# Bifurcations of limit cycles in piecewise smooth Hamiltonian system with
boundary perturbation

Nanasaheb Phatangare  Department of Mathematics, Fergusson College, Pune
Krishnat Masalkar  Department of Mathematics, Abasaheb Garware College, Pune
Subhash Kendre  Department of Mathematics, Savitribai Phule Pune University,
Pune

###### Abstract

In this paper, the general planar piecewise smooth Hamiltonian system with
period annulus around the center at the origin is considered. We obtain the
expressions for the first order and the second order Melnikov functions of
it’s general second order perturbation, which can be used to find the number
of limit cycles bifurcated from periodic orbits. Further, we have shown that
the number of limit cycles of the system
$\dot{X}=\begin{cases}(H_{y}^{+},-H_{x}^{+})&\mbox{if}\leavevmode\nobreak\
y>\varepsilon f(x)\\\ (H_{y}^{-},-H_{x}^{-})&\mbox{if}\leavevmode\nobreak\
y<\varepsilon f(x)\end{cases}$ equals to the number of positive zeros of $f$
when at $\varepsilon=0$ the system has a period annulus around the origin.

###### keywords:

Hamiltonian, Piecewise smooth system , Limit cycle , Melnikov function ,
Poincare Map.

††journal: Journal

###### Proof.

##  1 Introduction

Motions of many nonsmooth processes such as impact switching, sliding and
other discrete state transitions are modelled into piecewise smooth dynamical
systems rather than the smooth dynamical systems. Recently piecewise smooth
dynamical systems are of great interest. It has many applications in physical
processes such as electrical circuits, impact oscillators, dry friction
oscillators, relay control systems, modelling of irregular heartbeats etc.[1].
In many scientific applications systems with self sustained oscillations are
modelled where limit cycles plays an important role. Limit cycles bifurcations
in case of smooth dynamical systems is very well studied, whereas the non-
smooth systems have been studied recently.

Averaging theory, Melnikov theory and normal form theory are well known
techniques used to study the limit cycle bifurcation of planar smooth
differential systems [2, 3], whereas the techniques for piecewise smooth
systems are in the process of development [4, 5].

In [6] authors considered a piecewise linear differential systems (PLDS)
having center-focus type singularity with switching manifold $y=0$, in which
limit cycle bifurcation of the system is studied when the switching manifold
is $y=\varepsilon$. Also, in [7] C. Zou and J. Yang studied PLDS with saddle-
centre type singularity at the origin and switching curve $y=b\sin x$, in
which it has been shown that the number of limit cycles bifurcated from the
period annulus of the system with $b=0$ is equal to the number of positive
zeros of $\sin x$. Note that the system considered in [7] is symmetric about
the $y$-axis and zeros of switching curve $y=b\sin x$ are also symmetric about
the $y$-axis. In [8] authors studied the same system as in [7] by considering
the switching curve
$y=bx(x^{2}-x_{1}^{2})(x^{2}-x_{2}^{2})\cdots(x^{2}-x_{m}^{2})$, wherein it is
proved that the number of limit cycles bifurcated from the period annulus of
the system at $b=0$ is equal to $m$, where $m$ is a positive integer. In [9],
authors studied the number of limit cycles bifurcated from the origin of the
perturbation of a planar piecewise smooth system with centre-centre type
singularity at the origin. Further, in [10], normal forms of some planar
piecewise smooth systems with center-center type singularity of order $(k,l)$
at the origin are considered and their limit cycles bifurcation from the
origin under higher order perturbations have been studied. It is natural to
think about the limit cycles bifurcation of these normal forms when the
separation boundary is an analytic function. In [11] Xia Liu and Maoan Han
considered the first order perturbation of a planar piecewise smooth
Hamiltonian system. If the unperturbed system has a period annulus centered at
the origin, then using the first order Melnikov function, the number of limit
cycles bifurcated from the periodic annulus are studied.

In this paper we have obtained the second order Melnikov function for the
piecewise Hamiltonian system with second order perturbation and the separation
boundary $y=0$. We also considered a general piecewise smooth Hamiltonian
system with perturbed separation boundary $y=\varepsilon f(x)$ when $f$ is a
$C^{2}$ function.

The paper is organized as follows: In Section 2 we give some perliminary
concepts about Melnikov theory, limit cycles and stability of limit cycles.
Section 3 is devoted to investigate the first order and second order Melnikov
functions for piecewise smooth Hamiltonian systems with second order
perturbation. Section 4 deals with the general piecewise smooth Hamiltonian
system with boundaty perturbation. Finally, in Section 5, we give some
application of piecewise smooth Hamiltonian systems with boundary
perturbation.

##  2 Preliminaries

Consider a $C^{\infty}$ smooth system of the form

$\displaystyle\dot{X}=\begin{cases}H_{y}+\varepsilon
f(x,y,\varepsilon,\delta)\\\ -H_{x}+\varepsilon
g(x,y,\varepsilon,\delta)\end{cases},$ (2.1)

where $H,f,g$ are $C^{\infty}$ smooth functions for
$\varepsilon\in\mathbb{R},\delta\in K\subset\mathbb{R}^{m}$ with $K$ compact
and $H_{x}(x,y)=\dfrac{\partial H}{\partial x}(x,y),H_{y}(x,y)=\dfrac{\partial
H}{\partial y}(x,y)$. For $\varepsilon=0$, the system (2.1) becomes
Hamiltonian system

$\displaystyle\dot{X}=\begin{cases}H_{y}\\\ -H_{x}\end{cases}.$ (2.2)

Suppose that the system (2.2) has a period annulus
$\displaystyle\mathcal{A}=\\{\Gamma_{h}:H(x,y)=h,h\in(\alpha,\beta)\subset\mathbb{R}\\}$
such that $\Gamma_{h}\rightarrow\Gamma_{\alpha}$ as $h\rightarrow\alpha,$
which is an elementary center point for the system and
$\Gamma_{h}\rightarrow\Gamma_{\beta}$ as $h\rightarrow\beta,$ which is usually
a homoclinic loop consisting of a saddle point or heteroclinic loop containing
two saddle points. For some $h_{0}\in(\alpha,\beta)$, consider a periodic
orbit $\Gamma_{h_{0}}$ from the period annulus and a Poincare section
$S=\\{(a(h),0):h\in(h_{0}-\varepsilon_{0},h_{0}+\varepsilon_{0})\\}$, for some
$\varepsilon_{0}>0$, at the point $A(a(h_{0}),0)$ to $\Gamma_{h_{0}}$. Let
$\Gamma_{h_{0}\varepsilon}$ be the solution of (2.1) starting at
$A(a(h_{0}),0)$and let $B(b(h_{0},\varepsilon,\delta),0)$ be its first point
of intersection with the Poincare section. Then the Poincare map $\mathcal{P}$
maps $A(a(h_{0}),0)$ to $B(b(h_{0},\varepsilon,\delta),0)$. Note that
$H(A(a(h_{0}),0))=H(B(b(h_{0},\varepsilon,\delta),0))$ if and only if
$A(a(h_{0}),0)=B(b(h_{0},\varepsilon,\delta),0)$. Therefore we can use
difference map $H(B)-H(A)$ to investigate the number of limit cycles of (2.1)
bifurcated from $\Gamma_{h_{0}}$. Thus,

$\displaystyle H(B)-H(A)=$
$\displaystyle\int_{\widehat{AB}}dH=\int_{\widehat{AB}}(H_{x}dx+H_{y}dy)$
$\displaystyle=$ $\displaystyle\int_{0}^{\tau_{h_{0}}}[H_{x}(H_{y}+\varepsilon
f)+H_{y}(-H_{x}+\varepsilon
g)]dt=\varepsilon\int_{0}^{\tau_{h_{0}}}(H_{x}f+H_{y}g)dt$ $\displaystyle=$
$\displaystyle\varepsilon
F(h_{0},\varepsilon,\delta)=\sum_{k=1}^{\infty}M_{k}(h_{0},\delta)\varepsilon^{k},$
(2.3)

where $\tau_{h_{0}}$ is the time of flight along the trajectory $\widehat{AB}$
of (2.1) from $A$ to $B$ and

$\displaystyle
M_{k}(h_{0},\delta)=\dfrac{1}{(k-1)!}\dfrac{\partial^{(k-1)}F}{\partial\varepsilon^{k-1}}(h_{0},0,\delta).$

Here, $M_{k}(h_{0},\delta)$ is called as the $k$th order Melnikov function and
$F$ is called as a bifurcation function for the system (2.1).

Clearly, from equation (2) we have

$\displaystyle F(h_{0},0,\delta)=M_{1}(h_{0},\delta)=$
$\displaystyle\int_{\Gamma_{h_{0}}}(H_{x}f+H_{y}g)dt=\int_{\Gamma_{h_{0}}}(gdx-
fdy)=-\int\int_{Int(\Gamma_{h_{0}})}(f_{x}+g_{y})dxdy,$

where $Int(\Gamma_{h_{0}})$ is the region bounded by $\Gamma_{h_{0}}$. Here,
we say that the cyclicity of $\Gamma_{h_{0}}$ is $k$ if there exist
$\varepsilon_{0}$ such that (2.1) has at most $k$ limit cycles in some
neighborhood of $\Gamma_{h_{0}}$ for any $\delta\in K$ and for any
$0<\varepsilon<\varepsilon_{0}$ and that (2.1) has exactly $k$ limit cycles in
every neighbourhood of $\Gamma_{h_{0}}$ for some $(\varepsilon,\delta)$.

The following proposition states that the number of periodic solutions of
(2.1), called as limit cycles, in small neighbourhood for $\Gamma_{h_{0}}$ is
less than or equal to the number of isolated zeros of the first order Melnikov
function $M_{1}(h_{0},\delta)$.

######  Proposition 2.1.

[12] Let $\delta_{0}\in K$. Then we have the following:

  1. 1.

There is no limit cycle near $\Gamma_{h_{0}}$ for
$\varepsilon+|\delta-\delta_{0}|$ small, if $M_{1}(h_{0},\delta_{0})\neq 0$.

  2. 2.

