.

Assume $b_{2}^{+}\geq 0$ and $b_{2}^{-}\leq 0$. The origin of the cubic
switching Liénard system (7) is a nilpotent center if one of the following
conditions holds:

$\displaystyle\mathrm{I}:$ $\displaystyle\ \
a_{2}^{\pm}=a_{3}^{\pm}=b_{2}^{-}=0,\ b_{2}^{+}>0,\ b_{3}^{-}>0;$ (8)
$\displaystyle\mathrm{II}:$ $\displaystyle\ \ a_{2}^{\pm}=a_{3}^{\pm}=0,\
b_{2}^{+}>0,\ b_{2}^{-}<0;$ $\displaystyle\mathrm{III}:$ $\displaystyle\ \
a_{2}^{\pm}=b_{3}^{\pm}=a_{3}^{+}b_{2}^{-}-a_{3}^{-}b_{2}^{+}=0,\
a_{3}^{+}(b_{2}^{+}+b_{2}^{-})\neq 0,\ b_{2}^{+}>0,\ b_{2}^{-}<0;$
$\displaystyle\mathrm{IV}:$ $\displaystyle\ \ a_{2}^{-}=a_{2}^{+},\
a_{3}^{-}=-a_{3}^{+},\ b_{2}^{-}=-b_{2}^{+}<0,\ b_{3}^{-}=b_{3}^{+};$
$\displaystyle\mathrm{V}:$ $\displaystyle\ \
a_{2}^{\pm}=a_{3}^{\pm}=b_{2}^{+}=0,\ b_{2}^{-}<0,\ b_{3}^{+}>0;$
$\displaystyle\mathrm{VI}:$ $\displaystyle\ \ a_{2}^{\pm}=b_{2}^{\pm}=0,\
a_{3}^{-}=-a_{3}^{+},\ b_{3}^{-}=b_{3}^{+}>\tfrac{3}{4}(a_{3}^{+})^{2};$
$\displaystyle\mathrm{VII}:$ $\displaystyle\ \
a_{2}^{\pm}=a_{3}^{\pm}=b_{2}^{\pm}=0,\ b_{3}^{+}>0,\ b_{3}^{-}>0;$
$\displaystyle\mathrm{VIII}:$ $\displaystyle\ \ a_{2}^{-}=a_{2}^{+}\neq 0,\
a_{3}^{-}=-a_{3}^{+},\ b_{2}^{\pm}=0,\
b_{3}^{-}=b_{3}^{+}>\tfrac{1}{3}(a_{2}^{+})^{2}.$

The next result further characterizes the nilpotent center $(0,0)$ of the
cubic switching Liénard system (7) to be global.

######  Theorem 1.2.

Assume $b_{2}^{+}\geq 0$ and $b_{2}^{-}\leq 0$. The origin of the cubic
switching Liénard system (7) is a nilpotent global center if one of the
following conditions holds:

$\displaystyle\mathrm{G_{1}}:$ $\displaystyle\ \ a_{2}^{\pm}=a_{3}^{\pm}=0,\
b_{3}^{+}>0,\ b_{3}^{-}>0;$ (9) $\displaystyle\mathrm{G_{2}}:$ $\displaystyle\
\ a_{2}^{-}=a_{2}^{+},\ a_{3}^{\pm}=0,\ b_{2}^{-}=-b_{2}^{+},\
b_{3}^{-}=b_{3}^{+}>\tfrac{1}{2}(a_{2}^{+})^{2}.$

Moreover, regarding the bifurcation of limit cycles around the origin of (7),
we construct a perturbed system using the center condition $\mathrm{IV}$ and
obtain that the maximal number of bifurcating limit cycles from the nilpotent
origin of system (7) is $5$. This is a new lower bound on the limit cycles for
such cubic switching Liénard systems around a nilpotent singular point.

######  Theorem 1.3.

With the nilpotent center condition $\mathrm{IV}$ in Theorem 1.1, the cubic
switching Liénard system (7) can have at least $5$ limit cycles bifurcating
from the origin by higher-order cubic perturbations.

The rest of the paper is organized as follows. In the next section, we present
our high-order Poincaré-Lyapunov method with some formulas which are needed in
Section 3. Section 3 is devoted to derive the nilpotent center conditions at
the origin of system (7). The conditions on the global nilpotent center of
system (7) are obtained in Section 4. In Section 5, a perturbed system of (7)
is constructed to show the bifurcation of $5$ limit cycles from the origin of
(7). Finally, conclusion is drawn in Section 6.

##  2 The high-order Poincaré-Lyapunov method

We consider the following switching nilpotent systems divided by the $y$-axis,

$(\dot{x},\,\dot{y})=\left\\{\begin{aligned}
\Big{(}y+\sum\limits_{i+j=2}^{n}A_{ij}^{+}x^{i}y^{j},\
\sum\limits_{i+j=2}^{n}B_{ij}^{+}x^{i}y^{j}\Big{)},\ \ \text{if}\ \ x\geq
0,\\\\[0.0pt] \Big{(}y+\sum\limits_{i+j=2}^{n}A_{ij}^{-}x^{i}y^{i},\
\sum\limits_{i+j=2}^{n}B_{ij}^{-}x^{i}y^{j}\Big{)},\ \ \text{if}\ \
x<0,\end{aligned}\right.$ (10)

where $A_{ij}^{\pm}$ and $B_{ij}^{\pm}$ are parameters. We have the following
proposition for proving the nilpotent center conditions at the origin $(0,0)$,
which is the common nilpotent singular point in both the first and the second
systems of (10).

######  Proposition 2.1.

Assume that the origin $(0,0)$ of the switching nilpotent system (10) is
monodromic. If the first and second systems in (10) have respectively the
first integrals $I^{+}(x,y)$ and $I^{-}(x,y)$ near the origin, and either both
$I^{+}(0,y)$ and $I^{-}(0,y)$ are even functions in $x$ or $I^{+}(0,y)\equiv
I^{-}(0,y)$, then $(0,0)$ is a nilpotent center.

See [17] for more details about Proposition 2.1. In [26], the authors
redefined the symmetry of switching systems. By modifying the conditions, we
obtain the following result for proving the nilpotent center conditions of
(10) at the origin.

######  Proposition 2.2.

Assume that the nilpotent origin of the switching nilpotent system (10) is
monodromic. If the systems (10) are symmetric with respect to the $x$-axis,
i.e., the parameters on the right-hand side of (10) satisfy

$\displaystyle A_{i,2k}^{\pm}=B_{i,2k+1}^{\pm}=0,$ (11)

or the systems in (10) are symmetric with respect to the $y$-axis, i.e., the
parameters on the right-hand side of (10) satisfy

$\displaystyle A_{2k+1,j}^{+}=-A_{2k+1,j}^{-},\quad
A_{2k,j}^{+}=A_{2k,j}^{-},\quad B_{2k+1,j}^{+}=B_{2k+1,j}^{-},\quad
B_{2k,j}^{+}=-B_{2k,j}^{-},$ (12)

then the origin of (10) is a nilpotent center.

Now we introduce our higher-order Poincaré-Lyapunov method to study the
switching system (10), which establishes the relation between the unperturbed
systems and the perturbed systems based on Bogdanov-Takens (B-T) bifurcation
theory. Hence, we consider the following perturbed system of (10),

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}\vartheta_{1}\varepsilon
x+y+\displaystyle\sum\limits_{i+j=2}^{n}A_{ij}^{+}x^{i}y^{j}+\displaystyle\sum\limits_{k=1,i+j=2}^{n}P_{kij}^{+}\varepsilon^{k}x^{i}y^{j}\\\\[8.61108pt]
-\varepsilon^{2}x+\vartheta_{2}\varepsilon
y+\displaystyle\sum\limits_{i+j=2}^{n}B_{ij}^{+}x^{i}y^{j}+\displaystyle\sum\limits_{k=1,i+j=2}^{n}Q_{kij}^{+}\varepsilon^{k}x^{i}y^{j}\end{array}\right),\
\ &\text{if}\ \ x\geq 0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}\vartheta_{1}\varepsilon
x+y+\displaystyle\sum\limits_{i+j=2}^{n}A_{ij}^{-}x^{i}y^{i}+\displaystyle\sum\limits_{k=1,i+j=2}^{n}P_{kij}^{-}\varepsilon^{k}x^{i}y^{j}\\\\[8.61108pt]
-\varepsilon^{2}x+\vartheta_{2}\varepsilon
y+\displaystyle\sum\limits_{i+j=2}^{n}B_{ij}^{-}x^{i}y^{j}+\displaystyle\sum\limits_{k=1,i+j=2}^{n}Q_{kij}^{-}\varepsilon^{k}x^{i}y^{j}\end{array}\right),\
\ &\text{if}\ \ x<0,\end{aligned}\right.$ (13)

where $-\varepsilon^{2}x$ is called unfolding with sufficiently small
$\varepsilon>0$,
$\lambda^{\pm}=(\vartheta_{1,2},A_{ij}^{\pm},B_{ij}^{\pm},P_{ij}^{\pm},Q_{ij}^{\pm})$
represent two parameter vectors. For convenience, we denote that (10) is the
limit system of (13).

Note that the linear perturbation terms used in (13) are $\varepsilon^{2}x$
rather than $\varepsilon x$ can avoid the $\sqrt{\varepsilon}$ and
$\frac{1}{\varepsilon^{k}}$ perturbation terms ($k\in\mathbb{Q^{+}}$) in later
transformed systems. Based on the relation established for these two systems
(10) and (13), and the B-T bifurcation theory, we directly have the following
lemma. More detailed discussions on this subject can be found in [7].

######  Lemma 2.3.

Assume that the origin of (10) is monodromic, and that the two systems in (10)
are limit systems of (13). If the linear-type elementary singular point at the
origin of (13) becomes a linear-type center, then the origin of (10) is a
nilpotent center.

We give the following example to illustrate our basic idea, as it is known
that the nilpotent polynomial smooth systems can be transformed to the normal
form (see [20]),

$\displaystyle\dot{x}=$ $\displaystyle\ y,$ (14) $\displaystyle\dot{y}=$
$\displaystyle\
a_{k}\big{(}1+h(x)\big{)}x^{k}+b_{k}y\big{(}1+g(x)\big{)}x^{k-1}+y^{2}p(x,y),$

where $k\geq 2$, $h(x)$, $g(x)$ and $p(x,y)$ are real analytic functions
satisfying $h(0)=g(0)=0$. For example, we consider a codimension-2 symmetric
B-T bifurcation of the cubic normal form (14), given in the following form:

$\displaystyle\dot{x}=$ $\displaystyle\ y,$ (15) $\displaystyle\dot{y}=$
$\displaystyle\ \varepsilon_{1}x+\varepsilon_{2}y+a_{3}x^{3}+b_{3}x^{2}y,$

where the terms $\varepsilon_{1}x+\varepsilon_{2}y$ are called unfolding with
small $\varepsilon_{1}$ and $\varepsilon_{2}$. Note that if
$\varepsilon_{1}<0$, the system (15) can have Hopf bifurcation near the origin
from the bifurcation line $\varepsilon_{2}=0$. It is easy to check that the
elementary origin of (15) is a center if and only if $b_{3}=0$. The isolated
elementary origin of (15) is reduced to a nilpotent point when
$\varepsilon_{1}\rightarrow 0^{-}$ and $\varepsilon_{2}\rightarrow 0$.
Furthermore, it can be verified that the nilpotent monodromic origin (when
$a_{3}<0$) of system (15) without unfolding is also a center if $b_{3}=0$.

The difficulty arising from the problem of distinguishing a center from a
focus in switching nilpotent system (10) is that the problem may be not
algebraically solvable. That is, it does not have an infinite sequence of
independent polynomials involving the coefficients of the systems such that
the Lyapunov constants vanish simultaneously, which guarantees the existence
of a center. We will show that the higher-order Poincaré-Lyapunov method we
develop for the switching polynomial system (13) to determine nilpotent center
can overcome the difficulty. More details are described below.

To achieve this, introducing the transformation
$(x,y,t)\rightarrow(\varepsilon^{3}x,\varepsilon^{2}y,\frac{t}{\varepsilon})$
into (13), we obtain

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}\vartheta_{2}x-y+\displaystyle\sum\limits_{k=1,i+j=2}^{n}\widetilde{A_{kij}}^{+}(\lambda^{+})\varepsilon^{k}x^{i}y^{j}\\\\[10.76385pt]
x+\vartheta_{1}y+\displaystyle\sum\limits_{k-1,i+j=2}^{n}\widetilde{B_{kij}}^{+}(\lambda^{+})\varepsilon^{k}x^{i}y^{j}\end{array}\right),\
\ &\text{if}\ \ y\geq 0,\\\\[2.15277pt]
&\left(\begin{array}[]{c}\vartheta_{2}x-y+\displaystyle\sum\limits_{k=1,i+j=2}^{n}\widetilde{A_{kij}}^{-}(\lambda^{-})\varepsilon^{k}x^{i}y^{j}\\\\[10.76385pt]
x+\vartheta_{1}y+\displaystyle\sum\limits_{k=1,i+j=2}^{n}\widetilde{B_{kij}}^{-}(\lambda^{-})\varepsilon^{k}x^{i}y^{j}\end{array}\right),\
\ &\text{if}\ \ y<0,\end{aligned}\right.$ (16)

where $\widetilde{A_{kij}}^{\pm}(\lambda^{\pm})$ and
$\widetilde{B_{kij}}^{\pm}(\lambda^{\pm})$ are the polynomials in the
parameter vector $\lambda^{\pm}$.

With the polar coordinates transformation: $x=r\cos\theta$ and
$y=r\sin\theta$, the perturbation system (16) becomes the form,

$\frac{{\rm d}r}{{\rm d}\theta}=\left\\{\begin{aligned}
&\frac{\displaystyle\sum\limits_{k=0,i+j=1}^{n}c_{kij}^{+}(\lambda^{+})\cos\theta^{i}\sin\theta^{j}\varepsilon^{k}r^{i+j}}{1+\displaystyle\sum\limits_{k=0,i+j=1}^{n}d_{kij}^{+}(\lambda^{+})\cos\theta^{i}\sin\theta^{j}\varepsilon^{k}r^{i+j-1}},\quad\text{if}\
\ \theta\in[0,\pi],\\\\[4.30554pt]
&\frac{\displaystyle\sum\limits_{k=0,i+j=1}^{n}c_{kij}^{-}(\lambda^{-})\cos\theta^{i}\sin\theta^{j}\varepsilon^{k}r^{i+j}}{1+\displaystyle\sum\limits_{k=0,i+j=1}^{n}d_{ij}^{-}(\lambda^{-})\cos\theta^{i}\sin\theta^{j}\varepsilon^{k}r^{i+j-1}},\quad\text{if}\
\ \theta\in(\pi,2\pi),\end{aligned}\right.$ (17)

where $c_{kij}^{\pm}(\lambda^{\pm})$ and $d_{kij}^{\pm}(\lambda^{\pm})$ are
the polynomials in the parameter vector $\lambda^{\pm}$.

Let $r^{+}(\xi,\lambda^{+},\varepsilon,\theta)=\sum_{k\geq
1}v_{k}^{+}(\lambda^{+},\varepsilon,\theta)\xi^{k}$ and
$r^{-}(\xi,\lambda^{-},\varepsilon,\theta)=\sum_{k\geq
1}v_{k}^{-}(\lambda^{+},\varepsilon,\theta)\xi^{k}$ be the solutions of the
first and second systems of (17) associated with the initial conditions
$r^{+}(\xi,\lambda^{+},\varepsilon,0)=r^{-}(\xi,\lambda^{-},\varepsilon,\pi)=\xi$.
We denote by

$\Pi^{+}(\xi)=r^{+}(\xi,\lambda^{+},\varepsilon,\pi)=\sum\limits_{k\geq
1}v_{k}^{+}(\lambda^{+},\varepsilon,\pi)\xi^{k}$

and

$\Pi^{-}(\xi)=r^{-}(\xi,\lambda^{+},\varepsilon,2\pi)=\sum\limits_{k\geq
1}v_{k}^{-}(\lambda^{-},\varepsilon,\pi)\xi^{k},$

the first half-return map $\Pi^{+}(\xi)$ and the second half-return map
$\Pi^{-}(\xi)$, respectively, where
$v_{k}^{\pm}(\lambda^{\pm},\varepsilon,\pi)$ are the coefficients in Taylor
expansions. However, it is extremely difficult to composite these two maps to
compute the displacement map of (17). We may follow the procedure in [17] and
introduce the transformation $(x,y,t)\rightarrow(x,-y,-t)$ into the piecewise
smooth system (16) to yield

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}-\vartheta_{2}x-y-\displaystyle\sum\limits_{k=1,i+j=2}^{n}\widetilde{A_{kij}}^{-}(\lambda^{-})\varepsilon^{k}x^{i}(-y)^{j}\\\\[10.76385pt]
x-\vartheta_{1}y+\displaystyle\sum\limits_{k-1,i+j=2}^{n}\widetilde{B_{kij}}^{-}(\lambda^{-})\varepsilon^{k}x^{i}(-y)^{j}\end{array}\right),\
\ &\text{if}\ \ y>0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}-\vartheta_{2}x-y+\displaystyle\sum\limits_{k=1,i+j=2}^{n}\widetilde{A_{kij}}^{+}(\lambda^{+})\varepsilon^{k}x^{i}(-y)^{j}\\\\[10.76385pt]
x-\vartheta_{1}y+\displaystyle\sum\limits_{k=1,i+j=2}^{n}\widetilde{B_{kij}}^{+}(\lambda^{+})\varepsilon^{k}x^{i}y^{j}\end{array}\right),\
\ &\text{if}\ \ y\leq 0.\end{aligned}\right.$ (18)

Then, the displacement function can be written as

$\displaystyle
d(\xi)=\Pi^{+}(\xi)-(\Pi^{-})^{-1}(\xi)=\Pi^{+}(\xi)-\Pi_{-}^{+}(\xi)=\sum\limits_{j\geq
1}V_{j}(\lambda^{\pm},\varepsilon,\pi)\xi^{j},$ (19)

where $\Pi_{-}^{+}(\xi)$ is the first half-return map of (18), and

$V_{j}(\lambda^{\pm},\varepsilon,\pi)=\sum_{k=1}^{\infty}V_{jk}\,\varepsilon^{k},\quad
j=1,2,\ldots,$ (20)

in which $V_{jk}$ denotes the $j$th $\varepsilon^{k}$-order Lyapunov constant,
see [7] for more details about the computation of the generalized Lyapunov
constants. Hence, we can derive the center conditions of system (13) by
vanishing the $\varepsilon$ terms in these generalized Lyapunov constants, and
then by Lemma 2.3 we derive the algebraic conditions characterizing the
nilpotent center of system (10). Thus, we prove that these conditions are
necessary for $(0,0)$ of (10) be a nilpotent center. In general, these
nilpotent center conditions can be satisfied by appropriately choosing the
perturbation coefficients $P_{kij}^{\pm}$ and $Q_{kij}^{\pm}$.

Further, we generalize our method to study the bifurcation of limit cycles
from the nilpotent center of the switching system (10). Actually, by B-T
bifurcation theory, we may add the linear perturbation term
$-\varepsilon^{2}x$ to the switching system (10) such that the system has a
linear-type center at the origin. Then we perturb such system by adding higher
$\varepsilon$-order terms, and compute the generalized Lyapunov constants of
the perturbed system. Finally, by using the higher $\varepsilon$-order
Lyapunov constants [41, 43], we find the bifurcation of limit cycles from the
center as many as possible. More precisely, with the result in [40], we derive
the following lemma giving the sufficient conditions for the existence of
small-amplitude limit cycles around the origin of (10).

######  Lemma 2.4 ([Lemma 4, [40]).

If there exists a critical point $\lambda_{*}=(a_{1c},a_{2c},\cdots,a_{nc})$
such that a set of $\varepsilon^{k}$-order Lyapunov constants satisfies
$V_{1k}(\lambda_{*})=V_{2k}(\lambda_{*})=\cdots=V_{nk}(\lambda_{*})=0$,
$V_{n+1,k}(\lambda_{*})\neq 0$ and

$\begin{array}[]{l}{\rm{det}}\bigg{[}\displaystyle\frac{\partial(V_{1k},V_{2k},\cdots,V_{nk})}{\partial(a_{1c},a_{2c},\cdots,a_{nc})}(\lambda_{*})\bigg{]}\neq
0,\end{array}$ (21)

then small appropriate perturbations about $\lambda=\lambda_{*}$ lead to that
the switching system (16) has exactly $n$ limit cycles bifurcating from
$(0,0)$.

##  3 The proof of Theorem 1.1

From the results given in the precious section, we know that the nilpotent
center of (7) at the origin can be combined by a monodromic singular point or
a cusp, i.e. the parameters satisfy the condition $\Xi_{1}$ or $\Xi_{2}$.
Firstly, assume that the origin in the first smooth system of (7) is a cusp,
then we have $b_{2}^{+}\neq 0$. If $b_{2}^{-}\neq 0$, then the origin in the
second system of (7) is also a cusp with multiplicity two. If $b_{2}^{-}=0$,
we have the following statement: The origin of the second system of (7) is a
monodromic singular point with multiplicity three if and only if one of the
following conditions holds:

$\displaystyle a_{2}^{-}=b_{2}^{-}=0,\quad
b_{3}^{-}>\dfrac{3}{4}\,(a_{3}^{-})^{2};$ (22) $\displaystyle
b_{2}^{-}=0,\quad a_{2}^{-}\neq 0,\quad
b_{3}^{-}>\dfrac{1}{3}\,(a_{2}^{-})^{2}.$

We only consider $b_{2}^{+}>0$ and $b_{2}^{-}\leq 0$ because the origin of (7)
cannot be monodromic when either $b_{2}^{+}<0$ or $b_{2}^{-}>0$, which are
illustrated in the two phase portraits of system (7), as depicted in Figure 1.
It is shown that when $b_{2}^{+}<0$ the first system of (7) has two
seperatrices connecting the origin and two singular points in the right half
Poincaré disc, see Figure 1. Similarly when $b_{2}^{-}>0$ the second system of
(7) has two seperatrices connecting the origin and the other two ones in the
left half Poincaré disc, see Figure 1. Thus, the origin cannot be a center
when either $b_{2}^{+}<0$ or $b_{2}^{-}>0$.

######  Example 3.1.

Two different phase portraits of the cubic switching Liénard system (7), as
depicted in the Poincaré disc, show that the origin is a cusp for
$b_{2}^{+}=-1$ (see Figure 1(a)) or $b_{2}^{-}=3$ (see Figure 1(b)).

Figure 1: Two phase portraits of the cubic switching Liénard system (7)
showing that the origin is a cusp for (a) $a_{2}^{\pm}=b_{3}^{\pm}=1$,
$a_{3}^{\pm}=2$, $b_{2}^{+}=-1$ and $b_{2}^{-}=-2$; and (b)
$a_{2}^{\pm}=a_{3}^{\pm}=b_{3}^{+}=1$, $b_{3}^{-}=-1$, $b_{2}^{+}=2$ and
$b_{2}^{-}=3$.

Next, we discuss how to apply the higher-order Poincaré-Lyapunov method to
derive the nilpotent center conditions for the origin of system (7). With
perturbations added to system (7), we obtain the following perturbed systems,

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}y-(a_{2}^{+}x^{2}+a_{3}^{+}x^{3})+\displaystyle\sum_{k=1}\varepsilon^{k}P_{k}^{+}(x)\\\\[12.91663pt]
-\varepsilon^{2}x-(b_{2}^{+}x^{2}+b_{3}^{+}x^{3})+\displaystyle\sum_{k=1}\varepsilon^{k}Q_{k}^{+}(x)\end{array}\right),&\text{if}\
\ x\geq 0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}y-(a_{2}^{-}x^{2}+a_{3}^{-}x^{3})+\displaystyle\sum_{k=1}\varepsilon^{k}P_{k}^{-}(x)\\\\[12.91663pt]
-\varepsilon^{2}x-(b_{2}^{-}x^{2}+b_{3}^{-}x^{3})+\displaystyle\sum_{k=1}\varepsilon^{k}Q_{k}^{-}(x)\end{array}\right),&\text{if}\
\ x<0,\end{aligned}\right.$ (23)

where

$P_{k}^{\pm}(x)=p_{k2}^{\pm}\,x^{2}+p_{k3}^{\pm}\,x^{3},\quad
Q_{k}^{\pm}(x)=q_{k2}^{\pm}\,x^{2}+q_{k3}^{\pm}\,x^{3},$

in which $p_{ki}^{\pm}$, $q_{ki}^{\pm}$ are real parameters. It is very
difficult for computing the generalized Lyapuov constants of (23) with large
number of parameters. So we only consider the $\varepsilon^{2}$-order
perturbations. Further, introducing the transformation
$(x,y,t)\rightarrow(\varepsilon^{3}x,\varepsilon^{2}y,\frac{t}{\varepsilon})$
into system (23), we obtain

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}-y-(b_{2}^{+}+q_{22}^{+}\varepsilon^{2})y^{2}-(b_{3}^{+}\varepsilon^{2}+q_{23}^{+}\varepsilon^{4})y^{3}\\\\[4.30554pt]
x-(a_{2}^{+}\varepsilon+p_{22}^{+}\varepsilon^{3})y^{2}-(a_{3}^{+}\varepsilon^{3}+p_{23}^{+}\varepsilon^{5})y^{3}\end{array}\right),&\text{if}\
\ y\geq 0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}-y-(b_{2}^{-}+q_{22}^{-}\varepsilon^{2})y^{2}-(b_{3}^{-}\varepsilon^{2}+q_{23}^{-}\varepsilon^{4})y^{3}\\\\[4.30554pt]
x-(a_{2}^{-}\varepsilon+p_{22}^{-}\varepsilon^{3})y^{2}-(a_{3}^{-}\varepsilon^{3}+p_{23}^{-}\varepsilon^{5})y^{3}\end{array}\right),&\text{if}\
\ y<0.\end{aligned}\right.$ (24)

Then, we consider two cases: $b_{2}^{-}=0$ and $b_{2}^{-}<0$ to obtain the
nilpotent center conditions by considering each $i$th $\varepsilon^{k}$-order
Lyapunov constant.

###  3.1 Case 1: $b_{2}^{-}=0$

We use the higher-order Poincaré-Lyapunov method to compute the generalized
Lyapunov constants associated with the origin of (24). The first two
generalized Lyapunov constants are $V_{1}(\varepsilon)=0$ and

$V_{2}(\varepsilon)=\frac{4}{3}\big{[}(a_{2}^{+}-a_{2}^{-})+(p_{22}^{+}-p_{22}^{-})\varepsilon^{2}\big{]}\,\varepsilon.$
(25)

By (22) we consider the two subcases: (i) $a_{2}^{-}=0$,
$b_{3}^{-}>\frac{3}{4}(a_{3}^{-})^{2}$, and (ii) $a_{2}^{-}\neq 0$,
$b_{3}^{-}>\frac{1}{3}(a_{2}^{-})^{2}$.

  1. (i)

$a_{2}^{-}=0$, $b_{3}^{-}>\frac{3}{4}(a_{3}^{-})^{2}$. Setting the
$\varepsilon$-order and $\varepsilon^{3}$-order Lyapunov constants in
$V_{2}(\varepsilon)$ zero yields the necessary center conditions $a_{2}^{+}=0$
and $p_{22}^{-}=p_{22}^{+}$. Then, the $3$rd generalized Lyapunov constant is
obtained as

$V_{3}(\varepsilon)=-\,\frac{\pi}{8}\big{[}3a_{3}^{+}+3a_{3}^{-}-2b_{2}^{+}p_{22}^{+}+(3p_{23}^{+}+3p_{23}^{-}-2p_{22}^{+}q_{22}^{+}-2p_{22}^{+}q_{22}^{-})\varepsilon^{2}\big{]}\,\varepsilon^{3}.$

Letting the $\varepsilon^{3}$-order and $\varepsilon^{5}$-order terms in
$V_{3}(\varepsilon)$ equal zero we obtain the following conditions,

$\displaystyle p_{22}^{+}=$ $\displaystyle\
\frac{3(a_{3}^{+}+a_{3}^{-})}{2b_{2}^{+}},$ $\displaystyle p_{23}^{-}=$
$\displaystyle\
\frac{1}{b_{2}^{+}}\big{(}\\!-b_{2}^{+}p_{23}^{+}+a_{3}^{+}q_{22}^{+}+a_{3}^{-}q_{22}^{+}+a_{3}^{+}q_{22}^{-}+a_{3}^{-}q_{22}^{-}\big{)}.$

Then, we have the 4th generalized Lyapunov constant, given by

$V_{4}(\varepsilon)=\frac{4}{15b_{2}^{+}}\,\big{[}4a_{3}^{-}(b_{2}^{+})^{2}-M_{1}\varepsilon^{2}+M_{2}\varepsilon^{4}\big{]}\,\varepsilon^{3},$

where

$\displaystyle M_{1}=$ $\displaystyle\
3a_{3}^{+}b_{3}^{+}+3a_{3}^{-}b_{3}^{+}-3a_{3}^{+}b_{3}^{-}-3a_{3}^{-}b_{3}^{-}-4(b_{2}^{+})^{2}p_{23}^{-}-4a_{3}^{-}b_{2}^{+}q_{22}^{+}+4a_{3}^{+}b_{2}^{+}q_{22}^{-},$
$\displaystyle M_{2}=$ $\displaystyle\
4b_{2}^{+}p_{23}^{-}q_{22}^{+}+4b_{2}^{+}p_{23}^{-}q_{22}^{-}-4a_{3}^{+}q_{22}^{+}q_{22}^{-}-4a_{3}^{-}q_{22}^{+}q_{22}^{-}-4a_{3}^{+}(q_{22}^{-})^{2}-4a_{3}^{-}(q_{22}^{-})^{2}$
$\displaystyle-3a_{3}^{+}q_{23}^{+}-3a_{3}^{-}q_{23}^{+}+3a_{3}^{+}q_{23}^{-}+3a_{3}^{-}q_{23}^{-}.$