There is exactly one (at least one ) limit cycle
$\Gamma(h_{0},\varepsilon,\delta)$ for $\varepsilon+|\delta-\delta_{0}|$,
which approaches $\Gamma_{h_{0}}$ as
$(\varepsilon,\delta)\rightarrow(0,\delta_{0})$ if
$M_{1}(h_{0},\delta_{0})=0,\leavevmode\nobreak\ \dfrac{\partial
M_{1}}{\partial h}(h_{0},\delta_{0})\neq 0$ (if $h_{0}$ is a zero of
$M(h,\delta_{0})$ with odd multiplicity, respectively).

  3. 3.

If there exist $0\leq j\leq k$ such that $M_{1}(h_{0},\delta_{0})=0$ and
$\dfrac{\partial^{j}M_{1}}{\partial h^{j}}(h_{0},\delta)\neq 0$ then atmost
$k$ limit cycles of (2.1) are bifurcated form $\Gamma_{h_{0}}$.

Now we have the following result about the stability of limit cycles using the
first order Melnikov function.

######  Proposition 2.2.

The limit cycle of (2.1) bifurcated from the periodic orbit of (2.2) passing
through $(a(h),0)$ of the Poincare section is stable if and only if

$\dfrac{dM_{1}}{dh}-M_{1}\dfrac{H_{xx}(a(h),0)}{H_{x}(a(h),0)^{2}}<0,$

where $M_{1}$ is the first order Melnikov function for (2.1).

###### Proof.

From equation (2), we have

$\displaystyle H(b(h,\varepsilon,\delta),0)-H(a(h),0)=\varepsilon
M_{1}(h,\delta)+o(\varepsilon^{2}).$

By Taylor’s expansion in powers of $\varepsilon$, we have

$\displaystyle H(a(h),0)+\varepsilon H_{x}(a(h),0)\left(\dfrac{\partial
b}{\partial\varepsilon}\right)_{\varepsilon=0}+o(\varepsilon^{2})-H(a(h),0)$
$\displaystyle=\varepsilon M_{1}(h,\delta)+o(\varepsilon^{2}).$ (2.4)

Equating $\varepsilon$ order terms in equation (2.4) on both sides we get

$\displaystyle H_{x}(a(h),0)\left(\dfrac{\partial
b}{\partial\varepsilon}\right)_{\varepsilon=0}$
$\displaystyle=M_{1}(h,\delta).$

Now if $\mathcal{P}_{\varepsilon}$ is the Poincare map of system (2.1) then we
have $\mathcal{P}_{\varepsilon}(a(h))=b(h,\varepsilon,\delta)$. Hence

$\displaystyle\left(\dfrac{\partial\mathcal{P}_{\varepsilon}}{\partial\varepsilon}\right)_{\varepsilon=0}=\dfrac{M_{1}(h,\delta)}{H_{x}(a(h),0)}.$
(2.5)

Now differentiang (2.5) with respect to $h$ we get,

$\displaystyle\dfrac{d}{dx}\left[\left(\dfrac{\partial\mathcal{P}_{\varepsilon}}{\partial\varepsilon}\right)_{\varepsilon=0}\right]a^{\prime}(h)$
$\displaystyle=\dfrac{1}{H_{x}(a(h),0)}\dfrac{dM_{1}}{dh}-M_{1}\dfrac{H_{xx}(a(h),0)}{(H_{x}(a(h),0))^{2}}a^{\prime}(h).$
(2.6)

Since $H(a(h),0)=h$, we have $H_{x}(a(h),0)a^{\prime}(h)=1$. Therefore from
(2.6) we get,

$\displaystyle\dfrac{\partial}{\partial\varepsilon}\left[\left(\dfrac{d\mathcal{P}_{\varepsilon}}{dx}\right)\right]_{\varepsilon=0}$
$\displaystyle=\dfrac{dM_{1}}{dh}-M_{1}\dfrac{H_{xx}(a(h),0)}{H_{x}(a(h),0)^{2}}.$

Now if $0<\varepsilon\ll 1$, then we have

$\displaystyle\dfrac{d\mathcal{P}_{\varepsilon}}{dx}-\dfrac{d\mathcal{P}_{0}}{dx}$
$\displaystyle\approx\varepsilon\left(\dfrac{dM_{1}}{dh}-M_{1}\dfrac{H_{xx}(a(h),0)}{H_{x}(a(h),0)^{2}}\right).$

But $\mathcal{P}_{0}$ is poincare return map for (2.2). Hence,
$\mathcal{P}_{0}(a(h))=a(h)$, which imply that

$\dfrac{d\mathcal{P}_{0}}{dx}(a(h))a^{\prime}(h)=a^{\prime}(h).$

Hence,

$\displaystyle\dfrac{d\mathcal{P}_{\varepsilon}}{dx}-1$
$\displaystyle\approx\varepsilon\left(\dfrac{dM_{1}}{dh}-M_{1}\dfrac{H_{xx}(a(h),0)}{H_{x}(a(h),0)^{2}}\right).$

Thus, the limit cycle passing through $A(a(h),0)$ is stable if and only if

$\dfrac{dM_{1}}{dh}-M_{1}\dfrac{H_{xx}(a(h),0)}{H_{x}(a(h),0)^{2}}<0.$

∎

General planar piecewise smooth differential system with two zones and
switching manifold $\Sigma=\varphi^{-1}(0)$ is given by

$\displaystyle\dot{X}=$
$\displaystyle\begin{cases}(X_{1}^{+}(x,y),\leavevmode\nobreak\
\leavevmode\nobreak\ X_{2}^{+}(x,y)),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (x,y)\in\Sigma^{+}\\\
(X_{1}^{-}(x,y),\leavevmode\nobreak\ \leavevmode\nobreak\
X_{2}^{-}(x,y))\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(x,y)\in\Sigma^{-}\end{cases},$ (2.7)

where $X_{i}^{\pm},f^{\pm},g^{\pm},i=1,2$ and $\varphi$ are sufficiently
smooth functions on some open region containing the origin with $0$ as a
regular value of $\varphi$ and
$\Sigma^{+}=\varphi^{-1}(0,\infty),\leavevmode\nobreak\ \leavevmode\nobreak\
\Sigma^{-}=\varphi^{-1}(-\infty,0)$ are two zones of (2.7).

Now we denote $X^{\pm}=:(X_{1}^{\pm},X_{2}^{\pm})$ ,
$X^{\pm}\varphi=:<X^{\pm},\nabla\varphi>$ and
$(X^{\pm})^{k}\varphi=:<X^{\pm},\nabla(X^{\pm})^{k-1}\varphi>$, where $<,>$ is
an Euclidean dot product.

We say that a point $p\in\Sigma$ is a $k$th contact point for the vector field
$X$ if $(X^{k}\varphi)(p)\neq 0$ and $(X^{l}\varphi)(p)=0$ for
$l=1,\cdots,k-1$. A point $p\in\Sigma$ is a $(k,l)$-contact singularity of
$X^{\pm}$ if $X_{0}$ is a $k$th contact point for $X^{+}$ and is a $l$th
contact point for $X^{-}$.

##  3 Perturbation of Piecewise Smooth Hamiltonian System

Recently in [11], authors studied the number of limit cycles of the peicewise
smooth Hamiltonian system

$\displaystyle\dot{X}=$
$\displaystyle(H_{y}^{+}(x,y),-H_{x}^{+}(x,y))+\varepsilon(f^{+}(x,y,\varepsilon,\delta),g^{+}(x,y,\varepsilon,\delta)),$
$\displaystyle(x,y)\in\Sigma^{+}$ (3.1) $\displaystyle\dot{X}=$
$\displaystyle(H_{y}^{-}(x,y),-H_{x}^{-}(x,y))+\varepsilon(f^{-}(x,y,\varepsilon,\delta),g^{-}(x,y,\varepsilon,\delta)),$
$\displaystyle(x,y)\in\Sigma^{-}$ (3.2)

i.e.,

$\displaystyle\dot{X}=$
$\displaystyle\begin{cases}(H_{y}^{+}(x,y),-H_{x}^{+}(x,y))+\varepsilon(f^{+}(x,y,\varepsilon,\delta),g^{+}(x,y,\varepsilon,\delta)),&(x,y)\in\Sigma^{+}\\\
(H_{y}^{-}(x,y),-H_{x}^{-}(x,y))+\varepsilon(f^{-}(x,y,\varepsilon,\delta),g^{-}(x,y,\varepsilon,\delta)),&(x,y)\in\Sigma^{-}\end{cases}.$
(3.3)

System (3.3) is a perturbation of the Hamiltonian system

$\displaystyle\dot{X}=$
$\displaystyle(H_{y}^{+}(x,y),-H_{x}^{+}(x,y)),(x,y)\in\Sigma^{+}$ (3.4)
$\displaystyle\dot{X}=$
$\displaystyle(H_{y}^{-}(x,y),-H_{x}^{-}(x,y)),(x,y)\in\Sigma^{-}$ (3.5)

or

$\displaystyle\dot{X}=$
$\displaystyle\begin{cases}(H_{y}^{+}(x,y),-H_{x}^{+}(x,y)),&(x,y)\in\Sigma^{+}\\\
(H_{y}^{-}(x,y),-H_{x}^{-}(x,y)),&(x,y)\in\Sigma^{-}\end{cases}.$ (3.6)

Suppose that the switching manifold for (3.3) and (3.6) is
$\Sigma=\varphi^{-1}(0)$, where $\varphi(x,y)=y$. Assume that the system (3.6)
has a period annulus around the origin in some open region $V$. Let
$L_{+}=V\cap\\{(x,0):x>0\\}$ and $L_{-}=V\cap\\{(x,0):x<0\\}$. Let
$\Gamma^{+}_{r}:H^{+}(x,y)=r,\leavevmode\nobreak\ y\geq 0$ be a trajectory of
(3.4) which starts at $P(r)=(p(r),0)$ on $L_{+}$, ends at the point
$P_{1}(r)=(p_{1}(r),0)$ on $L_{-}$ with the time of flight $t^{+}(r)$. Then
the Poincare half return map $\mathcal{P}^{+}:L^{+}\rightarrow L^{-}$ is given
by

$\displaystyle\mathcal{P}^{+}(p(r))=p_{1}(r).$

Let $\Gamma^{-}_{r}:H^{-}(x,y)=s,y\leq 0$ be the trajectory of (3.5) starting
at $P_{1}(r)$ on $L_{-}$ and ending at the point $P(r)$ with time of flight
$t^{-}(r)$. Therefore the next half return map
$\mathcal{P}^{-}:L^{-}\rightarrow L^{+}$ is given by

$\displaystyle\mathcal{P}^{-}(p_{1}(r))=p(r).$

Figure 1: Trajectories showing the difference map

Let $\Gamma^{+}_{r\varepsilon}$ be a trajectory of (3.1) starting at $P(r)$
and meeting the first time on $L_{-}$ at the point
$P_{2}(p_{2}(r,\varepsilon),0)$ and let $\Gamma^{-}_{r\varepsilon}$ be the
trajectory of the system (3.2) starting at $P_{2}$ and the meeting first time
on $L_{+}$ at the point $Q(q(r,\varepsilon),0)$ (Fig1).