Letting each $\varepsilon$ term in $V_{4}(\varepsilon)$ be zero, we obtain the
conditions:

$\displaystyle a_{3}^{-}=$ $\displaystyle\ 0,$ $\displaystyle p_{23}^{-}=$
$\displaystyle\
\frac{3a_{3}^{+}b_{3}^{+}-3a_{3}^{+}b_{3}^{-}+4a_{3}^{+}b_{2}^{+}q_{22}^{-}}{4(b_{2}^{+})^{2}},$
$\displaystyle q_{23}^{-}=$ $\displaystyle\
\frac{1}{b_{2}^{+}}\big{(}\\!-b_{3}^{+}q_{22}^{+}+b_{3}^{-}q_{22}^{+}-b_{3}^{+}q_{22}^{-}+b_{3}^{-}q_{22}^{-}+b_{2}^{+}q_{23}^{+}\big{)}.$

Then, we have

$V_{5}(\varepsilon)=-\frac{5\pi}{256(b_{2}^{+})^{3}}a_{3}^{+}\,(b_{2}^{+}+q_{22}^{+}\varepsilon^{2}+q_{22}^{-}\varepsilon^{2})\big{[}2(b_{2}^{+})^{2}(b_{3}^{+}+7b_{3}^{-})+M_{3}\varepsilon^{2}\big{]}\,\varepsilon^{5},$

where

$\displaystyle M_{3}=$ $\displaystyle\
9(b_{3}^{+})^{2}-18b_{3}^{+}b_{3}^{-}+9(b_{3}^{-})^{2}-14b_{2}^{+}b_{3}^{+}q_{22}^{+}+14b_{2}^{+}b_{3}^{-}q_{22}^{+}-2b_{2}^{+}b_{3}^{+}q_{22}^{-}$
$\displaystyle+2b_{2}^{+}b_{3}^{-}q_{22}^{-}+16(b_{2}^{+})^{2}q_{23}^{+}.$

$a_{3}^{+}=0$ (leading to $V_{5}(\varepsilon)=0$) gives the condition I.
Otherwise, if $a_{3}^{+}\neq 0$, setting the other $\varepsilon$ terms in
$V_{5}(\varepsilon)$ zero we obtain

$\displaystyle b_{3}^{+}=-\,7b_{3}^{-},\quad
q_{23}^{+}=-\,\frac{1}{(b_{2}^{+})^{2}}\big{[}36(b_{3}^{-})^{2}+7b_{2}^{+}b_{3}^{-}q_{22}^{+}+b_{2}^{+}b_{3}^{-}q_{22}^{-}\big{]}.$

Further, under the above conditions, we have the 6th generalized Lyapunov
constant given by

$\displaystyle V_{6}(\varepsilon)=$
$\displaystyle-\frac{128}{315(b_{2}^{+})^{2}}a_{3}^{+}b_{3}^{-}\,(b_{2}^{+}+q_{22}^{+}\varepsilon^{2}+q_{22}^{-}\varepsilon^{2})^{3}\varepsilon^{5}\neq
0$

when $a_{3}^{+}\neq 0$.

  1. (ii)

$a_{2}^{-}\neq 0$, $b_{3}^{-}>\frac{1}{3}(a_{2}^{-})^{2}$. Setting the
$\varepsilon$-order and $\varepsilon^{3}$-order terms in $V_{2}(\varepsilon)$
zero yields the necessary center conditions $a_{2}^{-}=a_{2}^{+}\neq 0$ and
$p_{22}^{-}=p_{22}^{+}$. Then, the $3$rd generalized Lyapunov constant is
given by

$\displaystyle V_{3}(\varepsilon)=$ $\displaystyle\
\frac{\pi}{8}\big{[}2a_{2}^{-}b_{2}^{+}-(3a_{3}^{+}+3a_{3}^{-}-2b_{2}^{+}p_{22}^{+}-2a_{2}^{+}q_{22}^{+}-2a_{2}^{+}q_{22}^{-})\varepsilon^{2}-(3p_{23}^{+}+3p_{23}^{-}$
$\displaystyle-2p_{22}^{+}q_{22}^{+}-2p_{22}^{+}q_{22}^{-})\varepsilon^{4}\big{]}\varepsilon\neq
0.$

###  3.2 Case 2: $b_{2}^{-}<0$

Similarly, setting each $\varepsilon$ term in $V_{2}(\varepsilon)$ zero yields
the necessary center conditions $a_{2}^{-}=a_{2}^{+}$ and
$p_{22}^{-}=p_{22}^{+}$. Then, we obtain the $3$rd generalized Lyapunov
constant,

$\displaystyle V_{3}(\varepsilon)=$ $\displaystyle\
\frac{\pi}{8}\big{[}2a_{2}^{+}(b_{2}^{+}+b_{2}^{-})-(3a_{3}^{+}+3a_{3}^{-}-2b_{2}^{+}p_{22}^{+}-2b_{2}^{-}p_{22}^{+}-2a_{2}^{+}q_{22}^{+}-2a_{2}^{+}q_{22}^{-})\varepsilon^{2}$
$\displaystyle-(3p_{23}^{+}+3p_{23}^{-}-2p_{22}^{+}q_{22}^{+}-2p_{22}^{+}q_{22}^{-})\varepsilon^{4}\big{]}\varepsilon.$

Considering the $\varepsilon$-order term in $V_{3}(\varepsilon)$, we have two
subcases: (i) $a_{2}^{+}=0$, and (ii) $b_{2}^{+}+b_{2}^{-}=0$.

  1. (i)

Assume that $a_{2}^{+}=0$ and $b_{2}^{+}+b_{2}^{-}\neq 0$. We let the
$\varepsilon^{3}$-order and $\varepsilon^{5}$-order terms in
$V_{3}(\varepsilon)$ equal zero to obtain the conditions,

$\displaystyle p_{22}^{+}=$ $\displaystyle\
\frac{3(a_{3}^{+}+a_{3}^{-})}{2(b_{2}^{+}+b_{2}^{-})},$ $\displaystyle
p_{23}^{-}=$ $\displaystyle\
\frac{1}{b_{2}^{+}+b_{2}^{-}}\big{(}b_{2}^{+}p_{23}^{+}+b_{2}^{-}p_{23}^{+}-a_{3}^{+}q_{22}^{+}-a_{3}^{-}q_{22}^{+}-a_{3}^{+}q_{22}^{-}-a_{3}^{-}q_{22}^{-}\big{)}.$

Then, the 4th generalized Lyapunov constant becomes

$\displaystyle V_{4}(\varepsilon)=$ $\displaystyle\
\frac{-\,4}{15(b_{2}^{+}+b_{2}^{-})}\,\big{[}4(b_{2}^{+}+b_{2}^{-})(a_{3}^{-}b_{2}^{+}-a_{3}^{+}b_{2}^{-})-M_{4}\varepsilon^{2}-M_{5}\varepsilon^{4}\big{]}\varepsilon^{3},$

where

$\displaystyle M_{4}=$ $\displaystyle\
3a_{3}^{+}b_{3}^{+}+3a_{3}^{-}b_{3}^{+}-3a_{3}^{+}b_{3}^{-}-3a_{3}^{-}b_{3}^{-}+4(b_{2}^{+})^{2}p_{23}^{+}+8b_{2}^{+}b_{2}^{-}p_{23}^{+}+4(b_{2}^{-})^{2}p_{23}^{+}$
$\displaystyle-4a_{3}^{+}b_{2}^{+}q_{22}^{+}-8a_{3}^{-}b_{2}^{+}q_{22}^{+}-4a_{3}^{-}b_{2}^{-}q_{22}^{+}-4a_{3}^{-}b_{2}^{+}q_{22}^{-}+4a_{3}^{+}b_{2}^{-}q_{22}^{-},$
$\displaystyle M_{5}=$ $\displaystyle\
4b_{2}^{+}p_{23}^{+}q_{22}^{+}+4b_{2}^{-}p_{23}^{+}q_{22}^{+}-4a_{3}^{+}(q_{22}^{+})^{2}-4a_{3}^{-}(q_{22}^{+})^{2}+4b_{2}^{+}p_{23}^{+}q_{22}^{-}+4b_{2}^{-}p_{23}^{+}q_{22}^{-}$
$\displaystyle-4a_{3}^{+}q_{22}^{+}q_{22}^{-}-4a_{3}^{-}q_{22}^{+}q_{22}^{-}+3a_{3}^{+}q_{23}^{+}+3a_{3}^{-}q_{23}^{+}-3a_{3}^{+}q_{23}^{-}-3a_{3}^{-}q_{23}^{-}.$

Thus, we have

$a_{3}^{-}=\frac{a_{3}^{+}b_{2}^{-}}{b_{2}^{+}},\quad
p_{23}^{+}=\frac{-3a_{3}^{+}b_{2}^{+}+3a_{3}^{+}b_{3}^{-}+4a_{3}^{+}b_{2}^{+}q_{22}^{+}+4a_{3}^{+}b_{2}^{-}q_{22}^{+}}{4b_{2}^{+}(b_{2}^{+}+b_{2}^{-})}$

by setting the $\varepsilon^{3}$-order and $\varepsilon^{5}$-order terms in
$V_{4}(\varepsilon)$ zero. Then, $M_{5}$ is simplified as

$M_{5}=-\frac{3}{b_{2}^{+}}a_{3}^{+}(b_{3}^{+}q_{22}^{+}-b_{3}^{-}q_{22}^{+}+b_{3}^{+}q_{22}^{-}-b_{3}^{-}q_{22}^{-}-b_{2}^{+}q_{23}^{+}-b_{2}^{-}q_{23}^{+}+b_{2}^{+}q_{23}^{-}+b_{2}^{-}q_{23}^{-}).$

(i.1) If $a_{3}^{+}=0$, we obtain $M_{5}=0$, leading to a condition included
in the condition II.

(i.2) If $a_{3}^{+}\neq 0$, we have

$q_{23}^{+}=\frac{1}{b_{2}^{+}+b_{2}^{-}}(b_{3}^{+}q_{22}^{+}-b_{3}^{-}q_{22}^{+}+b_{3}^{+}q_{22}^{-}-b_{3}^{-}q_{22}^{-}+b_{2}^{+}q_{23}^{-}+b_{2}^{-}q_{23}^{-})$

by setting $M_{5}=0$. Further, we obtain the $5$th generalized Lyapunov
constant,

$\displaystyle V_{5}(\varepsilon)=$
$\displaystyle-\frac{5a_{3}^{+}\pi}{256b_{2}^{+}(b_{2}^{+}+b_{2}^{-})^{2}}\,\big{[}b_{2}^{+}+b_{2}^{-}+(q_{22}^{+}+q_{22}^{-})\varepsilon^{2}\big{]}\big{[}2(b_{2}^{+}+b_{2}^{-})M_{6}+M_{7}\varepsilon^{2}\big{]}\varepsilon^{5}$

where

$\displaystyle M_{6}=$ $\displaystyle\
b_{3}^{+}(b_{2}^{+}+7b_{2}^{-})+b_{3}^{-}(7b_{2}^{+}+b_{2}^{-}),$
$\displaystyle M_{7}=$ $\displaystyle\
9(b_{3}^{+})^{2}-18b_{3}^{+}b_{3}^{-}+9(b_{3}^{-})^{2}+2b_{2}^{+}b_{3}^{+}q_{22}^{+}+2b_{2}^{-}b_{3}^{+}q_{22}^{+}-2b_{2}^{+}b_{3}^{-}q_{22}^{+}-2b_{2}^{-}b_{3}^{-}q_{22}^{+}$
$\displaystyle+14b_{2}^{+}b_{3}^{+}q_{22}^{-}+14b_{2}^{-}b_{3}^{+}q_{22}^{-}-14b_{2}^{+}b_{3}^{-}q_{22}^{-}-14b_{2}^{-}b_{3}^{-}q_{22}^{-}+16(b_{2}^{+})^{2}q_{23}^{-}$
$\displaystyle+32b_{2}^{+}b_{2}^{-}q_{23}^{-}+16(b_{2}^{-})^{2}q_{23}^{-}.$

Since $a_{3}^{+}(b_{2}^{+}+b_{2}^{-})\neq 0$, the only possibility for
$V_{5}(\varepsilon)=0$ is $M_{6}=M_{7}=0$. Considering $M_{6}=0$, if
$b_{2}^{+}+7b_{2}^{-}\neq 0$, we have
$b_{3}^{+}=\frac{-(7b_{2}^{+}+b_{2}^{-})b_{3}^{-}}{b_{2}^{+}+7b_{2}^{-}}$; and
if $b_{2}^{+}=-7b_{2}^{-}$, we obtain $M_{6}=-48b_{2}^{-}b_{3}^{-}=0$ which
yields $b_{3}^{-}=0$ due to $b_{2}^{-}<0$. Thus, we consider the following two
subcases.

(i.2.1) If
$b_{3}^{+}=\frac{-(7b_{2}^{+}+b_{2}^{-})b_{3}^{-}}{b_{2}^{+}+7b_{2}^{-}}$,
setting $M_{7}=0$ yields

$\displaystyle
q_{23}^{-}=\frac{1}{(b_{2}^{+}+7b_{2}^{-})^{2}}[-36(b_{3}^{-})^{2}+b_{2}^{+}b_{3}^{-}q_{22}^{+}+7b_{2}^{-}b_{3}^{-}q_{22}^{+}+7b_{2}^{+}b_{3}^{-}q_{22}^{-}+49b_{2}^{-}b_{3}^{-}q_{22}^{-}].$

Then, we obtain the $6$th generalized Lyapunov constant,

$\displaystyle V_{6}(\varepsilon)=$
$\displaystyle-\frac{128}{315b_{2}^{+}(b_{2}^{+}+7b_{2}^{-})}a_{3}^{+}b_{3}^{-}\big{[}b_{2}^{+}+b_{2}^{-}+(q_{22}^{+}+q_{22}^{-})\varepsilon^{2}\big{]}^{3}\varepsilon^{5}.$

If $b_{3}^{-}=0$, we have $V_{6}(\varepsilon)=0$, and this necessary condition
is included in the condition III. Otherwise, we have $V_{6}(\varepsilon)\neq
0$ when $b_{2}^{+}+b_{2}^{-}\neq 0$.