Then the Poincare map $\mathcal{P}_{\varepsilon}$ for (3.3) defined on $L_{+}$
is given by

$\displaystyle\mathcal{P}_{\varepsilon}(p(r))=q(r,\varepsilon).$

Observe that $\Gamma^{+}_{r\varepsilon}\cup\Gamma^{-}_{r\varepsilon}$ forms a
closed trajectory of the system (3.3) if and only if $p(r)=q(r,\varepsilon)$.
But $p(r)=q(r,\varepsilon)$ is equivalent to
$H^{+}(p(r),0)=H^{+}(q(r,\varepsilon),0)$. Hence, analogous to the case of
smooth differential system, we use the difference map

$\displaystyle H^{+}(Q)-H^{+}(P)=$ $\displaystyle\varepsilon
F(r,\varepsilon,\delta)=\sum_{k=1}^{\infty}M_{k}(r,\delta)\varepsilon^{k},$

where $M_{k}(r,\delta)$ is called as the $k$th order Melnikov function for
system (3.3) and $F$ is a bifurcation function.

Similar to the Proposition 2.1 and Proposition 2.2, we can state the
conditions for cyclicity and stability of limit cycles for the system (3.3).

######  Proposition 3.1.

Assume that the system (3.6) has a period annulus with center at the origin.
Let $M_{1}(r)$ be the first order Melnikov function for the system (3.3). Then
we have the following:

  1. 1.

If there exist $0\leq j\leq k$ such that $M_{1}(r_{0},\delta)=0$ and
$\dfrac{\partial^{j}M_{1}}{\partial r^{j}}(r_{0},\delta)\neq 0$, then at most
$k$ limit cycles of (3.3) are bifurcated form $\Gamma_{r_{0}}$, where
$\Gamma_{r_{0}}$ is a periodic orbit of (3.6) through $r_{0}$.

  2. 2.

Limit cycle of (3.3) bifurcated from the periodic orbit of (3.6) passing
through $(p(r),0)$ of the Poincare section is stable if and only if

$\dfrac{dM_{1}}{dr}-M_{1}(r)\dfrac{H_{xx}^{+}(p(r),0)}{H_{x}^{+}(p(r),0)^{2}}<0.$

###### Proof.

Proof is similar to that of Proposition 2.1 and Proposition 2.2.∎

In [11], the first order Melnikov function for (3.3) when $f^{\pm},g^{\pm}$
are independent of $\varepsilon$ and $\delta$, is given by the following
proposition.

######  Proposition 3.2.

[11] If the system (3.6) has a period annulus then the first order Melnikov
function for the system (3.3) is

$\displaystyle M_{1}(r)=$
$\displaystyle\dfrac{H^{+}_{x}(P)}{H_{x}^{-}(P)}\left[\dfrac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}\int_{\widehat{PP_{1}}}(g^{+}dx-f^{+}dy)+\int_{\widehat{P_{1}P}}(g^{-}dx-f^{-}dy)\right],$

where $\widehat{PP_{1}}$ denote the path along the trajectory $\Gamma^{+}_{r}$
and $\widehat{P_{1}P}$ denote the path along the trajectory $\Gamma^{-}_{r}$.

In this section, we first derive the expressions for first order and second
order Melnikov functions for piecewise smooth perturebed Hamiltonian system

$\displaystyle\dot{X}=$ $\displaystyle(H_{y}^{+}(x,y)+\varepsilon
f_{1}^{+}(x,y)+\varepsilon^{2}f_{2}^{+}(x,y),-H_{x}^{+}(x,y)+\varepsilon
g_{1}^{+}(x,y)+\varepsilon^{2}g_{2}^{+}(x,y)),\leavevmode\nobreak\ y>0$ (3.7)
$\displaystyle\dot{X}=$ $\displaystyle(H_{y}^{-}(x,y)+\varepsilon
f_{1}^{-}(x,y)+\varepsilon^{2}f_{2}^{-}(x,y),-H_{x}^{-}(x,y)+\varepsilon
g_{1}^{-}(x,y)+\varepsilon^{2}g_{2}^{-}(x,y)),\leavevmode\nobreak\ y<0$ (3.8)

or

$\displaystyle\dot{X}=$ $\displaystyle\begin{cases}(H_{y}^{+}(x,y)+\varepsilon
f_{1}^{+}(x,y)+\varepsilon^{2}f_{2}^{+}(x,y),-H_{x}^{+}(x,y)+\varepsilon
g_{1}^{+}(x,y)+\varepsilon^{2}g_{2}^{+}(x,y)),\leavevmode\nobreak\ y>0\\\
(H_{y}^{-}(x,y)+\varepsilon
f_{1}^{-}(x,y)+\varepsilon^{2}f_{2}^{-}(x,y),-H_{x}^{-}(x,y)+\varepsilon
g_{1}^{-}(x,y)+\varepsilon^{2}g_{2}^{-}(x,y)),\leavevmode\nobreak\
y<0\end{cases},$ (3.9)

under the assumption that the uperturbed system (3.6) has a period annulus
around the origin.

######  Theorem 3.1.

If the system (3.6) has a period annulus around the origin, then the first
order Melnikov functions for the system (3.9) is given by

$\displaystyle M_{1}(r)=$
$\displaystyle\dfrac{H_{x}^{+}(P)}{H_{x}^{-}(P)}\left(\dfrac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\omega_{1}^{+}+\int_{\Gamma_{r}^{-}}\omega_{1}^{-}\right).$
(3.10)

Further, if $M_{1}\equiv 0$ then the second order Melnikov function $M_{2}$
for the system (3.9) is given by

$\displaystyle M_{2}(r)\frac{H_{x}^{-}(P)}{H_{x}^{+}(P)}=$
$\displaystyle\left(\int_{\Gamma_{r}^{-}}\omega_{2}^{-}+\dfrac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\omega_{2}^{+}\right)$
$\displaystyle+\left(K^{-}(P(r))\int_{\Gamma_{r}^{-}}\frac{\omega_{1}^{-}}{H_{y}^{-}}+\dfrac{H_{x}^{-}(P_{1})K^{+}(P(r))}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\frac{\omega_{1}^{+}}{H_{y}^{+}}\right)$
$\displaystyle-\left(\int_{\Gamma^{-}_{r}}\frac{f_{1}^{-}\omega_{1}^{-}}{H_{y}^{-}}+\frac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\frac{f_{1}^{+}\omega_{1}^{+}}{H_{y}^{+}}\right)+\frac{1}{2}\left(H_{xx}^{-}(P_{1})-\frac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}H_{xx}^{+}(P_{1})\right)\sigma^{2},$
(3.11)

where

$\displaystyle\omega_{i}^{\pm}=g_{i}^{\pm}dx-f_{i}^{\pm}dy\leavevmode\nobreak\
\mbox{for}\leavevmode\nobreak\ i=1,2;\leavevmode\nobreak\
\sigma=\dfrac{1}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\omega_{1}^{+},\leavevmode\nobreak\
\leavevmode\nobreak\
K^{\pm}=\frac{H_{x}^{\pm}f_{1}^{\pm}+H_{y}^{\pm}g_{1}^{\pm}}{H_{x}^{\pm}}.$

We prove Theorem 3.1 in sequence of following Lemmas.

######  Lemma 3.1.

Difference map for the system (3.9) can be expressed as,

$\displaystyle H^{+}(Q(q(r,\varepsilon),0)-H^{+}(P(p(r),0))=\varepsilon
H_{x}^{+}(P)\rho+\dfrac{\varepsilon^{2}}{2}\left[H_{xx}^{+}(P)\rho^{2}+H_{x}^{+}(P)\eta\right]+o(\varepsilon^{3}),$
(3.12)

where
$\displaystyle\rho=\left[\frac{\partial}{\partial\varepsilon}\left(q(r,\varepsilon)\right)\right]_{\varepsilon=0}$
and
$\displaystyle\eta=\left[\frac{\partial^{2}q(r,\varepsilon)}{\partial\varepsilon^{2}}\right]_{\varepsilon=0}$.

###### Proof.

Let $\displaystyle\sigma=\left[\frac{\partial
p_{2}(r,\varepsilon)}{\partial\varepsilon}\right]_{\varepsilon=0},\tau=\left[\frac{\partial^{2}p_{2}(r,\varepsilon)}{\partial\varepsilon^{2}}\right]_{\varepsilon=0},\rho=\left[\frac{\partial
q(r,\varepsilon)}{\partial\varepsilon}\right]_{\varepsilon=0}$ and
$\displaystyle\eta=\left[\frac{\partial^{2}q(r,\varepsilon)}{\partial\varepsilon^{2}}\right]_{\varepsilon=0}.$
The difference map for the system (3.9) is

$\displaystyle H^{+}(Q(r,\varepsilon))-H^{+}(P(r))=$ $\displaystyle
L_{1}+L_{2}+L_{3}+L_{4},$ (3.13)

where

$\displaystyle L_{1}=H^{+}(Q(r))-H^{-}(Q(r)),$ $\displaystyle
L_{2}=H^{-}(Q(r))-H^{-}(P_{2}(r)),$ $\displaystyle
L_{3}=H^{-}(P_{2}(r))-H^{+}(P_{2}(r)),$ $\displaystyle
L_{4}=H^{+}(P_{2}(r))-H^{+}(P(r)).$

Now by the Taylor’s series expansion in powers of $\varepsilon$ we have

$\displaystyle\sum_{i=1}^{4}L_{i}(r,\varepsilon)=$
$\displaystyle\sum_{i=1}^{4}\left[\varepsilon\left(\frac{\partial
L_{i}}{\partial\varepsilon}\right)_{\varepsilon=0}+\frac{\varepsilon^{2}}{2}\left(\frac{\partial^{2}L_{i}}{\partial\varepsilon^{2}}\right)_{\varepsilon=0}+o(\varepsilon^{3})\right].$

Since

$\displaystyle L_{1}=$ $\displaystyle
H^{+}(Q)-H^{-}(Q)=H^{+}(q(r,\varepsilon),0)-H^{-}(q(r,\varepsilon),0),$

we get

$\displaystyle\left[\dfrac{\partial
L_{1}}{\partial\varepsilon}\right]_{\varepsilon=0}=$
$\displaystyle(H_{x}^{+}(P)-H_{x}^{-}(P))\rho,$ (3.14)
$\displaystyle\mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\left[\dfrac{\partial^{2}L_{1}}{\partial\varepsilon^{2}}\right]_{\varepsilon=0}=$
$\displaystyle(H_{xx}^{+}(P)-H_{xx}^{-}(P))\rho^{2}+(H_{x}^{+}(P)-H_{x}^{-}(P))\eta.$
(3.15)

Similarly,

$\displaystyle L_{3}=$ $\displaystyle
H^{-}(P_{2})-H^{+}(P_{2})=H^{-}(p_{2}(r,\varepsilon),0)-H^{+}(p_{2}(r,\varepsilon),0)$

imply that

$\displaystyle\left[\frac{\partial
L_{3}}{\partial\varepsilon}\right]_{\varepsilon=0}=$
$\displaystyle(H_{x}^{-}(P_{1})-H_{x}^{+}(P_{1}))\sigma,$ (3.16)
$\displaystyle\mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\left[\frac{\partial^{2}L_{3}}{\partial\varepsilon^{2}}\right]_{\varepsilon=0}=$
$\displaystyle(H_{xx}^{-}(P_{1})-H_{xx}^{+}(P_{1}))\sigma^{2}+(H_{x}^{-}(P_{1})-H_{x}^{+}(P_{1}))\tau.$
(3.17)

Also,

$\displaystyle L_{2}=$ $\displaystyle
H^{-}(Q)-H^{-}(P_{2})=H^{-}(q(r,\varepsilon),0)-H^{-}(p_{2}(r,\varepsilon),0)$

gives us

$\displaystyle\left[\frac{\partial
L_{2}}{\partial\varepsilon}\right]_{\varepsilon=0}=$ $\displaystyle
H_{x}^{-}(P)\rho-H_{x}^{-}(P_{1})\sigma,$ (3.18)
$\displaystyle\mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\left[\frac{\partial^{2}L_{2}}{\partial\varepsilon^{2}}\right]_{\varepsilon=0}=$
$\displaystyle
H_{xx}^{-}(P)\rho^{2}-H_{xx}^{-}(P_{1})\sigma^{2}+H_{x}^{-}(P)\eta-
H_{x}^{-}(P_{1})\tau.$ (3.19)

Further,

$\displaystyle
L_{4}=H^{+}(P_{2})-H^{+}(P)=H^{+}((p_{2}(r,\varepsilon),0))-H^{+}(p(r),0),$

so that

$\displaystyle\left[\frac{\partial
L_{4}}{\partial\varepsilon}\right]_{\varepsilon=0}=$ $\displaystyle
H_{x}^{+}(P_{1})\sigma,$ (3.20) $\displaystyle\mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\left[\frac{\partial^{2}L_{4}}{\partial\varepsilon^{2}}\right]_{\varepsilon=0}=$
$\displaystyle H_{xx}^{+}(P_{1})\sigma^{2}+H_{x}^{+}(P_{1})\tau.$ (3.21)

Hence, from equations (3.13)-(3.21) we get equation (3.12). ∎

######  Lemma 3.2.

The expression for $L_{4}$ is given by

$\displaystyle L_{4}=$
$\displaystyle\varepsilon\int_{\Gamma^{+}_{r}}\omega_{1}^{+}+\varepsilon^{2}\left(\int_{\Gamma^{+}_{r}}\omega_{2}^{+}+K^{+}(P(r))\int_{\Gamma^{+}_{r}}\frac{\omega_{1}^{+}}{H_{y}^{+}}-\int_{\Gamma_{r}^{+}}\frac{f_{1}^{+}\omega_{1}^{+}}{H_{y}^{+}}\right)+o(\varepsilon^{3}),$
(3.22)

where $\displaystyle\omega_{i}^{+}=g^{+}_{i}dx-f^{+}_{i}dy,i=1,2$.

###### Proof.

We have,

$\displaystyle L_{4}=$ $\displaystyle
H^{+}(P_{2})-H^{+}(P)=\int_{\widehat{PP_{2}}}dH^{+}=\int_{\widehat{PP_{2}}}H^{+}_{x}dx+H^{+}_{y}dy$
$\displaystyle=$
$\displaystyle\int_{\widehat{PP_{2}}}\left[H^{+}_{x}(H_{y}^{+}+\varepsilon
f_{1}^{+}+\varepsilon^{2}f_{2}^{+})+H^{+}_{y}(-H_{x}^{+}+\varepsilon
g_{1}^{+}+\varepsilon^{2}g_{2}^{+})\right]dt$ $\displaystyle=$
$\displaystyle\varepsilon\int_{\widehat{PP_{2}}}(H^{+}_{x}f_{1}^{+}+H_{y}^{+}g_{1}^{+})dt+\varepsilon^{2}\int_{\widehat{PP_{2}}}(H_{x}^{+}f_{2}^{+}+H_{y}^{+}g_{2}^{+})dt$
$\displaystyle=$
$\displaystyle\varepsilon\int_{\widehat{PP_{2}}}(H^{+}_{x}f_{1}^{+}+H_{y}^{+}g_{1}^{+})dt+\varepsilon^{2}\int_{\widehat{PP_{1}}}(H_{x}^{+}f_{2}^{+}+H_{y}^{+}g_{2}^{+})dt+o(\varepsilon^{3})$
$\displaystyle=$
$\displaystyle\varepsilon\int_{\widehat{PP_{2}}}(H^{+}_{x}f_{1}^{+}+H_{y}^{+}g_{1}^{+})dt+\varepsilon^{2}\int_{\Gamma_{r}^{+}}\omega_{2}^{+}+o(\varepsilon^{3}).$
(3.23)

Along the path $\widehat{PP_{2}}$, we have
$dt=\dfrac{dy}{\dot{y}}=\dfrac{dy}{-H_{x}^{+}+\varepsilon
g_{1}^{+}+\varepsilon^{2}g_{2}^{+}}=\dfrac{-1}{H_{x}^{+}}\left(1+\varepsilon\dfrac{g_{1}^{+}}{H_{x}^{+}}+o(\varepsilon^{2})\right)dy.$

Let us denote
$K^{+}=\dfrac{H_{x}^{+}f_{1}^{+}+H_{y}^{+}g_{1}^{+}}{H_{x}^{+}},K^{+}dy=-g_{1}^{+}dx+f_{1}^{+}dy=-\omega_{1}^{+}$
on $\Gamma_{r}^{+}$ and $R$ is the region bounded by
$\Gamma^{+}_{r\varepsilon}$ and
$\Gamma^{+}_{r}\cup\overrightarrow{P_{1}P_{2}}$, where
$\overrightarrow{P_{1}P_{2}}$ denote the line segment from $P_{1}$ to $P_{2}$.
Then

$\displaystyle\int_{\widehat{PP_{2}}}({H_{x}^{+}}{f_{1}}^{+}$
$\displaystyle+{H_{y}}^{+}{g_{1}}^{+})dt=\int_{\widehat{PP_{2}}}-\dfrac{{H_{x}^{+}}{f_{1}}^{+}+{H_{y}}^{+}{g_{1}}^{+}}{{H_{x}}^{+}}\left(1+\varepsilon\dfrac{{g_{1}}^{+}}{{H_{x}}^{+}}+o(\varepsilon^{2})\right)dy$
$\displaystyle=$
$\displaystyle-\int_{\widehat{PP_{2}}}\dfrac{{H_{x}}^{+}{f_{1}}^{+}+{H_{y}}^{+}{g_{1}}^{+}}{{H_{x}}^{+}}dy-\varepsilon\int_{\widehat{PP_{2}}}\dfrac{{g_{1}}^{+}({H_{x}}^{+}{f_{1}}^{+}+{H_{y}}^{+}{g_{1}}^{+})}{{{H_{x}}^{+}}^{2}}dy+o(\varepsilon^{2})$
$\displaystyle=$
$\displaystyle-\int_{\widehat{PP_{2}}}K^{+}dy-\varepsilon\left(\int_{\widehat{PP_{1}}}\dfrac{{g_{1}}^{+}K^{+}}{{H_{x}}^{+}}dy+o(\varepsilon)\right)+o(\varepsilon^{2})$
$\displaystyle=$
$\displaystyle-\int_{\widehat{PP_{1}}}K^{+}dy-\int_{\overrightarrow{P_{1}P_{2}}}K^{+}dy-\iint_{R}\dfrac{\partial
K^{+}}{\partial
x}dxdy-\varepsilon\left(\int_{\widehat{PP_{1}}}\dfrac{K^{+}{g_{1}}^{+}}{{H_{x}}^{+}}dy+o(\varepsilon)\right)+o(\varepsilon^{2})$
$\displaystyle=$
$\displaystyle\int_{\Gamma^{+}_{r}}\omega^{+}_{1}-\int_{\overrightarrow{P_{1}P_{2}}}K^{+}dy-\iint_{R}\dfrac{\partial
K^{+}}{\partial
x}dxdy-\varepsilon\int_{\Gamma^{+}_{r}}\dfrac{K^{+}{g_{1}}^{+}}{{H_{x}}^{+}}dy+o(\varepsilon^{2})$
$\displaystyle=$
$\displaystyle\int_{\Gamma^{+}_{r}}\omega^{+}_{1}-\iint_{R}\dfrac{\partial
K^{+}}{\partial
x}dxdy+\varepsilon\int_{\Gamma^{+}_{r}}\dfrac{{g_{1}}^{+}\omega_{1}^{+}}{{H_{x}}^{+}}+o(\varepsilon^{2}).$
(3.24)