(i.2.2) If $b_{3}^{-}=b_{2}^{+}+7b_{2}^{-}=0$ we have

$\displaystyle q_{23}^{-}=$
$\displaystyle\frac{1}{192(b_{2}^{-})^{2}}[-3(b_{3}^{+})^{2}+4b_{2}^{-}b_{3}^{+}q_{22}^{-}+28b_{2}^{-}b_{3}^{+}q_{22}^{-}]$

from $M_{7}=0$. Then the $6$th generalized Lyapunov constant has the following
form

$\displaystyle V_{6}(\varepsilon)=$
$\displaystyle-\frac{8}{6615(b_{2}^{-})^{2}}a_{3}^{+}b_{3}^{+}\big{[}6b_{2}^{-}+(q_{22}^{+}+q_{22}^{-})\varepsilon^{2}\big{]}^{3}\varepsilon^{5}\neq
0.$

Assume that $b_{3}^{+}=0$, we have $V_{6}(\varepsilon)=0$. Combining the
necessary conditions from (i.2.1) and (i.2.2), we obtain the condition III.
Otherwise we have $V_{6}(\varepsilon)\neq 0$ when $a_{3}\neq 0$.

  1. (ii)

Assume that $b_{2}^{-}+b_{2}^{+}=0$. Consider two subcases: (ii.1)
$a_{2}^{+}=0$ and (ii.2) $a_{2}^{+}\neq 0$.

(ii.1) If $a_{2}^{+}=0$, setting each $\varepsilon$ term in
$V_{3}(\varepsilon)$ zero we have

$\displaystyle a_{3}^{-}=$ $\displaystyle-a_{3}^{+},$ $\displaystyle
p_{23}^{-}=$ $\displaystyle\
\frac{1}{3}(-3p_{23}^{+}+2p_{22}^{+}q_{22}^{+}+2p_{22}^{+}q_{22}^{-}).$

Then, the 4th generalized Lyapunov constant is given by

$\displaystyle V_{4}(\varepsilon)=$
$\displaystyle-\frac{8}{45}\,\big{(}M_{8}+M_{9}\varepsilon^{2}\big{)}\,\varepsilon^{5},$

where

$\displaystyle M_{8}=$
$\displaystyle-3b_{3}^{+}p_{22}^{+}+3b_{3}^{-}p_{22}^{+}-6a_{3}^{+}q_{22}^{+}+4b_{2}^{+}p_{22}^{+}q_{22}^{+}-6a_{3}^{+}q_{22}^{-}+4b_{2}^{+}p_{22}^{+}q_{22}^{-},$
$\displaystyle M_{9}=$ $\displaystyle\
6p_{23}^{+}q_{22}^{+}-4p_{22}^{+}(q_{22}^{+})^{2}+6p_{23}^{+}q_{22}^{-}-4p_{22}^{+}q_{22}^{+}q_{22}^{-}+3p_{22}^{+}q_{23}^{+}-3p_{22}^{+}q_{23}^{-}.$

(ii.1.1) If $q_{22}^{-}=-q_{22}^{+}$, it follows from $M_{8}=M_{9}=0$ that
either $b_{3}^{-}=b_{3}^{+}$, $q_{23}^{-}=q_{23}^{+}$, or $p_{22}=0$. The
first choice leads to a necessity condition included in the condition IV. For
the second choice, we have the 5th generalized Lyapunov constant,

$\displaystyle V_{5}(\varepsilon)=$ $\displaystyle\
\frac{15}{64}\,\big{(}a_{3}^{+}+p_{23}^{+}\varepsilon^{2}\big{)}\big{(}b_{3}^{+}-b_{3}^{-}+q_{23}^{+}\varepsilon^{2}-q_{23}^{-}\varepsilon^{2}\big{)}\,\varepsilon^{5},$

which yields $a_{3}^{+}=p_{23}^{+}=0$ by setting $V_{5}(\varepsilon)=0$,
leading to a condition included in the condition II. Combining the subcases
(i.1) and (ii.1.1) we have the condition II.

(ii.1.2) If $q_{22}^{-}\neq-q_{22}^{+}$, we have

$\displaystyle a_{3}^{+}=$ $\displaystyle\
\frac{1}{6(q_{22}^{+}+q_{22}^{-})}(-3b_{3}^{+}p_{22}^{+}+3b_{3}^{-}p_{22}^{+}+4b_{2}^{+}p_{22}^{+}q_{22}^{+}+4b_{2}^{+}p_{22}^{+}q_{22}^{-}),$
$\displaystyle p_{23}^{+}=$ $\displaystyle\
\frac{1}{6(q_{22}^{+}+q_{22}^{-})}\big{[}4p_{22}^{+}(q_{22}^{+})^{2}+4p_{22}^{+}q_{22}^{+}q_{22}^{-}-3p_{22}^{+}q_{23}^{+}+3p_{22}^{+}q_{23}^{-}\big{]}$

by setting $M_{8}=M_{9}=0$. Then, we obtain the 5th generalized Lyapunov
constant,

$\displaystyle V_{5}(\varepsilon)=$
$\displaystyle-\frac{5\pi}{384(q_{22}^{+}+q_{22}^{-})}p_{22}^{+}\big{(}M_{10}+2M_{11}\varepsilon^{2}+M_{12}\varepsilon^{2}\big{)}\,\varepsilon^{5},$

where

$\displaystyle M_{10}=$ $\displaystyle\
3(b_{3}^{+}-b_{3}^{-})(3b_{3}^{+}-3b_{3}^{-}-4b_{2}^{+}q_{22}^{+}-4b_{2}^{+}q_{22}^{-}),$
$\displaystyle M_{11}=$ $\displaystyle\
b_{3}^{+}(q_{22}^{+})^{2}+7b_{3}^{-}(q_{22}^{+})^{2}+8b_{3}^{+}q_{22}^{+}q_{22}^{-}+8b_{3}^{-}q_{22}^{+}q_{22}^{-}+7b_{3}^{+}(q_{22}^{-})^{2}+b_{3}^{-}(q_{22}^{-})^{2}+9b_{3}^{+}q_{23}^{+}$
$\displaystyle-9b_{3}^{-}q_{23}^{+}-6b_{2}^{+}q_{22}^{+}q_{23}^{+}-6b_{2}^{+}q_{22}^{-}q_{23}^{+}-9b_{3}^{+}q_{23}^{-}+9b_{3}^{-}q_{23}^{-}+6b_{2}^{+}q_{22}^{+}q_{23}^{-}+6b_{2}^{+}q_{22}^{-}q_{23}^{-},$
$\displaystyle M_{12}=$ $\displaystyle\
2(q_{22}^{+})^{2}q_{23}^{+}+16q_{22}^{+}q_{22}^{-}q_{23}^{+}+14(q_{22}^{-})^{2}q_{23}^{+}+9(q_{23}^{+})^{2}+14(q_{22}^{+})^{2}q_{23}^{-}+16q_{22}^{+}q_{22}^{-}q_{23}^{-}$
$\displaystyle+2(q_{22}^{-})^{2}q_{23}^{-}-18q_{23}^{+}q_{23}^{-}+9(q_{23}^{-})^{2}.$

If $p_{22}^{+}=0$ or
$b_{2}^{+}=\frac{3(b_{3}^{-}-b_{3}^{+})}{4(q_{22}^{+}+q_{22}^{-})}$, we have
$a_{3}^{+}=0$, which is included in the condition II. Otherwise, we have
$b_{3}^{-}=b_{3}^{+}$ from $M_{10}$, which leads to a necessity condition
included in the condition IV.

(ii.2) If $a_{2}^{+}\neq 0$, letting the $\varepsilon^{3}$-order and
$\varepsilon^{5}$-order terms in $V_{3}(\varepsilon)$ equal zero we obtain the
conditions:

$\displaystyle q_{22}^{-}=$ $\displaystyle\
\frac{3a_{3}^{+}+3a_{3}^{-}-2a_{2}^{+}q_{22}^{+}}{2a_{2}^{+}},$ $\displaystyle
p_{23}^{-}=$
$\displaystyle-\frac{a_{3}^{+}p_{22}^{+}+a_{3}^{-}p_{22}^{+}-a_{2}^{+}p_{23}^{+}}{a_{2}^{+}}.$

Then, we have the 4th generalized Lyapunov constant, given by

$\displaystyle V_{4}(\varepsilon)=$ $\displaystyle\
\frac{8}{15a_{2}^{+}}\,\big{(}a_{2}^{+}M_{13}+M_{14}\varepsilon^{2}+M_{15}\varepsilon^{4}\big{)}\,\varepsilon^{3},$

where

$\displaystyle M_{13}=$
$\displaystyle-2a_{3}^{+}b_{2}^{+}-2a_{3}^{-}b_{2}^{+}+a_{2}^{+}b_{3}^{+}-a_{2}^{+}b_{3}^{-},$
$\displaystyle M_{14}=$ $\displaystyle\
3(a_{3}^{+})^{2}+3a_{3}^{+}a_{3}^{-}-2a_{3}^{+}b_{2}^{+}p_{22}^{+}-2a_{3}^{-}b_{2}^{+}p_{22}^{+}+a_{2}^{+}b_{3}^{+}p_{22}^{+}-a_{2}^{+}b_{3}^{-}p_{22}^{+}$
$\displaystyle-2a_{2}^{+}a_{3}^{+}q_{22}^{+}-2a_{2}^{+}a_{3}^{-}q_{22}^{+}+(a_{2}^{+})^{2}q_{23}^{+}-(a_{2}^{+})^{2}q_{23}^{-},$
$\displaystyle M_{15}=$ $\displaystyle\
3a_{3}^{+}p_{23}^{+}+3a_{3}^{-}p_{23}^{+}-2a_{3}^{+}p_{22}^{+}q_{22}^{+}-2a_{3}^{-}p_{22}^{+}q_{22}^{+}+a_{2}^{+}p_{22}^{+}q_{23}^{+}-a_{2}^{+}p_{22}^{+}q_{23}^{-}.$

Setting $M_{13}=M_{14}=0$, we obtain the conditions:

$\displaystyle b_{3}^{-}=$ $\displaystyle\
\frac{1}{a_{2}^{+}}(-2a_{3}^{+}b_{2}^{+}-2a_{3}^{-}b_{2}^{+}+a_{2}^{+}b_{3}^{+}),$
$\displaystyle q_{23}^{-}=$ $\displaystyle\
\frac{1}{(a_{2}^{+})^{2}}\big{[}3(a_{3}^{+})^{2}+3a_{3}^{+}a_{3}^{-}-2a_{2}^{+}a_{3}^{+}q_{22}^{+}-2a_{2}^{+}a_{3}^{-}q_{22}^{+}+(a_{2}^{+})^{2}q_{23}^{+}\big{]}.$

Then, two subcases follow from $M_{15}=0$: (ii.2.1) $a_{3}^{+}+a_{3}^{-}=0$
and (ii.2.2) $p_{23}^{+}=\frac{a_{3}^{+}p_{22}^{+}}{a_{2}^{+}}$.

(ii.2.1) If $a_{3}^{+}+a_{3}^{-}=0$, which is combined with the necessary
conditions from (ii.1.1) to yield the condition IV.

(ii.2.2) If $p_{23}^{+}=\frac{a_{3}^{+}p_{22}^{+}}{a_{2}^{+}}$, then we have
$V_{4}(\varepsilon)=0$ and obtain the $5$th generalized Lyapunov constant,

$\displaystyle V_{5}(\varepsilon)=$
$\displaystyle-\frac{5\pi}{128(a_{2}^{+})^{3}}\,(a_{3}^{+}+a_{3}^{-})(a_{2}^{+}+p_{22}^{+}\varepsilon^{2})\big{[}2a_{2}^{+}M_{16}+M_{17}\varepsilon^{2}\big{]}\,\varepsilon^{5},$

where

$\displaystyle M_{16}=$
$\displaystyle-7a_{3}^{+}b_{2}^{+}-a_{3}^{-}b_{2}^{+}+4a_{2}^{+}b_{3}^{+},$
$\displaystyle M_{17}=$ $\displaystyle\
21(a_{3}^{+})^{2}+3a_{3}^{+}a_{3}^{-}-14a_{2}^{+}a_{3}^{+}q_{22}-2a_{2}^{+}a_{3}^{-}q_{22}^{+}+8(a_{2}^{+})^{2}q_{23}^{+}.$

Setting $M_{16}=M_{17}=0$ results in

$\displaystyle b_{3}^{+}=$ $\displaystyle\
\frac{7a_{3}^{+}b_{2}^{+}+a_{3}^{-}b_{2}^{+}}{4a_{2}^{+}},$ $\displaystyle
q_{23}^{+}=$
$\displaystyle-\frac{1}{q_{23}^{+}}\big{[}21(a_{3}^{+})^{2}+3a_{3}^{+}a_{3}^{-}-14a_{2}^{+}a_{3}^{+}q_{22}^{+}-2a_{2}^{+}a_{3}^{-}q_{22}^{+}\big{]},$

under which the $6$th generalized Lyapunov constant becomes

$\displaystyle V_{6}(\varepsilon)=$ $\displaystyle\
\frac{8}{105(a_{2}^{+})^{4}}(a_{3}^{+}+a_{3}^{-})^{3}(a_{2}^{+}+p_{22}^{+}\varepsilon^{2})(2a_{2}^{+}b_{2}^{+}-3a_{3}^{+}\varepsilon^{2}+2a_{2}^{+}q_{22}^{+}\varepsilon^{2})\,\varepsilon^{7}\neq
0$

with $a_{3}^{+}+a_{3}^{-}\neq 0$.

The above results give the necessary nilpotent center conditions for system
(7) at the origin when the origin of the first system of (7) is a cusp. Now we
consider the center conditions associated with the origin of the first system
of (7) being a monodromic singular point with multiplicity three if and only
if one of the following holds:

$\displaystyle a_{2}^{+}=b_{2}^{+}=0,\quad
b_{3}^{+}>\dfrac{3}{4}\,(a_{3}^{+})^{2};$ (26) $\displaystyle
b_{2}^{+}=0,\quad a_{2}^{+}\neq 0,\quad
b_{3}^{+}>\dfrac{1}{3}\,(a_{2}^{+})^{2}.$

If $b_{2}^{-}<0$, the origin in the second system of (7) is a cusp, which is a
similar case to that discussed above, leading to the condition V in Theorem
1.1. Hence, we assume that one of the conditions in (22) holds. Then, the
origin in the second system of (7) is also a monodromic singular point.

Setting the second generalized Lyapunov constant (25) zero, we find two cases:
either $a_{2}^{+}=a_{2}^{-}=0$ or $a_{2}^{+}a_{2}^{-}\neq 0$ when the
conditions in (22) and (26) are satisfied. The detailed analysis for the two
cases are given below.