Now suppose that $R=R_{1}\cup R_{2}$, where $R_{1}$ is the region bounded by
$\Gamma^{+}_{r},\Gamma^{+}_{r\varepsilon},x=p_{1}(r)$ and $x=p(r)$ whereas
$R_{2}$ is bounded by $y=0,\Gamma^{+}_{r\varepsilon},x=p_{2}(r,\varepsilon)$
and $x=p_{1}(r)$. Note that, since the radial distance $z$ form $P_{1}(r)$ to
the point on $\Gamma^{+}_{r\varepsilon}$ in $R_{2}$ is of order $\varepsilon$,
we have
$\displaystyle\int\int_{R_{2}}dxdy=\int_{0}^{\frac{\pi}{2}}\int_{0}^{z}rdrd\theta=\frac{\pi}{4}z^{2}=o(\varepsilon^{2})$.
Therefore

$\displaystyle\displaystyle\iint_{R_{2}}\dfrac{\partial K^{+}}{\partial
x}dxdy=o(\varepsilon^{2}).$ (3.25)

Let us represent $\Gamma_{r\varepsilon}^{+}$ and $\Gamma_{r}^{+}$ by
$y_{\varepsilon}=y(x,\varepsilon),y_{0}=y(x,0)$ respectively. Put
$y(x,s)=y_{0}(x)+s(y_{\varepsilon}(x)-y_{0}(x)),\leavevmode\nobreak\
\leavevmode\nobreak\ p_{1}(r)\leq x\leq p(r),\leavevmode\nobreak\
\leavevmode\nobreak\ 0\leq s\leq 1.$ Hence, area element for region $R_{1}$
becomes $dydx=(y_{\varepsilon}(x)-y_{0}(x))dsdx$. Therefore

$\displaystyle\iint_{R_{1}}\dfrac{\partial K^{+}}{\partial x}dxdy=$
$\displaystyle\int_{p_{1}(r)}^{p(r)}\left(\int_{0}^{1}\dfrac{\partial
K^{+}}{\partial x}(y_{\varepsilon}(x)-y_{0}(x))ds\right)dx$ $\displaystyle=$
$\displaystyle\varepsilon\int_{p_{1}(r)}^{p(r)}\left(\int_{0}^{1}\dfrac{\partial
K^{+}}{\partial x}\left(\frac{\partial
y_{\varepsilon}}{\partial\varepsilon}\right)_{\varepsilon=0}ds\right)dx+o(\varepsilon^{2})$
$\displaystyle=$
$\displaystyle\varepsilon\int_{p_{1}(r)}^{p(r)}\left(\int_{0}^{1}\dfrac{\partial
K^{+}}{\partial x}(x,y_{0}(x))\left(\frac{\partial
y_{\varepsilon}}{\partial\varepsilon}\right)_{\varepsilon=0}ds\right)dx+o(\varepsilon^{2})$
$\displaystyle=$
$\displaystyle\varepsilon\int_{p_{1}(r)}^{p(r)}\dfrac{\partial K^{+}}{\partial
x}(x,y_{0})\left(\frac{\partial
y_{\varepsilon}}{\partial\varepsilon}\right)_{\varepsilon=0}dx+o(\varepsilon^{2}).$
(3.26)

Now along $\Gamma^{+}_{r\varepsilon}$, we have

$\displaystyle\dot{y_{\varepsilon}}=\dfrac{\partial y_{\varepsilon}}{\partial
x}\dot{x}=-{H_{x}}^{+}+\varepsilon{g_{1}}^{+}+\varepsilon^{2}{g_{2}}^{+}\Rightarrow\dfrac{\partial
y_{\varepsilon}}{\partial
x}=\dfrac{-{H_{x}}^{+}+\varepsilon{g_{1}}^{+}+\varepsilon^{2}{g_{2}}^{+}}{{H_{y}}^{+}+\varepsilon{f_{1}}^{+}+\varepsilon^{2}{f_{2}}^{+}}$
$\displaystyle\Rightarrow$ $\displaystyle
y_{\varepsilon}=\int_{p_{1}(r)}^{x}\dfrac{-{H_{x}}^{+}+\varepsilon{g_{1}}^{+}+\varepsilon^{2}{g_{2}}^{+}}{{H_{y}}^{+}+\varepsilon{f_{1}}^{+}+\varepsilon^{2}{f_{2}}^{+}}ds=\int_{p_{1}(r)}^{x}\dfrac{-{H_{x}}^{+}}{{H_{y}}^{+}}ds+\varepsilon\int_{p_{1}(r)}^{x}\dfrac{{H_{x}}^{+}{f_{1}}^{+}+{H_{y}}^{+}{g_{1}}^{+}}{{{H_{y}}^{+}}^{2}}ds+o(\varepsilon^{2}).$

Note here that last integrals is taken along $\Gamma_{r}^{+}$. From above
expression we have

$\displaystyle\left(\dfrac{\partial
y_{\varepsilon}}{\partial\varepsilon}\right)_{\varepsilon=0}=$
$\displaystyle\int_{p_{1}(r)}^{x}\dfrac{{H_{x}}^{+}{f_{1}}^{+}+{H_{y}}^{+}{g_{1}}^{+}}{{{H_{y}}^{+}}^{2}}ds=\int_{p_{1}(r)}^{x}\dfrac{K^{+}{H_{x}}^{+}}{({H_{y}}^{+})^{2}}dx=I^{+}(x),\leavevmode\nobreak\
\mbox{(say)}.$ (3.27)

Therefore, from equations (3.25), (3) and (3.27), we get

$\displaystyle\iint_{R}\dfrac{\partial K^{+}}{\partial
x}dxdy=\varepsilon\int_{p_{1}(r)}^{p(r)}\dfrac{\partial K^{+}}{\partial
x}(x,y_{0})I^{+}dx+o(\varepsilon^{2}).$

Now using integration by parts, we have

$\displaystyle\iint_{R}\dfrac{\partial K^{+}}{\partial
x}dxdy=\varepsilon\left(K^{+}(P(r))\int_{p_{1}(r)}^{p(r)}\frac{K^{+}H_{x}^{+}}{(H_{y}^{+})^{2}}dx\right)-\varepsilon\int_{p_{1}(r)}^{p(r)}\frac{(K^{+})^{2}H_{x}^{+}}{(H_{y}^{+})^{2}}dx+o(\varepsilon^{2})$
$\displaystyle=-\varepsilon
K^{+}(P(r))\int_{\Gamma_{r}^{+}}\frac{\omega_{1}^{+}}{H_{y}^{+}}+\varepsilon\int_{\Gamma_{r}^{+}}\frac{f_{1}^{+}\omega_{1}^{+}}{H_{y}^{+}}+\varepsilon\int_{\Gamma_{r}^{+}}\frac{g_{1}^{+}\omega_{1}^{+}}{H_{x}^{+}}+o(\varepsilon^{2}).$
(3.28)

Hence, from (3), (3) and (3) we obtain the formula for $L_{4}$. ∎

######  Lemma 3.3.

The expression for $L_{2}$ is given by

$\displaystyle L_{2}=$
$\displaystyle\varepsilon\int_{\Gamma^{-}_{r}}\omega_{1}^{-}+\varepsilon^{2}\left(\int_{\Gamma^{-}_{r}}\omega_{2}^{-}+K^{-}(P(r))\int_{\Gamma^{-}_{r}}\frac{\omega_{1}^{-}}{H_{y}^{-}}-\int_{\Gamma_{r}^{-}}\frac{f_{1}^{-}\omega_{1}^{-}}{H_{y}^{-}}\right)+o(\varepsilon^{3}),$
(3.29)

where $\displaystyle\omega_{i}^{-}=g^{-}_{i}dx-f^{-}_{i}dy,i=1,2$.

###### Proof.

Proof is similar to the proof of Lemma 3.2. ∎

###### Proof of Theorem 3.1.

Comparing coefficients of $\varepsilon$ and $\varepsilon^{2}$ in the
expression for $L_{4}$ obtained in Lemma (3.2) and from expressions (3.20),
(3.21), we have

$\displaystyle\sigma=\dfrac{1}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\omega_{1}^{+},\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}$ (3.30)
$\displaystyle\dfrac{1}{2}H_{x}^{+}(P_{1})\tau=$
$\displaystyle\int_{\Gamma^{+}_{r}}\omega_{2}^{+}+K^{+}(P(r))\int_{\Gamma^{+}_{r}}\frac{\omega_{1}^{+}}{H_{y}^{+}}-\int_{\Gamma_{r}^{+}}\frac{f_{1}^{+}\omega_{1}^{+}}{H_{y}^{+}}-\dfrac{1}{2}H_{xx}^{+}(P_{1})\sigma^{2}.$
(3.31)

Similarly, from Lemma (3.3) and expressions (3.18), (3.19) we get

$\displaystyle H_{x}^{-}(P)\rho=$ $\displaystyle
H_{x}^{-}(P_{1})\sigma+\int_{\Gamma_{r}^{-}}\omega_{1}^{-}=\dfrac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\omega_{1}^{+}+\int_{\Gamma_{r}^{-}}\omega_{1}^{-},\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}$ (3.32)
$\displaystyle\frac{1}{2}H_{x}^{-}(P)\eta=$
$\displaystyle\int_{\Gamma^{-}_{r}}\omega_{2}^{-}+K^{-}(P(r))\int_{\Gamma^{-}_{r}}\frac{\omega_{1}^{-}}{H_{y}^{-}}-\int_{\Gamma_{r}^{-}}\frac{f_{1}^{-}\omega_{1}^{-}}{H_{y}^{-}}-\frac{1}{2}H_{xx}^{-}(P)\rho^{2}+\frac{1}{2}H_{xx}^{-}(P_{1})\sigma^{2}+\frac{1}{2}H_{x}^{-}(P_{1})\tau.$
(3.33)

From (3.12) and (3.32) we get the first order Melnikov function.