###  3.3 Case 3: $a_{2}^{+}=a_{2}^{-}=0$

The second generalized Lyapunov constant becomes

$\displaystyle V_{2}(\varepsilon)=$ $\displaystyle\
\frac{4}{3}\,(p_{22}^{+}-p_{22}^{-})\,\varepsilon^{3}.$

Setting $V_{2}(\varepsilon)=0$ yields the condition $p_{22}^{-}=p_{22}^{+}$.
Then, the $3$rd generalized Lyapunov constant is given by

$\displaystyle V_{3}(\varepsilon)=$
$\displaystyle-\frac{\pi}{8}\big{[}3(a_{3}^{+}+a_{3}^{-})+(3p_{23}^{+}+3p_{23}^{-}-2p_{22}^{+}q_{22}^{+}-2p_{22}^{+}q_{22}^{-})\varepsilon^{2}\big{]}\,\varepsilon^{3}.$

Letting the $\varepsilon^{3}$-order and $\varepsilon^{5}$-order terms in
$V_{3}(\varepsilon)$ equal zero we obtain the conditions,

$\displaystyle a_{3}^{-}=$ $\displaystyle-a_{3}^{+},$ $\displaystyle
p_{23}^{-}=$ $\displaystyle\
\frac{1}{3}\big{(}-3p_{23}^{+}+2p_{22}^{+}q_{22}^{+}+2p_{22}^{+}q_{22}^{-}\big{)}.$

Then, we have the 4th generalized Lyapunov constant, given by

$\displaystyle V_{4}(\varepsilon)=$
$\displaystyle\frac{8}{45}\,\big{[}3(b_{3}^{+}+b_{3}^{-})p_{22}^{+}+6a_{3}^{+}(q_{22}^{+}+q_{22}^{-})+M_{18}\varepsilon^{2}\big{]}\,\varepsilon^{5},$

where

$\displaystyle
M_{18}=6p_{23}^{+}q_{22}^{+}-4p_{22}^{+}(q_{22}^{+})^{2}+6p_{23}^{+}q_{22}^{-}-4p_{22}^{+}q_{22}^{+}q_{22}^{-}+3p_{22}^{+}q_{23}^{+}-3p_{22}^{+}q_{23}^{-}.$

Setting the $\varepsilon^{5}$-order term in $V_{4}(\varepsilon)$ zero yields
three necessary center conditions: (i) $a_{3}^{+}=0$ and
$b_{3}^{-}=b_{3}^{+}$, which leads to conditions included in the condition VI;
(ii) $a_{3}^{+}=p_{22}^{+}=0$, giving the condition VII; and (iii)
$a_{3}^{+}\neq 0$ and
$q_{22}^{-}=\frac{1}{2a_{3}^{+}}(-b_{3}^{+}p_{22}^{+}+b_{3}^{-}p_{22}^{+}-2a_{3}^{+}q_{22}^{+})$.

For the case (iii), by $M_{18}=0$ we have

$\displaystyle
q_{23}^{+}=\frac{1}{3a_{3}^{+}}(3b_{3}^{+}p_{23}^{+}-3b_{3}^{-}p_{23}^{+}-2b_{3}^{+}p_{22}^{+}q_{22}^{+}+2b_{3}^{-}p_{22}^{+}q_{22}^{+}+3a_{3}^{+}q_{23}^{-}).$

Then, we have the 5th generalized Lyapunov constant, given by

$\displaystyle V_{5}(\varepsilon)=$
$\displaystyle-\frac{5\pi}{1152(a_{3}^{+})^{2}}(b_{3}^{+}-b_{3}^{-})\,\big{[}54(a_{3}^{+})^{3}-a_{3}^{+}M_{19}\varepsilon^{2}-M_{20}\varepsilon^{4}\big{]}\,\varepsilon^{5},$

where

$\displaystyle M_{19}=$
$\displaystyle-7b_{3}^{+}(p_{22}^{+})^{2}-b_{3}^{-}(p_{22}^{+})^{2}-36a_{3}^{+}p_{23}^{+}+12a_{3}^{+}p_{22}^{+}q_{22}^{+},$
$\displaystyle M_{20}=$
$\displaystyle-21b_{3}^{+}(p_{22}^{+})^{2}p_{23}^{+}+21b_{3}^{-}(p_{22}^{+})^{2}p_{23}^{+}-54a_{3}^{+}(p_{23}^{+})^{2}+14b_{3}^{+}(p_{22}^{+})^{3}q_{22}^{+}$
$\displaystyle-14b_{3}^{-}(p_{22}^{+})^{3}q_{22}^{+}+36a_{3}^{+}p_{22}^{+}p_{23}^{+}q_{22}^{+}-24a_{3}^{+}(p_{22}^{+})^{2}q_{23}^{-}.$

Since $a_{3}^{+}\neq 0$, we have $b_{3}^{+}=b_{3}^{-}$ from
$V_{5}(\varepsilon)=0$. Combining the conditions from the subcases (i) and
(iii), we have the condition VI.

###  3.4 Case 4: $a_{2}^{+}a_{2}^{-}\neq 0$

Setting each $\varepsilon^{k}$-order term in $V_{2}(\varepsilon)$ zero yields
the necessary center conditions $a_{2}^{-}=a_{2}^{+}$ and
$p_{22}^{-}=p_{22}^{+}$. Then, the $3$rd generalized Lyapunov constant is
given by

$\displaystyle V_{3}(\varepsilon)=$
$\displaystyle-\frac{\pi}{8}\big{[}3a_{3}^{+}+3a_{3}^{-}-2a_{2}^{+}q_{22}^{+}-2a_{2}^{+}q_{22}^{-}+(3p_{23}^{+}+3p_{23}^{-}-2p_{22}^{+}q_{22}^{+}-2p_{22}^{+}q_{22}^{-})\,\varepsilon^{2}\big{]}\,\varepsilon^{3}.$

Letting the $\varepsilon^{3}$-order and $\varepsilon^{5}$-order terms in
$V_{3}(\varepsilon)$ equal zero we obtain the conditions,

$\displaystyle a_{3}^{-}=$ $\displaystyle\
\frac{1}{3}(-3a_{3}^{+}+2a_{2}^{+}q_{22}^{+}+2a_{2}^{+}q_{22}^{-}),$
$\displaystyle p_{23}^{-}=$ $\displaystyle\
\frac{1}{3}\big{(}3p_{23}^{+}+2p_{22}^{+}q_{22}^{+}+2p_{22}^{+}q_{22}^{-}\big{)}.$

Then, 4th generalized Lyapunov constant becomes

$\displaystyle V_{4}(\varepsilon)=$ $\displaystyle\
\frac{1}{90}\,\big{[}48a_{2}^{+}(b_{3}^{+}+b_{3}^{-})+16M_{21}\varepsilon^{2}+225a_{2}^{+}p_{23}^{+}\pi\varepsilon^{3}+16M_{22}\varepsilon^{4}+225p_{22}^{+}p_{23}^{+}\pi\varepsilon^{5}\big{]}\,\varepsilon^{3},$

where

$\displaystyle M_{21}=$ $\displaystyle\
3b_{3}^{+}p_{22}^{+}-3b_{3}^{-}p_{22}^{+}+6a_{3}^{+}q_{22}^{+}-4a_{2}^{+}(q_{22}^{+})^{2}+6a_{3}^{+}q_{22}^{-}-4a_{2}^{+}q_{22}^{+}q_{22}^{-}+3a_{2}^{+}q_{23}^{+}-3a_{2}^{+}q_{23}^{-},$
$\displaystyle M_{22}=$ $\displaystyle\
6p_{23}^{+}q_{22}^{+}-4p_{22}^{+}(q_{22}^{+})^{2}-6p_{23}^{+}q_{22}^{-}-4p_{22}^{+}q_{22}^{+}q_{22}^{-}+3p_{22}^{+}q_{23}^{+}-3p_{22}^{+}q_{23}^{-}.$

Letting the $\varepsilon^{k}$-order terms ($k=3,5,6,7$) in
$V_{4}(\varepsilon)$ equal zero we obtain the following conditions:

$\displaystyle b_{3}^{-}=b_{3}^{+},\quad p_{23}^{+}=0,\quad
q_{23}^{-}=\frac{1}{3a_{2}^{+}}\big{[}6a_{3}^{+}q_{22}^{+}-4a_{2}^{+}(q_{22}^{+})^{2}+6a_{3}^{+}q_{22}^{-}-4a_{2}^{+}q_{22}^{+}q_{22}^{-}+3a_{2}^{+}q_{23}^{+}\big{]}.$

Then, $M_{22}$ is reduced to

$M_{22}=-\frac{6}{a_{2}^{+}}a_{3}^{+}p_{22}^{+}(q_{22}^{+}+q_{22}^{-}).$

If $q_{22}^{-}=-q_{22}^{+}$, we obtain $a_{3}^{-}=-a_{3}^{+}$, yielding the
condition VIII. Otherwise, we obtain the 5th generalized Lyapunov constant,
given by

$\displaystyle V_{5}(\varepsilon)=$
$\displaystyle-\frac{1}{1440a_{2}^{+}}(q_{22}^{+}+q_{22}^{-})\,\big{[}300(a_{2}^{+})^{2}b_{3}^{+}\pi+25\pi
M_{23}\varepsilon^{2}-6144a_{2}^{+}a_{3}^{+}p_{22}^{+}\varepsilon^{3}$
$\displaystyle-50p_{22}^{+}\pi
M_{24}\varepsilon^{4}-6144a_{3}^{+}(p_{22}^{+})^{2}\pi\varepsilon^{5}\big{]}\,\varepsilon^{5},$

where

$\displaystyle M_{23}=$ $\displaystyle\
27(a_{3}^{+})^{2}+12a_{2}^{+}b_{3}^{+}p_{22}^{+}-15a_{2}^{+}a_{3}^{+}q_{22}^{+}-2(a_{2}^{+})^{2}(q_{22}^{+})^{2}+3a_{2}^{+}a_{3}^{+}q_{22}^{-}$
$\displaystyle-2(a_{2}^{+})^{2}q_{22}^{+}q_{22}^{-}+12(a_{2}^{+})^{2}q_{23}^{+},$
$\displaystyle M_{24}=$ $\displaystyle\
9a_{3}^{+}q_{22}^{+}+a_{2}^{+}(q_{22}^{+})^{2}-12a_{3}^{+}q_{22}^{-}+a_{2}^{+}q_{22}^{+}q_{22}^{-}-6a_{2}^{+}q_{23}^{+}.$

With $b_{3}^{+}>\frac{1}{3}(a_{2}^{+})^{2}$, we have $V_{5}(\varepsilon)\neq
0$ when $a_{3}^{+}p_{22}^{+}=0$ and $q_{22}^{+}+q_{22}^{-}\neq 0$.

To this end, we finish the proof for the necessity of the conditions in
Theorem 1.1. Next, we prove the sufficiency of these eight nilpotent center
conditions.

If the condition I in Theorem 1.1 holds, (7) is reduced to

$(\dot{x},\,\dot{y})=\left\\{\begin{aligned} &\Big{(}y,\
-(b_{2}^{+}x^{2}+b_{3}^{+}x^{3})\Big{)},\ \ &\text{if}\ \ x\geq 0,\\\\[0.0pt]
&\Big{(}y,\ -b_{3}^{-}x^{3}\Big{)},\ \ &\text{if}\ \ x<0.\end{aligned}\right.$
(27)

The two systems in (27) are Hamiltonian systems, having respectively the
Hamiltonian functions,

$\displaystyle
I^{+}(x,y)=\frac{1}{2}y^{2}+\frac{1}{3}b_{2}^{+}x^{3}+\frac{1}{4}b_{3}^{+}x^{4}\quad\textrm{and}\quad
I^{-}(x,y)=\frac{1}{2}y^{2}+\frac{1}{4}b_{3}^{-}x^{4},$

which implies that the condition $I^{+}(0,y)\equiv I^{-}(0,y)$ in Proposition
2.1 is satisfied. So the origin of system (27) is a nilpotent center.

Similar to the proof for the condition I, we can derive the Hamiltonian
functions (which are polynomials similar to the above $I^{\pm}(x,y)$) for
system (7) with the condition II, or V or VII to show that the origin of (7)
is a nilpotent center for these three cases.

If the condition III in Theorem 1.1 holds, (7) becomes

$(\dot{x},\,\dot{y})=\left\\{\begin{aligned} &\big{(}y-a_{3}^{+}x^{3},\
-b_{2}^{+}x^{2}\big{)},\ \ &\text{if}\ \ x\geq 0,\\\\[0.0pt]
&\big{(}y-a_{3}^{-}x^{3},\ -b_{2}^{-}x^{2}\big{)},\ \ &\text{if}\ \
x<0,\end{aligned}\right.$ (28)

with $-\,\frac{a_{3}^{-}}{a_{3}^{+}}=-\,\frac{b_{2}^{-}}{b_{2}^{+}}\equiv
h>0,\,\neq 1$. Then, system (28) can be rewritten as

$(\dot{x},\,\dot{y})=\left\\{\begin{aligned} &\big{(}y-a_{3}^{+}x^{3},\
-b_{2}^{+}x^{2}\big{)},\ \ &\text{if}\ \ x\geq 0,\\\\[0.0pt]
&\big{(}y+h\,a_{3}^{+}x^{3},\ h\,b_{2}^{+}x^{2}\big{)},\ \ &\text{if}\ \
x<0.\end{aligned}\right.$ (29)

Then, we apply the integrating factors:

$\mu^{+}=\dfrac{9(a_{3}^{+})^{2}}{3(a_{3}^{+})^{2}x^{3}-3a_{3}^{+}y-b_{2}^{+}}\quad\textrm{and}\quad\mu^{-}=\dfrac{9(a_{3}^{+})^{2}}{-3(a_{3}^{+})^{2}hx^{3}-3a_{3}^{+}y-b_{2}^{+}}$

respectively to the first and second systems of (29) to obtain the first
integrals:

$\left\\{\begin{array}[]{ll}I^{+}(x,y)=-\,3a_{3}^{+}y+b_{2}^{+}\ln|\\!-\\!3(a_{3}^{+})^{2}x^{3}+3a_{3}^{+}y+b_{2}^{+}|,&\textrm{for}\
\ x\geq 0,\\\\[4.30554pt]
I^{-}(x,y)=-\,3a_{3}^{+}y+b_{2}^{+}\ln|3h(a_{3}^{+})^{2}x^{3}+3a_{3}^{+}y+b_{2}^{+}|,&\textrm{for}\
\ x<0,\end{array}\right.$

which clearly shows that $I^{+}(0,y)\equiv I^{-}(0,y)$, implying that the
origin of the system (29) and so the system (28) is a center.

If the condition IV in Theorem 1.1 holds, (7) can be rewritten as

$(\dot{x},\,\dot{y})=\left\\{\begin{aligned}
&\Big{(}y-(a_{2}^{+}x^{2}+a_{3}^{+}x^{3}),\
-(b_{2}^{+}x^{2}+b_{3}^{+}x^{3})\Big{)},\ \ &\text{if}\ \ x\geq 0,\\\\[0.0pt]
&\Big{(}y-(a_{2}^{+}x^{2}-a_{3}^{+}x^{3}),\
-(-b_{2}^{+}x^{2}+b_{3}^{+}x^{3})\Big{)},\ \ &\text{if}\ \
x<0,\end{aligned}\right.$ (30)

which is symmetric with respect to the $y$-axis, and by Proposition 2.2 we
know that the origin of (30) is a nilpotent center. Similarly, since (7) is
symmetric with respect to the $y$-axis when the condition VI or VIII holds,
the origin of system (7) is a nilpotent center.

In the following, we present examples for the above systems (27), (28) and
(30) to show global phase portraits with a nilpotent center at the origin.

######  Example 3.2.

The global phase portraits of the switching Liénard systems (27), (28) and
(30) with three sets of parameter values show a nilpotent center at the
origin, as illustrated in the Poincaré disc of Figure 2.

Figure 2: Phase portraits for (a) system (27) with
$b_{2}^{+}=1,\,b_{3}^{+}=-2,\,b_{3}^{-}=1$, (b) system (28) with
$a_{3}^{+}=1,\,b_{2}^{+}=2,\,a_{3}^{-}=-2,\,b_{2}^{-}=-4$, and (c) system (30)
with $a_{2}^{+}=1,\,a_{3}^{+}=2,\,b_{2}^{+}=1,\,b_{3}^{+}=-2$.

##  4 The proof of Theorem 1.2

We recall the notion of Poincaré compactification of switching differential
systems. This compactification identifies $\mathbb{R}^{2}$ with the interior
of the closed unit Poincaré disc $\mathbb{D}^{2}$ centered at the origin of
coordinates, and extends the switching differential systems to the boundary of
$\mathbb{D}^{2}$, which is called the circle of the infinity. The switching
manifold is indicated by the dash line $``---"$ in $\mathbb{D}^{2}$. The
singular points on the boundary of Poincaré disc are called infinite singular
points. More details about the Poincaré compactification for switching
polynomial systems can be found in [8].

With the results in [29], we have the proposition which characterizes the
global center of switching polynomial systems.

######  Proposition 4.1.

[Propositions 5, [29]] Consider the switching polynomial system (10) which has
a unique finite center at the singular point. Assume that the infinity of (10)
is not filled of singular points. Then, the center is global if and only if
all the infinite singular points in $\mathbb{D}^{2}$, if they exist, are such
that the local phase portrait of each infinite singular point is consisting of
two hyperbolic sectors, where the two separatrices are on the circle of the
infinity.

Suppose the origin of the switching Liénard system (7) is the unique
monodromic finite singular point. Then, the remaining finite singular points,
given by

$\displaystyle
e^{+}=(x^{+},y^{+})=\bigg{(}\\!-\frac{b_{2}^{+}}{b_{3}^{+}},-\frac{(b_{2}^{+})^{2}(a_{3}^{+}b_{2}^{+}-a_{2}^{+}b_{3}^{+})}{(b_{3}^{+})^{3}}\bigg{)}$

and

$\displaystyle
e^{-}=(x^{-},y^{-})=\bigg{(}\\!-\frac{b_{2}^{-}}{b_{3}^{-}},-\frac{(b_{2}^{-})^{2}(a_{3}^{-}b_{2}^{-}-a_{2}^{-}b_{3}^{-})}{(b_{3}^{-})^{3}}\bigg{)}.$

in the first and the second systems of (7) must be virtual, i.e., $x^{+}<0$
and $x^{-}>0$. Hence, we have the necessity conditions:

$\displaystyle G_{1}^{+}=$ $\displaystyle\ \\{b_{2}^{+}=0,\ b_{3}^{+}\neq
0\\},$ (31) $\displaystyle G_{2}^{+}=$ $\displaystyle\ \\{b_{2}^{+}>0,\
b_{3}^{+}\geq 0\\}$

and

$\displaystyle G_{1}^{-}=$ $\displaystyle\ \\{b_{2}^{-}=0,\ b_{3}^{-}\neq
0\\},$ (32) $\displaystyle G_{2}^{-}=$ $\displaystyle\ \\{b_{2}^{-}<0,\
b_{3}^{-}\geq 0\\}.$

Next, we consider the infinite singular point of (7). For studying the
dynamical behaviours in the circle of the infinity, we use the local charts:
$U_{k}=\\{(z_{1},z_{2})\in\mathbb{D}^{2}:z_{k}>0\\}$ and
$W_{k}=\\{(z_{1},z_{2})\in\mathbb{D}^{2}:z_{k}<0\\}$, $k=1,2$, with the
corresponding diffeomorphisms,

$\phi_{k}:~{}U_{k}\rightarrow\mathbb{R}^{2},\qquad\psi_{k}:~{}W_{k}\rightarrow\mathbb{R}^{2},$
(33)

defined by
$\phi_{k}(z_{1},z_{2})=\big{(}\frac{z_{2}}{z_{1}},\frac{1}{z_{1}}\big{)}=(u,w)$
and
$\psi_{k}(z_{1},z_{2})=\big{(}\frac{z_{1}}{z_{2}},\frac{1}{z_{2}}\big{)}=(u,w)$.
Here, the coordinates $(u,w)$ play different roles in the distinct local
charts. Thus, the corresponding vector fields of (7) in the local charts
$U_{1}$ and $W_{1}$ are given by

$\left(\begin{array}[]{cc}\dot{u}\\\
\dot{w}\end{array}\right)=\left(\begin{array}[]{c}-b_{3}^{+}+a_{3}^{+}u-b_{2}^{+}w+a_{2}^{+}uw-u^{2}w^{2}\\\\[2.15277pt]
w(a_{3}^{+}+a_{2}^{+}w-uw^{2})\end{array}\right)$ (34)

and

$\left(\begin{array}[]{cc}\dot{u}\\\
\dot{w}\end{array}\right)=\left(\begin{array}[]{c}-b_{3}^{-}+a_{3}^{-}u-b_{2}^{-}w+a_{2}^{-}uw-u^{2}w^{2}\\\\[2.15277pt]
w(a_{3}^{-}+a_{2}^{-}w-uw^{2})\end{array}\right),$ (35)

respectively. We derive the possible infinite singular points
$\big{(}\frac{b_{3}^{\pm}}{a_{3}^{\pm}},0\big{)}$ or
$\big{(}\frac{b_{2}^{\pm}}{a_{2}^{\pm}},0\big{)}$ which are nodes. It follows
from Proposition 4.1 that the origin of (7) cannot be a global center in these
cases. Hence, the systems (34) and (35) cannot have these infinite singular
points when the origin of (7) is a global center. Therefore, the conditions
(31) and (32) become

$\displaystyle G_{1}^{+}=$ $\displaystyle\
\\{a_{2}^{+}=a_{3}^{+}=b_{3}^{+}=0,\,\,b_{2}^{+}>0\\},$ (36) $\displaystyle
G_{2}^{+}=$ $\displaystyle\
\\{a_{3}^{+}=0,\,\,b_{2}^{+}>0,\,\,b_{3}^{+}>0\\},$ $\displaystyle G_{3}^{+}=$
$\displaystyle\ \\{a_{3}^{+}=0,\,\,b_{2}^{+}=0,\,\,b_{3}^{+}\neq 0\\}$

and

$\displaystyle G_{1}^{-}=$ $\displaystyle\
\\{a_{2}^{-}=a_{3}^{-}=b_{3}^{-}=0,\,\,b_{2}^{-}<0\\},$ (37) $\displaystyle
G_{2}^{-}=$ $\displaystyle\
\\{a_{3}^{-}=0,\,\,b_{2}^{-}<0,\,\,b_{3}^{-}>0\\},$ $\displaystyle G_{3}^{-}=$
$\displaystyle\ \\{a_{3}^{-}=0,\,\,b_{2}^{-}=0,\,\,b_{3}^{-}\neq 0\\}.$

Then we consider the following cases.

(i) Firstly, we consider the case $G_{1}^{+}\cap G_{1}^{-}$, for which system
(7) becomes a quadratic switching Liénard system. Since the expression of the
quadratic switching system (7) in $W_{2}$ is the one in $U_{2}$ multiplied by
$-1$, the flows in the local phase portrait around the origin of $W_{2}$ have
the opposite direction compared to the ones in $U_{2}$. Thus, we only need to
study the dynamics around the origin in $U_{2}$. Then, in the local chart
$U_{2}$ we have the following system,

$\left(\begin{array}[]{cc}\dot{u}\\\
\dot{w}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}w+b_{2}^{+}u^{3}\\\\[2.15277pt]
b_{2}^{+}u^{2}w\end{array}\right),&\text{if}\ \ u\geq 0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}w+b_{2}^{-}u^{3}\\\\[2.15277pt]
b_{2}^{-}u^{2}w\end{array}\right),&\text{if}\ \ u<0,\end{aligned}\right.$ (38)

where both the two origins of the first and the second system of (38) are
nilpotent singular points. Note that if the origin of $U_{2}$ is an infinite
singular point, it must come from the combination of the two origins of the
first and the second systems of (38). Since $b_{2}^{+}>0$ and $b_{2}^{-}<0$,
it is easy to verify that the origin of the first system of (38) is an
unstable node while the origin of the second one of (38) is a stable node.
Thus, it follows from Proposition 4.1 that the origin of (7) cannot be a
global center.

The above analysis shows that the infinite singular point of the quadratic
switching Liénard system (7) in $U_{2}$ does not consist of two hyperbolic
sectors. This implies that three is the lowest degree of the first and the
second systems of the switching Liénard system (7) to have a global nilpotent
center. Thus, we only need to consider the case $G_{i}^{+}\cap G_{j}^{-}$,
$i,j=2,3$.

(ii) For the case $G_{i}^{+}\cap G_{j}^{-}$, $i,j=2,3$, we obtain the
following system from (7),

$\left(\begin{array}[]{cc}\dot{u}\\\
\dot{w}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}b_{3}^{+}u^{4}-a_{2}^{+}u^{2}w+b_{2}^{+}u^{3}w+w^{2}\\\\[2.15277pt]
u^{2}w(b_{3}^{+}u+b_{2}^{+}w)\end{array}\right),&\text{if}\ \ u\geq
0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}b_{3}^{-}u^{4}-a_{2}^{-}u^{2}w+b_{2}^{-}u^{3}w+w^{2}\\\\[2.15277pt]
u^{2}w(b_{3}^{-}u+b_{2}^{-}w)\end{array}\right),&\text{if}\ \
u<0,\end{aligned}\right.$ (39)

in $U_{2}$ and $W_{2}$. Since system (7) is a cubic switching system, the
local qualitative property of the origin of $W_{2}$ has the same sense with
respect to the one in $U_{2}$. Obviously, the origins of the first and second
system in (39) in $U_{2}$ are singular points which are linearly zero. In
order to understand the local qualitative properties around these two singular
points, we apply a direct blow-up $(u,v)\rightarrow(u,w)$ with $v=\frac{w}{u}$
to eliminate the common factor $u$, yielding

$\left(\begin{array}[]{cc}\dot{u}\\\
\dot{v}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}u(b_{3}^{+}u^{2}-a_{2}^{+}uv+b_{2}^{+}u^{2}v+v^{2})\\\\[2.15277pt]
(a_{2}^{+}u-v)v^{2}\end{array}\right),&\text{if}\ \ u\geq 0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}u(b_{3}^{-}u^{2}-a_{2}^{-}uv+b_{2}^{-}u^{2}v+v^{2})\\\\[2.15277pt]
(a_{2}^{-}u-v)v^{2}\end{array}\right),&\text{if}\ \
u<0,\end{aligned}\right.\vspace{0.10in}$ (40)

where the two origins are singular points with linearly zero. We do a further
blow-up $(u,v)\rightarrow(u,V)$ with $V=\frac{v}{u}$, to obtain the following
system,

$\left(\begin{array}[]{cc}\dot{u}\\\
\dot{V}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}u(b_{3}^{+}-a_{2}^{+}V+b_{2}^{+}uV+V^{2})\\\\[2.15277pt]
-V(b_{3}^{+}-2a_{2}^{+}V+b_{2}^{+}uV+2V^{2})\end{array}\right),&\text{if}\ \
u\geq 0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}u(b_{3}^{-}-a_{2}^{-}V+b_{2}^{-}uV+V^{2})\\\\[2.15277pt]
-V(b_{3}^{-}-2a_{2}^{-}V+b_{2}^{-}uV+2V^{2})\end{array}\right),&\text{if}\ \
u<0,\end{aligned}\right.$ (41)

where the common factor $u^{2}$ has been eliminated.

When $(a_{2}^{+})^{2}-2b_{3}^{+}<0$, for $u=0$ the first system of (41) has a
saddle at $E_{1}=(0,0)$. Going back through the change of variables we get
that the local phase portrait of the right origin in $U_{2}$ has two
hyperbolic sectors, see Figure 3. When $(a_{2}^{+})^{2}=2b_{3}^{+}\neq 0$, for
$u=0$ the first system of (41) has two singular points at $E_{1}=(0,0)$ (a
saddle) and at $E_{2}=(\frac{a_{2}^{+}}{2},0)$ (a saddle-node). Again, going
back through the change of variables we find that the local phase portrait of
the right origin in $U_{2}$ has two hyperbolic and two parabolic sectors, see
Figure 4. When $(a_{2}^{+})^{2}-2b_{3}^{+}>0$, for $u=0$ the first system of
(41) has three singular points at $E_{1}=(0,0)$ (a saddle),
$E_{2,3}=\big{(}\frac{1}{2}(a_{2}^{+}\pm((a_{2}^{+})^{2}-2b_{3}^{+})^{1/2},0\big{)}$
(a saddle and a node). Going back through the change of variables we obtain
that the local qualitative property of the right origin in $U_{2}$ has two
hyperbolic and two parabolic sectors, see Figure 5.

\begin{overpic}[width=361.92029pt]{Fig3.pdf} \put(11.0,-5.0){$(a)$}
\put(47.0,-5.0){$(b)$} \put(83.0,-5.0){$(c)$} \end{overpic} Figure 3: Local
phase portraits around the origin of the local chart $U_{2}$ with
$(a_{2}^{+})^{2}-2b_{3}^{+}<0$ for (a) the first system of (41), (b) the first
system of (40), and (c) the first system of (39).
\begin{overpic}[width=361.92029pt]{Fig4.pdf} \put(11.0,-5.0){$(a)$}
\put(47.0,-5.0){$(b)$} \put(83.0,-5.0){$(c)$} \end{overpic} Figure 4: Local
phase portraits around the origin of the local chart $U_{2}$ with
$(a_{2}^{+})^{2}-2b_{3}^{+}=0$ for (a) the first system of (41), (b) the first
system of (40), and (c) the first system of (39).
\begin{overpic}[width=361.92029pt]{Fig5.pdf} \put(11.0,-5.0){$(a)$}
\put(47.0,-5.0){$(b)$} \put(83.0,-5.0){$(c)$} \end{overpic} Figure 5: Local
phase portraits at the origin of the local chart $U_{2}$ with
$(a_{2}^{+})^{2}-2b_{3}^{+}>0$ for (a) the first system of (41), (b) the first
system (40), and (c) the first system of (39).