Now if the first order Melnikov function is identically zero, then from
equation (3.12) we have $\rho\equiv 0$. Hence, from (3.12) and (3.33), the
second order Melnikov function is

$\displaystyle
M_{2}(r)={H_{x}}^{+}(P)\eta=\dfrac{{H_{x}}^{+}(P)}{{H_{x}}^{-}(P)}\left(\int_{\Gamma^{-}_{r}}{\omega_{2}}^{-}-\int_{\Gamma^{-}_{r}}\frac{{f_{1}}^{-}{\omega_{1}}^{-}}{{H_{y}}^{-}}+K^{-}(P(r))\int_{\Gamma_{r}^{-}}\frac{{\omega_{1}}^{-}}{{H_{y}}^{-}}+\frac{1}{2}{H_{xx}}^{-}(P_{1})\sigma^{2}+\frac{1}{2}{H_{x}}^{-}(P_{1})\tau\right).$

and hence by substituting $\tau$ from (3.31) we get the required expression
for $M_{2}$. ∎

If the Hamiltonian system (3.6) is extended smoothly on the boundary $y=0$,
then it becomes a smooth Hamiltonian system. In this case the first order and
second order Melnikov function are simple line integrals of one forms.
Further, if perturbation of this system is also smooth, then the first order
and second order Melnikov functions obtained from Theorem 3.1 are well known
integrals of one forms as in the following corollary.

######  Corollary 1.

If in the system (3.9) $\displaystyle\lim_{y\rightarrow
0^{+}}{H_{x}}^{+}(x,y)=\lim_{y\rightarrow 0^{-}}{H_{x}}^{-}(x,y)$ and
$\displaystyle\lim_{y\rightarrow 0^{+}}{H_{y}}^{+}(x,y)=\lim_{y\rightarrow
0^{-}}{H_{y}}^{-}(x,y)$ for all $x\in\mathbb{R}$, then the first order and the
second order Melnikov functions are given by

$\displaystyle M_{1}(r)$
$\displaystyle=\int_{\Gamma_{r}^{+}}\omega_{1}^{+}+\int_{\Gamma_{r}^{-}}\omega_{1}^{-}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}$ (3.34) $\displaystyle
M_{2}(r)$
$\displaystyle=\left(\int_{\Gamma_{r}^{-}}\omega_{2}^{-}+\int_{\Gamma_{r}^{+}}\omega_{2}^{+}\right)-\left(\int_{\Gamma^{-}_{r}}\frac{f_{1}^{-}\omega_{1}^{-}}{H_{y}^{-}}+\int_{\Gamma_{r}^{+}}\frac{f_{1}^{+}\omega_{1}^{+}}{H_{y}^{+}}\right)$
$\displaystyle+\left(K^{-}(P(r))\int_{\Gamma_{r}^{-}}\frac{\omega_{1}^{-}}{H_{y}^{-}}+K^{+}(P(r))\int_{\Gamma_{r}^{+}}\frac{\omega_{1}^{+}}{H_{y}^{+}}\right),$
(3.35)

respectively.

Further, if $\displaystyle\lim_{y\rightarrow
0^{+}}f_{i}^{+}(x,y)=\lim_{y\rightarrow 0^{-}}f_{i}^{-}(x,y)$ and
$\displaystyle\lim_{y\rightarrow 0^{+}}g_{i}^{+}(x,y)=\lim_{y\rightarrow
0^{-}}g_{i}^{-}(x,y)$ for $i=1,2$ and for all $x\in\mathbb{R}$, then the first
order and second order Melnikov functions are given by

$\displaystyle M_{1}(r)=\oint_{\Gamma_{r}}\omega_{1},\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
M_{2}(r)=\oint_{\Gamma_{r}}\omega_{2}-\oint_{\Gamma_{r}}\frac{f_{1}\omega_{1}}{H_{y}}+K(P(r))\oint_{\Gamma_{r}}\frac{\omega_{1}}{H_{y}},$
(3.36)

respectively,
where $\displaystyle
H_{x}(x,y)=\begin{cases}H_{x}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\ y>0\\\
H_{x}^{-}(x,y)&\mbox{if}\leavevmode\nobreak\ y<0\\\
\displaystyle\lim_{y\rightarrow
0^{+}}H_{x}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\
y=0\end{cases},\leavevmode\nobreak\
H_{y}(x,y)=\begin{cases}H_{y}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\ y>0\\\
H_{y}^{-}(x,y)&\mbox{if}\leavevmode\nobreak\ y<0\\\
\displaystyle\lim_{y\rightarrow
0^{+}}H_{y}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\ y=0\end{cases},\\\
f_{i}(x,y)=\begin{cases}f_{i}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\ y>0\\\
f_{i}^{-}(x,y)&\mbox{if}\leavevmode\nobreak\ y<0\\\
\displaystyle\lim_{y\rightarrow
0^{+}}f_{i}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\
y=0\end{cases},\leavevmode\nobreak\
g_{i}(x,y)=\begin{cases}g_{i}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\ y>0\\\
g_{i}^{-}(x,y)&\mbox{if}\leavevmode\nobreak\ y<0\\\
\displaystyle\lim_{y\rightarrow
0^{+}}g_{i}^{+}(x,y)&\mbox{if}\leavevmode\nobreak\ y=0\end{cases},\\\
\omega_{i}=g_{i}dx-f_{i}dy,\leavevmode\nobreak\ \leavevmode\nobreak\
K=\frac{H_{x}f_{1}+H_{y}g_{1}}{H_{x}}$ for $i=1,2$ and
$\Gamma_{r}=\Gamma_{r}^{+}\cup\Gamma_{r}^{-}$, a closed trajectory of the
unperturbed system.

###### Proof.

Proof follows from the equation (3.10) and (3.1). ∎

##  4 Piecewise Hamiltonian System with Boundary Perturbation

Now consider a piecewise Hamiltonian system with boundary perturbation,

$\displaystyle\dot{X}=$
$\displaystyle\begin{cases}(H_{y}^{+},-H_{x}^{+}),\leavevmode\nobreak\
y>\varepsilon f(x)\\\ (H_{y}^{-},-H_{x}^{-}),\leavevmode\nobreak\
y<\varepsilon f(x)\end{cases},$ (4.1)

where $H^{+},H^{-}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ are $C^{2}$ functions
and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a $C^{1}$ function.

Here $\Sigma=\\{(x,y)\in\mathbb{R}^{2}:y=\varepsilon f(x)\\}$ is a switching
manifold and $\Sigma^{\pm}=\\{(x,y)\in\mathbb{R}^{2}:\pm(y-\varepsilon
f(x))>0\\}$ are two zones saparated by $\Sigma$.

######  Remark 4.1.

According to the Filippov convension [13], the switching manifold $\Sigma$ is
divided into the following regions:

Crossing region $\Sigma_{c}=\\{(x,\varepsilon f(x)):H_{x}^{\pm}+\varepsilon
H_{y}^{\pm}f^{\prime}(x)>0\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
H_{x}^{\pm}+\varepsilon H_{y}^{\pm}f^{\prime}(x)<0\\}$,

Sliding region $\Sigma_{s}=\\{(x,\varepsilon f(x)):H_{x}^{+}+\varepsilon
H_{y}^{+}f^{\prime}(x)<0,H_{x}^{-}+\varepsilon H_{y}^{-}f^{\prime}(x)>0\\}$,
and

Escaping region $\Sigma_{e}=\\{(x,\varepsilon f(x)):H_{x}^{+}+\varepsilon
H_{y}^{+}f^{\prime}(x)>0,H_{x}^{-}+\varepsilon H_{y}^{-}f^{\prime}(x)<0\\}$.

Discontinuity induced bifurcations are studied according to these regions.

Using the analytic invertible change of variables $u=x,v=y-\varepsilon f(x)$
and renaming the variables $u$ by $x$ and $v$ by $y$, the system (4.1) becomes

$\displaystyle\dot{X}=$
$\displaystyle\begin{cases}(X^{+},Y^{+}),\leavevmode\nobreak\ y>0\\\
(X^{-},Y^{-}),\leavevmode\nobreak\ y<0\end{cases},$ (4.2)

where

$\displaystyle X^{+}=$ $\displaystyle H_{y}^{+}+\varepsilon
f(x)H^{+}_{yy}+\varepsilon^{2}\dfrac{1}{2}f(x)^{2}H^{+}_{yyy}+o(\varepsilon^{3}),$
$\displaystyle Y^{+}=$
$\displaystyle-H^{+}_{x}-\varepsilon(f^{\prime}(x)H^{+}_{y}+f(x)H^{+}_{xy})-\varepsilon^{2}f(x)\left(\dfrac{1}{2}f(x)H^{+}_{xyy}+f^{\prime}(x)H^{+}_{yy}\right)+o(\varepsilon^{3}),$
$\displaystyle X^{-}=$ $\displaystyle H_{y}^{-}+\varepsilon
f(x)H^{-}_{yy}+\varepsilon^{2}\dfrac{1}{2}f(x)^{2}H^{-}_{yyy}+o(\varepsilon^{3}),$
$\displaystyle Y^{-}=$
$\displaystyle-H^{-}_{x}-\varepsilon(f^{\prime}(x)H^{-}_{y}+f(x)H^{-}_{xy})-\varepsilon^{2}f(x)\left(\dfrac{1}{2}f(x)H^{-}_{xyy}+f^{\prime}(x)H^{-}_{yy}\right)+o(\varepsilon^{3}).$

######  Remark 4.2.

Note that if $\Phi(x,y)=(x,y-\varepsilon f(x))$, then $\Phi$ is
diffeormorphism. Therefore, systems (4.1) and (4.2) are topologically
equivalent and their flows are $C^{1}$ conjugate.

Melnikov function for the system (4.2) are given by the following theorem.