Similarly, we derive that the local qualitative property around the left
origin in $W_{2}$ has two hyperbolic sectors if and only if
$(a_{2}^{-})^{2}-2b_{3}^{-}<0$. Thus, the switching Liénard system (7) can
only have a global center at the origin if condition

$\displaystyle G=$ $\displaystyle\\{a_{3}^{\pm}=0,\ b_{2}^{+}\geq 0,\
b_{2}^{-}\leq 0,\ (a_{2}^{+})^{2}-2b_{3}^{+}<0,\
(a_{2}^{-})^{2}-2b_{3}^{-}<0\\}$ (42)

holds. Thus, by Theorem 1.1 we have proved Theorem 1.2.

In fact, combining the conditions I, II, V and VII of Theorem 1.1 and the
above condition $G$, we obtain the condition $G_{1}$ of Theorem 1.2. From the
condition III of Theorem 1.1, we have $a_{3}^{+}a_{3}^{-}\neq 0$ which
contradicts the condition $G$, and thus the origin of system (7) cannot be a
global center. From the conditions IV, VI and VIII of Theorem 1.1 together
with the condition $G$, we obtain the condition $G_{2}$ of Theorem 1.2.

##  5 The proof of Theorem 1.3

In this section, we prove the existence of $5$ small-amplitude limit cycles by
perturbing the nilpotent origin of the cubic switching Liénard system (7)
under the center condition IV. More precisely, with the center condition IV,
we perturb system (7) to obtain

$\left(\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\right)=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}y-(a_{2}^{+}x^{2}+a_{3}^{+}x^{3})+d\varepsilon^{9}+\displaystyle\sum_{k=1}^{6}\varepsilon^{k+1}\delta_{k}x+\sum_{k=1}^{5}\varepsilon^{k}(p_{k2}^{+}x^{2}+p_{k3}^{+}x^{3})\\\\[4.30554pt]
-\varepsilon^{2}x-(b_{2}^{+}x^{2}+b_{3}^{+}x^{3})+\displaystyle\sum_{k=1}^{5}\varepsilon^{k}(q_{k2}^{+}x^{2}+q_{k3}^{+}x^{3})\end{array}\right),&\text{if}\
\ x\geq 0,\\\\[4.30554pt]
&\left(\begin{array}[]{c}y-(-a_{2}^{+}x^{2}+a_{3}^{+}x^{3})+\displaystyle\sum_{k=1}^{6}\varepsilon^{k+1}\delta_{k}x+\sum_{k=1}^{5}\varepsilon^{k}(p_{k2}^{-}x^{2}+p_{k3}^{-}x^{3})\\\\[4.30554pt]
-\varepsilon^{2}x-(b_{2}^{+}x^{2}-b_{3}^{+}x^{3})+\displaystyle\sum_{k=1}^{5}\varepsilon^{k}(q_{k2}^{-}x^{2}+q_{k3}^{-}x^{3})\end{array}\right),&\text{if}\
\ x<0.\end{aligned}\right.$ (43)

Denote by one parameter vector

$\lambda^{*}=(a_{2}^{+},a_{3}^{+},b_{2}^{+},b_{3}^{+},d,\delta_{i},p_{j2}^{\pm},p_{j3}^{\pm},q_{j2}^{\pm},q_{j3}^{\pm}),$

$i=1,2,...,6$, $j=1,2,...,5$. Further, introducing the scaling
$(x,y,t)\rightarrow\big{(}\varepsilon^{2}y,\varepsilon^{3}x,\frac{t}{\varepsilon}\big{)}$
into (43), we obtain a system up to $\varepsilon^{6}$-order terms,

$\\!\\!\\!\left(\\!\begin{array}[]{cc}\dot{x}\\\
\dot{y}\end{array}\\!\right)\\!=\\!\left\\{\begin{aligned}
&\left(\begin{array}[]{c}-y-b_{2}^{+}y^{2}-b_{3}^{+}\varepsilon^{2}y^{3}-\displaystyle\sum_{k=1}^{5}\varepsilon^{k}q_{k2}^{+}y^{2}-\displaystyle\sum_{k=1}^{5}\varepsilon^{k+2}q_{k3}^{+}y^{3}\\\\[4.30554pt]
x+\displaystyle\sum_{k=1}^{6}\varepsilon^{k}\delta_{k}y-(a_{2}^{+}\varepsilon
y^{2}+a_{3}^{+}\varepsilon^{3}y^{3})-\displaystyle\sum_{k=1}^{5}\varepsilon^{k+1}p_{k2}^{+}y^{2}-\displaystyle\sum_{k=1}^{5}\varepsilon^{k+3}p_{k3}^{+}y^{3}\end{array}\right)\\!,\\!\\!&\text{if}\
y\geq 0,\\\
&\left(\\!\\!\\!\begin{array}[]{c}-y+b_{2}^{+}y^{2}-b_{3}^{+}\varepsilon^{2}y^{3}-\displaystyle\sum_{k=1}^{5}\varepsilon^{k}q_{k2}^{-}y^{2}-\displaystyle\sum_{k=1}^{5}\varepsilon^{k+2}q_{k3}^{-}y^{3}\\\\[4.30554pt]
x+\\!\displaystyle\sum_{k=1}^{6}\varepsilon^{k}\delta_{k}y\\!-\\!(a_{2}^{+}\varepsilon
y^{2}\\!-\\!a_{3}^{+}\varepsilon^{3}y^{3})\\!+\\!d\varepsilon^{6}\\!-\\!\displaystyle\sum_{k=1}^{5}\varepsilon^{k+1}p_{k2}^{-}y^{2}\\!-\\!\displaystyle\sum_{k=1}^{5}\varepsilon^{k+3}p_{k3}^{-}y^{3}\end{array}\\!\\!\\!\right)\\!,&\text{if}\
y<0.\end{aligned}\right.$ (44)

Note that we let the constant term be zero, i.e., $d=0$, when we compute
$\varepsilon^{k}$-order Lyapunov constants. By using the generalized Lyapunov
constants, we start from the $\varepsilon$-order terms to determine the number
of limit cycles. Thus, we need to find $\varepsilon^{k}$-order center
conditions such that all the $\varepsilon^{k}$-order Lyapunov constants being
zero when we want to obtain more limit cycles by using
$\varepsilon^{k+1}$-order terms. That is to say that the origin of the system
is a center up to $\varepsilon^{k}$-order, see [41]. Finally, we perturb the
coefficient $d$ of the constant term to obtain one more small-amplitude limit
cycle.

More precisely, to prove the existence of small-amplitude limit cycles around
the origin, we compute the higher-order Lyapunov constants $V_{jk}$ for
$j=1,2,\ldots,5$ and $k=1,2,\ldots,6$. We start from $k=1$. For a fixed $k$,
we choose appropriate parameter values such that as many as possible higher-
order Lyapunov constants vanish.

First, for $k=1$, we obtain that all $\varepsilon$-order Lyapunov constants
are zero when $\delta_{1}=0$, i.e., $V_{j1}=0$ for all $j$.

Next, for $k=2$, we have $V_{12}=\delta_{2}\pi=0$ when $\delta_{2}=0$. Then,
we solve $V_{22}=\frac{4}{3}(p_{12}^{-}-p_{12}^{+})=0$ to obtain

$p_{12}^{-}=p_{12}^{+},$

under which the 3rd $\varepsilon^{2}$-order Lyapunov constant becomes

$V_{32}=\,\frac{\pi}{4}\,a_{2}^{+}(q_{12}^{-}+q_{12}^{+}).$

Setting $q_{12}^{-}=-q_{12}^{+}$ yields $V_{j2}=0$ for all $j$. So, for $k=2$,
we can have $2$ limit cycles by perturbing $\delta_{2}$ and $p_{12}^{-}$ with
$V_{32}\neq 0$.

Now, consider $k=3$. We have $V_{13}=0$ with $\delta_{3}=0$ and
$V_{23}=\frac{4}{3}(p_{22}^{-}-p_{22}^{+})$. Solving $V_{23}=0$ we have

$p_{22}^{-}=p_{22}^{+}.\vspace{-0.10in}$

Further, the 3rd $\varepsilon^{3}$-order Lyapunov constant is obtained as

$V_{33}=\,\frac{\pi}{4}\,a_{2}^{+}(q_{22}^{-}+q_{22}^{+}).$

Setting $q_{22}^{-}=-q_{22}^{+}$ yields $V_{j3}=0$ for all $j$, which
indicates that for $k=3$ we can have $2$ limit cycles if choosing $V_{33}\neq
0$ and perturbing $\delta_{3}$ and $p_{22}^{-}$.

For $k=4$, solving $V_{24}=V_{34}=0$ we have

$p_{32}^{-}=p_{32}^{+}\quad\textrm{and}\quad
p_{13}^{-}=\frac{2}{3}a_{2}^{+}q_{32}^{-}+\frac{2}{3}a_{2}^{+}q_{32}^{+}-p_{13}^{+},$

under which the 4th $\varepsilon^{4}$-order Lyapunov constant becomes

$V_{44}=-\,\frac{8}{45}\,a_{2}^{+}(4q_{32}^{-}b_{2}^{+}+4b_{2}^{+}q_{32}^{+}+3q_{13}^{-}-3q_{13}^{+}).$

Setting $V_{44}=0$ we have

$q_{13}^{-}=-\frac{4}{3}b_{2}^{+}q_{32}^{-}-\frac{4}{3}b_{2}^{+}q_{32}^{+}-q_{13}^{+},$

leading to $V_{j4}=0$ for all $j$. So, for $k=4$ we can have $3$ limit cycles
by choosing $V_{44}\neq 0$ and perturbing $\delta_{4}$, $p_{32}^{-}$ and $\
p_{13}^{-}$.

For $k=5$, we similarly obtain

$p_{42}^{-}=p_{42}^{+}$

by solving $V_{25}=0$, and

$p_{23}^{-}=\frac{2}{3}q_{32}^{-}p_{12}^{+}+\frac{2}{3}q_{42}^{-}a_{2}^{+}+\frac{2}{3}a_{2}^{+}q_{42}^{+}+\frac{2}{3}p_{12}^{+}q_{32}^{+}-p_{23}^{+}$

by solving $V_{35}=0$. Then, the 4th $\varepsilon^{5}$-order Lyapunov constant
is given by

$V_{45}=-\frac{8}{45}a_{2}^{+}(4q_{32}^{-}q_{12}^{+}+4q_{42}^{-}b_{2}^{+}+4b_{2}^{+}q_{42}^{+}+4q_{12}^{+}q_{32}^{+}+3Q_{23}^{-}-3q_{23}^{+}).$

Setting

$q_{23}^{-}=-\frac{4}{3}q_{12}^{+}q_{32}^{-}-\frac{4}{3}q_{42}^{-}b_{2}^{+}-\frac{4}{3}b_{2}^{+}q_{42}^{+}-\frac{4}{3}q_{12}^{+}q_{32}^{+}+q_{23}^{+},$

under which $V_{j5}=0$ for all $j$. This implies that $3$ limit cycles can be
obtained for $k=4$ if choosing $V_{45}\neq 0$ and perturbing $\delta_{5}$,
$p_{42}^{-}$ and $p_{23}^{-}$.

Finally, for $k=6$, similarly by solving $V_{16}=V_{26}=V_{36}=0$ we obtain

$\begin{array}[]{ll}\delta_{6}=0,\\\\[4.30554pt]
p_{52}^{-}=p_{52}^{+},\\\\[4.30554pt]
p_{33}^{-}=\frac{2}{3}q_{32}^{-}p_{22}^{+}+\frac{2}{3}q_{42}^{-}p_{12}^{+}+\frac{2}{3}q_{52}^{-}a_{2}^{+}+\frac{2}{3}a_{2}^{+}q_{52}^{+}+\frac{2}{3}p_{12}^{+}q_{42}^{+}+\frac{2}{3}p_{22}^{+}q_{32}^{+}-p_{33}^{+}.\end{array}$

Then, we solve $V_{46}=0$ for $q_{33}^{-}$ to obtain

$\displaystyle q_{33}^{-}=$
$\displaystyle-\frac{4}{3a_{2}^{+}}(q_{32}^{-}a_{2}^{+}q_{22}^{+}+4q_{42}^{-}a_{2}^{+}q_{12}^{+}+4q_{52}^{-}a_{2}^{+}b_{2}^{+}+4a_{2}^{+}b_{2}^{+}q_{52}^{+}$
$\displaystyle+4a_{2}^{+}q_{12}^{+}q_{42}^{+}+4a_{2}^{+}q_{22}^{+}q_{32}^{+}-6q_{32}^{-}a_{3}^{+}-3a_{2}^{+}q_{33}^{+}-6a_{3}^{+}q_{32}^{+}),$

under which the 5th $\varepsilon^{6}$-order Lyapunov constant is reduced to

$V_{56}=-\dfrac{5\pi}{48}\,(2a_{2}^{+}b_{3}^{+}-3a_{3}^{+}b_{2}^{+})(q_{32}^{-}+q_{32}^{+}).$

So letting

$(2a_{2}^{+}b_{3}^{+}-3a_{3}^{+}b_{2}^{+})(q_{32}^{-}+q_{32}^{+})=0$

yields $V_{j6}=0$ for all $j$. Therefore, for $k=6$, we can obtain $4$ limit
cycles for choosing $V_{56}\neq 0$.

Summarizing the above results, $4$ small-amplitude limit cycles can be
obtained from the $\varepsilon^{6}$-order Lyapunov constants. Assume that
$p_{22}^{+}\neq 0$, a direct computation shows that

$\begin{array}[]{rl}&\det\left[\dfrac{\partial(V_{16},V_{26},V_{36},V_{46})}{\partial(\delta_{6},p_{52}^{-},q_{32}^{+},q_{33}^{+})}\right]=\dfrac{8\pi^{2}}{45}a_{2}^{+}p_{22}^{+}\neq
0,\end{array}$

which implies that there exists $4$ small-amplitude limit cycles by perturbing
the parameters, in backward, $\delta_{6}$, $p_{52}^{-}$, $q_{32}^{+}$ and
$q_{33}^{+}$.

One more small-amplitude limit cycle is obtained by perturbing $d$, leading to
a total $5$ small-amplitude limit cycles. For $d$ and $\varepsilon$
sufficiently small, the switching system (44) has a small sliding segment on
the switching manifold $y=0$ with the end points at $(0,0)$ and
$(x_{\varepsilon},0)$, where $x_{\varepsilon}=-d\varepsilon^{6}$ is the unique
root of the equation $\dot{y}=0$ in the second system of (44). Thus, the
sliding segment shrinks to $(0,0)$ when $\varepsilon$ goes to zero.