######  Theorem 4.2.

If the system (4.2) at $\varepsilon=0$ has a period annulus around the origin
and $H_{y}^{\pm}(P)=H_{y}^{\pm}(P_{1})$, then the first order Melnikov
function for (4.2) is

$\displaystyle\lambda
M_{1}(r)={H_{x}^{-}(P_{1})}\left(\dfrac{H_{y}^{+}(P_{1})}{H_{x}^{+}(P_{1})}-\dfrac{H_{y}^{-}(P_{1})}{H_{x}^{-}(P_{1})}\right)(f(p)-f(p_{1})).$
(4.3)

and the second order Melnikov function is given by

$\displaystyle\lambda M_{2}(r)=$
$\displaystyle\dfrac{H_{x}^{-}(P_{1})}{2}\left(\dfrac{H_{yy}^{+}(P_{1})}{H_{x}^{+}(P_{1})}-\dfrac{H_{yy}^{-}(P_{1})}{H_{x}^{-}(P_{1})}\right)((f(p(r)))^{2}-(f(p_{1}(r)))^{2})$
$\displaystyle+\dfrac{H_{x}^{-}(P_{1})K^{+}(P)-H_{x}^{+}(P_{1})K^{-}(P)}{H_{x}^{+}(P_{1})}\left(f(p(r))-f(p_{1}(r))\right)$
$\displaystyle-\left(K^{-}(P)\int_{\Gamma^{-}_{r}}\dfrac{f}{H_{y}^{-}}d(H_{y}^{-})+\dfrac{H_{x}^{-}(P_{1})K^{+}(P)}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}\dfrac{f}{H_{y}^{+}}d(H_{y}^{+})\right)$
$\displaystyle+\int_{\Gamma_{r}^{-}}H_{yy}^{-}\left[d\left(\frac{f^{2}}{2}\right)+\left(\frac{f^{2}}{{H_{y}}^{-}}\right)d(H_{y}^{-})\right]+\dfrac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}\int_{\Gamma_{r}^{+}}H_{yy}^{+}\left[d\left(\frac{f^{2}}{2}\right)+\left(\frac{f^{2}}{H_{y}^{+}}\right)d(H_{y}^{+})\right]$
$\displaystyle+\dfrac{1}{2}\left(H_{xx}^{-}(P_{1})-\dfrac{H_{x}^{-}(P_{1})}{H_{x}^{+}(P_{1})}H_{xx}^{+}(P_{1})\right)\sigma^{2},$
(4.4)

where $\lambda=\dfrac{H_{x}^{-}(P)}{H_{x}^{+}(P)}.$

###### Proof.

From (4.2), we have

$\displaystyle\int_{\Gamma^{+}_{r}}\omega_{1}^{+}=$
$\displaystyle\int_{\Gamma^{+}_{r}}(g_{1}^{+}dx-
f_{1}^{+}dy)=-\int_{\Gamma^{+}_{r}}d(fH_{y}^{+})=f(p)H_{y}^{+}(P)-f(p_{1})H_{y}^{+}(P_{1}),\leavevmode\nobreak\
\mbox{and}$ $\displaystyle\int_{\Gamma^{-}_{r}}\omega_{1}^{-}=$
$\displaystyle\int_{\Gamma^{-}(r)}(g_{1}^{-}dx-
f_{1}^{-}dy)=-\int_{\Gamma_{r}^{-}}d(fH_{y}^{-})=f(p_{1})H_{y}^{-}(P_{1})-f(p)H_{y}^{-}(P).$

Hence, by Theorem 3.1,

$\displaystyle M_{1}(r)=$
$\displaystyle\dfrac{H_{x}^{+}(P)}{H_{x}^{-}(P)}\left(f(p)\left(\dfrac{H_{x}^{-}(P_{1})H_{y}^{+}(P)}{H_{x}^{+}(P_{1})}-H_{y}^{-}(P)\right)-f(p_{1})\left(\dfrac{H_{x}^{-}(P_{1})H_{y}^{+}(P_{1})}{H_{x}^{+}(P_{1})}-H_{y}^{-}(P_{1})\right)\
\right).$ (4.5)

Now if $H_{y}^{+}(P)=H_{y}^{+}(P_{1})$ and $H_{y}^{-}(P)=H_{y}^{-}(P_{1})$,
then we get (4.3).

Again, from (4.2) we have
$\omega_{1}^{\pm}=-d(f(x)H_{y}^{\pm}),\omega_{2}^{\pm}=-d\left(\dfrac{f^{2}(x)H_{yy}^{\pm}}{2}\right).$
Hence, the formula for $M_{2}$ follows from (3.1).∎

######  Remark 4.3.

If the system (4.2) at $\varepsilon=0$ is smooth, then
$\dfrac{H_{y}^{+}(P_{1})}{H_{x}^{+}(P_{1})}-\dfrac{H_{y}^{-}(P_{1})}{H_{x}^{-}(P_{1})}=0$.
Thus, $M_{1}(r)=0$ if and only if $f(p(r))=f(p_{1}(r))$ or the system is
smooth. If $f(p(r))=f(p_{1}(r))$, the periodic orbit passing through $P(r)$ of
(4.2) at $\varepsilon=0$ is also a periodic orbit of (4.2). Hence, the number
of periodic orbits persists under perturbation equals to the number of roots
of $f(p(r))-f(p_{1}(r))$.

If the system (4.1) at $\varepsilon=0$ is smoothly extended on $y=0$, then its
period annulus persists under perturbation of switching boundary. In the
following corollary we obtain the first order and second order Melnikov
functions for such system.

######  Corollary 2.

If the system (4.1) at $\varepsilon=0$ is smooth then its period annulus
persists under smooth perturbation of switching manifold $y=0$.

###### Proof.

From (4.3) we have $\displaystyle
M_{1}(r)=\oint_{\Gamma_{r}}\omega_{1}=\oint_{\Gamma_{r}}-d(f(x)H_{y})=0.$

Now from (4.2) we have

$\displaystyle\lambda
M_{2}(r)=-K(P)\oint_{\Gamma_{r}}\frac{f}{H_{y}}dH_{y}+\oint_{\Gamma_{r}}H_{yy}\frac{f^{2}}{H_{y}}dH_{y}+\oint_{\Gamma_{r}}H_{yy}d\left(\frac{f^{2}}{2}\right),$
(4.6)

where $f,K,H_{y},H_{yy}$ and $\Gamma_{r}$ are as defined in Corollary 1.

Along $\Gamma_{r}$ we have

$\displaystyle\dfrac{f(x)}{H_{y}}dH_{y}=$
$\displaystyle\dfrac{f(x)}{H_{y}}(H_{yx}dx+H_{yy}dy)=\dfrac{f(x)}{H_{y}}(-H_{yx}\dfrac{H_{y}}{H_{x}}+H_{yy})dy$
$\displaystyle=$ $\displaystyle
f(x)\left(\dfrac{-H_{xy}}{H_{x}}+\dfrac{H_{yy}}{H_{y}}\right)dy=\dfrac{\partial}{\partial
y}\left(f(x)\log\left(\frac{H_{y}}{H_{x}}\right)\right)dy,$ (4.7)

so that $\displaystyle\oint_{\Gamma_{r}}\frac{f}{H_{y}}dH_{y}=0.$ Also,

$\displaystyle\oint
H_{yy}\left(f^{2}\frac{dH_{y}}{H_{y}}+d\left(\frac{f^{2}}{2}\right)\right)=\oint
H_{yy}\left(f^{2}\frac{\partial}{\partial
y}\left(\log\frac{H_{y}}{H_{x}}\right)-ff^{\prime}\frac{H_{y}}{H_{x}}\right)dy=\oint
H_{yy}\left(f^{2}v_{y}-ff^{\prime}u\right)dy,$

where $u=\frac{H_{y}}{H_{x}}\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\ v=\log(u)$. Integrating by
parts twice, we get

$\displaystyle\oint
H_{yy}\left(f^{2}\frac{dH_{y}}{H_{y}}+d\left(\frac{f^{2}}{2}\right)\right)=\oint
H\left(f^{2}v_{yyy}-ff^{\prime}u_{y}y\right)dy=r\oint\left((f^{2}v)_{yyy}-(ff^{\prime}u)_{yy}\right)dy=0.$
(4.8)

Hence from equation (4.6), (4) and (4.8), we get $M_{2}\equiv 0$. Thus, we
conclude that, no limit cycle is bifurcated from the period annulus of (4.1).∎

######  Remark 4.4.

In the above proof we consider an extension of the natural logarithmic
function on $\mathbb{R}\cup\\{+\infty,-\infty\\}$ as,
$\log(x)=\begin{cases}\ln x,x>0\\\ \ln(-x),x<0\\\ -\infty,x=0\\\
\infty,x=\pm\infty\end{cases}.$ This function is an antiderivative of the
functions $f_{1}(x)=\begin{cases}\dfrac{1}{x},x>0\\\ \infty,x=0\end{cases}$ on
$[0,\infty]$ and $f_{2}(x)=\begin{cases}\dfrac{1}{x},x<0\\\
-\infty,x=0\end{cases}$ on $[-\infty,0]$.

##  5 Applications

There are various types of planar piecewise smooth systems according to the
types of singularities in zones separated by the switching line with center at
the origin viz. center-center, saddle-center, center-focus. Here we discuss
the limit cycle bifurcation from period annulus due to perturbation of the
switching manifold of center- center and saddle-center type using the first
and second order Melnikov functions.

###  5.1 Boundary perturbation of center-center type system

Consider the piecewise Hamiltonian system

$\displaystyle\dot{X}=\begin{cases}(-1,2x)&\mbox{if}\leavevmode\nobreak\
y>0\\\ (1,2x)&\mbox{if}\leavevmode\nobreak\ y<0\end{cases}.$ (5.1)

System (5.1) has center at the origin (Fig.2(a)). Hamiltonian of the system
$\dot{X}=(-1,2x)$ is $H^{+}(x,y)=-y-x^{2}$. Trajectories of this system at
level $h=r^{2}$ are given by $y+x^{2}=-r^{2}$. Hamiltonian for
$\dot{X}=(1,2x)$ is $H^{-}(x,y)=y-x^{2}$ and its trajectories at the levels
$h=r^{2}$ are given by $y-x^{2}=r^{2}.$

Now consider the perturbed piecewise smooth system

$\displaystyle\dot{X}=\begin{cases}(-1+\varepsilon
f_{1}^{+}(x,y)+\varepsilon^{2}f_{2}^{+}(x,y),2x+\varepsilon
g_{1}^{+}(x,y)+\varepsilon^{2}g_{2}^{+}(x,y))&\mbox{if}\leavevmode\nobreak\
y>0\\\ (1+\varepsilon
f_{1}^{-}(x,y)+\varepsilon^{2}f_{2}^{-}(x,y),2x+\varepsilon
g_{1}^{-}(x,y)+\varepsilon^{2}g_{2}^{-}(x,y))&\mbox{if}\leavevmode\nobreak\
y<0,\end{cases}$ (5.2) where $\displaystyle f_{1}^{+}(x,y)=ax^{2}+bxy+cy^{2},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
f_{2}^{+}(x,y)=Ax^{2}+Bxy+Cy^{2},$ $\displaystyle
g_{1}^{+}(x,y)=dx^{2}+exy+fy^{2},$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ g_{2}^{-}(x,y)=Dx^{2}+Exy+Fy^{2},$
$\displaystyle f_{1}^{-}(x,y)=px^{2}+qxy+sy^{2},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
f_{2}^{-}(x,y)=Px^{2}+Qxy+Sy^{2},$ $\displaystyle
g_{1}^{-}(x,y)=lx^{2}+mxy+ny^{2},$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ g_{2}^{-}(x,y)=Lx^{2}+Mxy+Ny^{2}.$

and $a,b,c,d,e,f,p,q,s,l,m,n,A,B,C,D,E,F,P,Q,S,L,M$ and $N$ are real
constants.