For the point $(x_{e},0)$ on the switching manifold $y=0$ with
$x_{\varepsilon}<x_{e}\ll 1$, we define a bifurcation function,

$d(x_{e},\varepsilon)=\Pi^{+}(x_{e},\varepsilon)-\Pi_{-}^{+}(x_{e},\varepsilon),$
(45)

for which has two small positive constants $\varepsilon_{1}$ and
$\varepsilon_{2}$ such that
$\Pi^{+}(x_{e},\varepsilon):(0,\varepsilon_{1})\rightarrow(-\varepsilon_{1},0)$
is the first half-bifurcation function of (44), and
$\Pi_{-}^{+}(x_{\varepsilon},\varepsilon):(x_{\varepsilon},x_{\varepsilon}+\varepsilon_{2})\rightarrow(x_{\varepsilon}-\varepsilon_{2},x_{\varepsilon})$
is the second half-bifurcation function derived from the second system of (44)
by the transformation $(x,y,t)\rightarrow(x,-y,-t)$. By the polar coordinate
transformation $x=r\cos\theta$ and $y=r\sin\theta$, we have

$\Pi^{+}(x_{e},\varepsilon)=V_{1}^{+}(\varepsilon)x_{e}+O(x_{e}^{2})\quad\text{and}\quad\Pi_{-}^{+}(x_{e},\varepsilon)=V_{0}^{-}(\varepsilon)+V_{1}^{-}(\varepsilon)x_{e}+O(x_{e}^{2}),$
(46)

where

$V_{0}^{-}(\varepsilon)=2d\varepsilon^{6}+O(\varepsilon^{7})\quad\text{and}\quad
V_{1}^{\pm}(\varepsilon)=\pm\frac{1}{2}\delta_{6}\pi\varepsilon^{6}+O(\varepsilon^{7}),$
(47)

It follows from (45) and (46) that $V_{06}=-2d$ and
$V_{16}|_{d=0}=\delta_{6}\pi$. Hence, we obtain the $\varepsilon^{6}$-order
Lyapunov constants $V_{06}$, $V_{16}$, $V_{26}$, $V_{36}$ and $V_{46}$, which
are independent of the parameter vector $\lambda^{*}$. Therefore, we have
proved the existence of $5$ small-amplitude crossing limit cycles near the
origin of cubic switching Liénard system (7).

To end this section, we provide an example with exact critical parameter
vector $\lambda^{*}$ to demonstrate the existence of $5$ small-amplitude limit
cycles. To achieve this, we need to find $5$ positive roots which are solved
from the displacement equation (19):

$d(\xi)=V_{0}(\lambda^{*})+V_{1}(\lambda^{*})\xi+\cdots+V_{4}(\lambda^{*})\xi^{4}+V_{5}(\lambda^{*})\xi^{5}+O(\xi^{6})=0.$
(48)

As discussed above, we have
$V_{j}(\lambda^{*})=V_{j6}\varepsilon^{6}+O(\varepsilon^{6})$,
$j=0,1,\ldots,5$. Hence, (48) becomes

$d(\xi)=(V_{06}+V_{16}\xi+\cdots+V_{46}\xi^{4}+V_{56}\xi^{5})\varepsilon^{6}+O(\varepsilon^{7})=0.$
(49)

In general, it is a very challenging task to find the perturbed parameter
vector to give a numerical realization. However, for our system the parameters
in the displacement equation are linearly decoupled. Thus, we can obtain these
values by perturbing exactly one parameter for one higher-order Lyapunov
constant such that equation (48) has $5$ positive roots.

In the following, we construct a concrete example. First, note that the
critical parameter vector $\lambda^{*}$ given above satisfies that $V_{j6}=0$,
$j=0,1,\ldots,4$, but $V_{56}\neq 0$, and $V_{56}$ does not contain any non-
zero lower-order $\varepsilon$ terms. Thus, we can take perturbations in the
backward order: on $q_{32}^{-}$ for $V_{56}$, on $q_{33}^{-}$ for $V_{46}$, on
$p_{33}^{-}$ for $V_{36}$, on $p_{52}^{-}$ for $V_{26}$, on $\delta_{6}$ for
$V_{16}$, and on $d$ for $V_{06}$. In particular, with the help of Maple
taking the accuracy up to $60$ decimal points, we set the free parameters to
take the values:

$\begin{array}[]{llllllll}&a_{2}^{+}=b_{3}^{+}=p_{12}^{+}=p_{22}^{+}=p_{33}^{+}=p_{52}^{+}=q_{12}^{+}=q_{22}^{+}=q_{32}^{+}=q_{33}^{+}=q_{42}^{\pm}=q_{52}^{\pm}=1,\\\\[4.30554pt]
&a_{3}^{+}-3=b_{2}^{+}+2=q_{32}^{-}+2=0,\end{array}$ (50)

and choose the following values for the perturbed parameters:

$\begin{array}[]{rlrl}d^{*}\,=\\!\\!\\!\\!&d-5\times 10^{-31},&\delta_{6}^{*}\
=\\!\\!\\!\\!&\delta_{6}-3.2\times 10^{-21},\\\\[2.15277pt]
p_{52}^{-*}=\\!\\!\\!\\!&p_{52}^{+}+1.5\times
10^{-13},&p_{33}^{-*}=\\!\\!\\!\\!&p_{33}^{-}-8.5\times
10^{-8},\\\\[2.15277pt]
q_{33}^{-*}=\\!\\!\\!\\!&q_{33}^{-}+0.0188.\end{array}$ (51)

With the above perturbed parameter values, we obtain the perturbed
$\varepsilon^{6}$-order Lyapunov constants as follows:

$\begin{array}[]{llllllll}&V_{06}=-1.0\times 10^{-30},&V_{16}\approx
1.0053096491\times 10^{-20},\\\\[2.15277pt] &V_{26}\approx-2.0006218904\times
10^{-13},&V_{36}\approx 1.0013826592\times 10^{-7},\\\\[2.15277pt]
&V_{46}\approx-0.0100266667,&V_{56}\approx 6.5449846949,\end{array}$ (52)

for which the displacement equation (49) has $5$ positive roots:

$\\!\\!\\!\begin{array}[]{llllllll}\xi_{1}\\!\\!\\!&\approx
0.0015219219,&\xi_{2}\\!\\!\\!&\approx 7.3120695527\times
10^{-6},&\xi_{3}\\!\\!\\!&\approx 2.6762403416\times 10^{-6},\\\\[4.30554pt]
\xi_{4}\\!\\!\\!&\approx 5.1472016794\times 10^{-8},&\xi_{5}\\!\\!\\!&\approx
9.9669521943\times 10^{-11},\end{array}$ (53)

which are approximations of the amplitudes for the bifurcating $5$ small-
amplitude limit cycles.

##  6 Conclusion

In this paper, we have studied the nilpotent center problem and the limit
cycle bifurcation problem for switching nilpotent systems in $\mathbb{R}^{2}$.
We have developed a higher-order Poincaré-Lyapunov method to compute the
generalized Lyapunov constants for switching nilpotent systems. By using this
method, we derive the nilpotent center conditions for the origin of the
switching cubic polynomial Liénard systems. Further, we characterize all the
switching cubic polynomial Liénard systems to have a global nilpotent center.
Finally, we construct a perturbed system with one of the center conditions to
show the existence of $5$ small-amplitude limit cycles bifurcating from the
nilpotent center, which is a new lower bound of the maximal number of limit
cycles in such switching cubic Liénard systems with a nilpotent singular
point. The methodology developed in this paper can be applied to investigate
complex dynamics of other nonlinear systems with nilpotent singular points.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China,
No. 12001112 (T. Chen) and No. 12071198 (F. Li), Guangdong Basic and Applied
Basic Research Foundation, No. 2022A1515011827 (T. Chen), and the Natural
Sciences and Engineering Research Council of Canada, No. R2686A02 (P. Yu).

## References

  * [1] A. Buic$\rm{\breve{a}}$, J. Llibre, O. Makarenkov, Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator, SIAM J. Math. Anal. 40 (2009) 2478–2495.
  * [2] M. Caubergh, Hilbert’s sixteenth problem for polynomial Liénard equations, Qual. Theory Dyn. Syst. 11 (2012) 3–18.
  * [3] H. Chen, X. Chen, Dynamical analysis of a cubic Liénard system with global parameters, Nonlinearity 28 (2015) 3535–3562.
  * [4] H. Chen, X. Chen, Dynamical analysis of a cubic Liénard system with global parameters (II), Nonlinearity 29 (2016) 1798–1826.
  * [5] H. Chen, X. Chen, Dynamical analysis of a cubic Liénard system with global parameters III, Nonlinearity 33 (2020) 1443–1465.
  * [6] H. Chen, S. Duan, Y. Tang, J. Xie, Global dynamics of a mechanical system with dry friction, J. Differential Equations 265 (2018) 5490–5519.
  * [7] T. Chen, L. Huang, P. Yu, Center condition and bifurcation of limit cycles for quadratic switching systems with a nilpotent equilibrium point, J. Differential Equations 303 (2021) 326–368.
  * [8] T. Chen, J. Llibre, Nilpotent center in a continuous piecewise quadratic polynomial Hamiltonian vector field, Int. J. Bifurcation and Chaos 32 (2022) 2250116 (23 pages).
  * [9] L. A. Cherkas, Conditions for a Liénard equation to have a center, Differential Equations 12(2) (1977) 201–206.
  * [10] C. Christopher, An algebraic approach to the classification of centers in polynomial Liénard systems, J. Appl. Math. Mech. 229 (1999) 329–329.
  * [11] C. Christopher, S. Lynch, Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces, Nonlinearity 12 (1999) 1099–1112.
  * [12] B. Coll, R. Prohens, A. Gasull, The center problem for discontinuous Liénard differential equation, Int. J. Bifurcation and Chaos 9 (1999) 1751–1761.
  * [13] A. Colombo, P. Lamiani, L. Benadero, M. di Bernardo, Two-parameter bifurcation analysis of the buck converter, SIAM J. Appl. Dyn. Syst. 8 (2009) 1507–1522.
  * [14] F. Dumortier, J. Llibre, J. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, New York, 2006.
  * [15] I. García, Cyclicity of some symmetric nilpotent centers, J. Differential Equations 260 (2016) 5356–5377.
  * [16] A. Gasull, Differential equations that can be transformed into equations of Liénard type, in “$17^{\circ}$ Colóquio Brasileiro de Matemática, 1989.”
  * [17] A. Gasull, J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifurcation and Chaos 13 (2003) 1755–1765.
  * [18] I. García, H. Giacomini, J. Giné, J. Llibre, Analytic nilpotent centers as limits of nondegenerate centers revisited, J. Math. Anal. Appl. 441 (2016) 893–899.
  * [19] M. Han, Bifurcation Theory of Limit Cycles, Science Press, Beijing, 2013.
  * [20] M. Han, P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, Springer-Verlag, New York, 2012.
  * [21] P. Hirschberg, E. Knobloch, An unfolding of the Takens-Bogdanov singularity, Quarterly Appl. Math. 49 (1991) 281–287.
  * [22] J. Jiang, M. Han, P. Yu, S. Lynch, Limit cycles in two types of symmetric Liénard systems, Int. J. Bifurcation and Chaos 17 (2007) 2169–2174.
  * [23] J. Jiang, M. Han, Small-amplitude limit cycles of some Liénard-type systems, Nonlinear Anal. 71 (2009) 6373–6377.
  * [24] F. Li, H. Li, Y. Liu, New Double Bifurcation of Nilpotent Focus, Int. J. Bifurcation and Chaos 31 (2021) 2150053.
  * [25] F. Li, Y. Liu, Y. Liu, P. Yu, Bi-center problem and bifurcation of limit cycles from nilpotent singular points in $Z_{2}$-equivariant cubic vector fields, J. Differential Equations 265 (2018) 4965–4992.
  * [26] F. Li, P. Yu, Y. Tian, Y. Liu, Center and isochronous center conditions for switching systems associated with elementary singular points, Commun. Nonlinear Sci. Numer. Simul. 28 (2015) 81–97.
  * [27] A. Lins, W. de Melo, C.C. Pugh, On Liénard’s equation, in: Geometry and Topology, in: Lecture Notes in Mathematics, vol. 597, Springer, New York, 1977, pp. 335–357.
  * [28] J. Llibre, M.A. Teixeira, Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys. 66 (2015) 51–66.
  * [29] J. Llibre, C. Valls. Global centers of the generalized polynomial Liénard differential systems, J. Differential Equations 330 (2022) 66–80.
  * [30] Y. Liu, F. Li, Double bifurcation of nilpotent focus, Int. J. Bifurcation and Chaos 25 (2015) 1550036 (10 pages).
  * [31] Y. Liu, J. Li, Bifurcations of limit cycles created by a multiple nilpotent critical point of planar dynamical systems, Int. J. Bifurcation and Chaos 21 (2011) 497–504.
  * [32] P. De Maesschalck, F. Dumortier, Classical Liénard equations of degree $n\geq 6$ can have $[\frac{n-1}{2}]+2$ limit cycles, J. Differential Equations 250 (2011) 2162–2176.
  * [33] R.M. Martins, A.C. Mereu, Limit cycles in discontinuous classical Liénard equations, Nonlinear Anal., Real World Appl. 20 (2014) 67–73.
  * [34] L. Sheng, M. Han, V. Romanovsky, On the number of limit cycles by perturbing a piecewise smooth Liénard model, Int. J. Bifurcation and Chaos 26 (2016) 1650168.
  * [35] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998) 7–15.
  * [36] E. Stróżzyna, H. Żoła̧dek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations 179 (2002) 479–537.
  * [37] S. Tang, J. Liang, Y. Xiao, R.A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math. 72 (2012) 1061–1080.
  * [38] Y. Tian, M. Han, Hopf bifurcation for two types of Liénard systems, J. Differential Equations 251 (2011) 834–859.
  * [39] Y. Tian, M. Han, F. Xu, Bifurcations of small limit cycles in Liénard systems with cubic restoring terms, J. Differential Equations 267 (2019) 1561–1580.
  * [40] Y. Tian, P. Yu, Center conditions in a switching Bautin system, J. Differential Equations 259 (2015) 1203–1226.
  * [41] Y. Tian, P. Yu, Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis, J. Differential Equations 264 (2018) 5950–5976.
  * [42] J. Yang, L. Zhao, The cyclicity of period annuli for a class of cubic Hamiltonian systems with nilpotent singular points, J. Differential Equations 263 (2017) 5554–5581.
  * [43] P. Yu, M. Han, J. Li, An improvement on the number of limit cycles bifurcating from a non-degenerate center of homogeneous polynomial systems, Int. J. Bifurcation and Chaos 28 (2018) 1850078 (31 pages).
  * [44] P. Yu, F. Li, Bifurcation of limit cycles in a cubic-order planar system around a nilpotent critical point, J. Math. Anal. Appl. 453(2) (2017) 645–667.
  * [45] Z. Zhang, T. Ding, W. Huang, Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992.