From (3.10) we get

$\displaystyle
M_{1}(r)=r^{3}\left(\dfrac{8}{15}\left(4b-7f+7n-4q\right){r}^{2}+\dfrac{2}{3}\left(l-d\right)\right).$
(5.3)

Note that, $M_{1}(r)\equiv 0$ if

$\displaystyle 4b-7f+7n-4q=0,\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ d-l=0.$ (5.4)

Also, from (3.10) the second order Melnikov function is

$\displaystyle M_{2}(r)=$
$\displaystyle\left(-{\dfrac{2656\,sn}{315}}+{\dfrac{3712\,qs}{315}}+{\dfrac{3712\,bc}{315}}-{\frac{2656\,cf}{315}}\right){r}^{9}$
$\displaystyle+\left(\dfrac{8}{15}(p\left(7\,n-4\,q\right)-a\left(4\,b-7\,f\right))-\dfrac{184}{105}(ls+qm+np+af+be+cd)+\dfrac{96}{35}(pq+ab)\right){r}^{7}$
$\displaystyle-\dfrac{4}{15}\left(l\left(7\,n-4\,q\right)+d\left(4\,b-7\,f\right)\right){r}^{6}+\dfrac{4}{15}\left(14(N-F)+8(B-Q)+lp+ad\right){r}^{5}$
$\displaystyle+\dfrac{1}{3}\left(d^{2}-l^{2}\right){r}^{4}+\dfrac{2}{3}\left(L-D\right){r}^{3}.$
(5.5)

In the view of the conditions (5.4), the expression (5.1) becomes
$M_{2}(r)=r^{3}M_{2}^{\prime}(h)$, where $h=r^{2}$ and

$\displaystyle M_{2}^{\prime}(h)=$
$\displaystyle\dfrac{32}{315}\left(-29(7f-4b)(s+c)+120(sn+cf)\right)h^{3}$
$\displaystyle+\left(\dfrac{8}{15}(7f-4b)(p+a)-\dfrac{184}{105}(d(s+c)+qm+np+af+be)+\dfrac{96}{35}(pq+ab)\right)h^{2}$
$\displaystyle+\dfrac{4}{15}\left(14(N-F)+8(B-Q)+d(p+a)\right)h+\dfrac{2}{3}\left(L-D\right).$
(5.6)

Thus under the conditions (5.4), the system (5.2) can have at most three limit
cycles.

In particular, if $a=p,b=q,c=s,m=-e,f=n=2b,L=-D,F=N,B=Q$, then (5.1) becomes

$\displaystyle M^{\prime}_{2}(h)=$
$\displaystyle-\dfrac{640}{63}bch^{3}+\left(\dfrac{80}{21}ab+\dfrac{16}{3}b^{2}-\dfrac{368}{105}cd\right)h^{2}+\dfrac{8}{15}adh-\dfrac{4}{3}D.$
(5.7)

We can choose constants $a,b,c,d$ and $D$ such that (5.7) will have three
distinct positive roots. In particular, if $a=1,bc=1,d=-\dfrac{8800}{42}$ and
$D=-\dfrac{960}{21}$, then (5.7) becomes

$\displaystyle
M_{2}^{\prime}(h)=-\dfrac{640}{63}(h^{3}-6.000000005h^{2}+11h-6).$ (5.8)

Polynomial (5.8) has three positive zeros;
$h=1.000000003,1.999999980,3.000000022$. Consequently, the corresponding
system will have three limit cycles (Fig.2(c)).

Further, in (5.1), if
$4b=7f,e=m=c=0,a=p=s=n=1,q=-b,d=1/10,f=98.9,L=D,14(N-F)+8(B-Q)=0$, then

$\displaystyle
M^{\prime}_{2}(h)=\dfrac{256}{21}h^{3}-\dfrac{3680}{21}h^{2}+\dfrac{4}{75}h.$
(5.9)

Note that $(\ref{p1507})$ has two real positive zeros,
$h=\dfrac{115}{16}+\dfrac{3}{80}\sqrt{36733},\dfrac{115}{16}-\dfrac{3}{80}\sqrt{36733}$.
Therefore the corresponding system will have two limit cycles (Fig.2(b)).

(a) System (5.1)

(b) Two limit cycles

(c) Three limit cycles

Figure 2: Flow of the system (5.2)

In [10], authers characterize all planar piecewise smooth differential systems
having $(k,l)$ center at the origin. Here we mention the result;

######  Proposition 5.1.

[10] Let $k,l,r$ be positive integers and $\max\\{k,l\\}\leq
r\in\mathbb{N}\cup\\{\infty,\omega\\}$. Suppose that the system

$\displaystyle\dot{X}=\begin{cases}(F^{+}(x,y),G^{+}(x,y))&\mbox{if}\leavevmode\nobreak\
y>0\\\ (F^{-}(x,y),G^{-}(x,y))&\mbox{if}\leavevmode\nobreak\ y<0\end{cases}$
(5.10)

is piecewise smooth with $F^{\pm},G^{\pm}\in C^{r}$ and having
$(k,l)$-$\Sigma$-center at the origin, where $\Sigma$ is the $x$-axis. Then
there exists a $C^{r}$ diffeomorphism $h$ from period annulus of (5.10) to a
period annulus of (5.1), which maps $x$-axis to the $x$-axis.

We note that the system (5.1) is piecewise smooth Hamiltonian system with
Hamiltonian $H^{+}(x,y)=-y-x^{2},y>0$ and $H^{-}(x,y)=y-x^{2},y<0$. The
following proposition gives the information about the limit cycles bifurcated
from period annulus of this system due to perturbation of switching manifold.

######  Proposition 5.2.

The number of limit cycles for the system
$\dot{X}=\begin{cases}(-1,2x)&\mbox{if}\leavevmode\nobreak\ y>\varepsilon
f(x)\\\ (1,2x)&\mbox{if}\leavevmode\nobreak\ y<\varepsilon f(x)\end{cases}$ is
equal to the number of isolated positive zeros of $f_{o}(x)$, where
$f_{o}(x)=\dfrac{f(x)-f(-x)}{2}$.

###### Proof.

From (4.3) and (4.2), we get

$\displaystyle M_{1}(r)=$ $\displaystyle
2[f(-{r})-f({r})]=4f_{o}({r}),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{and}$ $\displaystyle M_{2}(r)=$
$\displaystyle\dfrac{f^{\prime}({r})}{{r}}M_{1}(r).$

Therefore the number of limit cycles bifurcated from period annulus of the
unperturbed system is same as the number of positive roots of $y=f_{o}(x)$. In
particular, if $f$ is an even function, then no limit cycles bifurcated.∎

######  Remark 5.1.

From Proposition 3.1, it is clear that the limit cycle of system (5.1) through
the point $({r},0)$ is stable if and only if
$\dfrac{dM_{1}}{dr}=\dfrac{-2}{{r}}{f_{e}}^{\prime}({r})<0$, where
$f_{e}(x)=\dfrac{f(x)+f(-x)}{2}.$

Since the system (5.1) is the normal form of planar piecewise smooth systems
of the center-center type, Proposition (5.2) also holds for the systems
(5.10).

###  5.2 Boundary perturbation of saddle-center type system

In [7, 8], authors studied the number of limit cyclce alongwith stability and
hyperbolicity of limit cycles of the system

$\displaystyle\dot{X}=\begin{cases}(y-a,x)&\mbox{if}\leavevmode\nobreak\
y>\varepsilon f(x)\\\ (-y,x)&\mbox{if}\leavevmode\nobreak\ y<\varepsilon
f(x)\end{cases}.$ (5.11)

Origin is singularity of this system of saddle-center type and $f(x)=\sin x$
or $f(x)=x(x^{2}-x_{1}^{2})(x^{2}-x_{2}^{2})...(x^{2}-x_{m}^{2})$. Also, note
that this system is piecewise linear Hamiltonian system with two zones. We
study the number of limit cycles bifurcated from the period annulus and
stability of above systems if we change the switching manifold $y=0$ to
$y=\varepsilon f(x)$, under the assumption that $f$ is sufficiently smooth.
Using dialtion $x=au,y=av$ and renaming the variables $u$ and $v$ by $x$ and
$y$ respectively, system (5.10) becomes

$\displaystyle\dot{X}=$
$\displaystyle\begin{cases}(y-1,x),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ y>\varepsilon f(ax)\\\
(-y,x),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ y<\varepsilon f(ax)\end{cases}.$ (5.12)

At $\varepsilon=0$, system (5.11) has a period annulus
$\displaystyle\mathcal{A}=\bigcup_{r\in[0,1]}(\Gamma_{r}^{+}\cup\Gamma_{r}^{-})$,
where

$\displaystyle\Gamma_{r}^{+}:H^{+}(x,y)=\dfrac{(y-1)^{2}}{2}-\dfrac{x^{2}}{2}=\frac{r}{2},y\geq
0\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\
\Gamma_{r}^{-}:H^{-}(x,y)=\dfrac{x^{2}}{2}+\dfrac{y^{2}}{2}=\frac{s}{2},y\leq
0.$

Now for any point $P(r)=(p(r),0),r\in[0,1]$ we have

$\displaystyle
H^{+}(P)=\dfrac{r}{2}=H^{-}(P)=\dfrac{s}{2}\Rightarrow\dfrac{(0-1)^{2}}{2}-\dfrac{(p(r))^{2}}{2}=\dfrac{r}{2}=-\dfrac{(p(r))^{2}}{2}-\dfrac{0^{2}}{2}=\dfrac{s}{2}\Rightarrow
p(r)=\sqrt{1-r}=\sqrt{s}.$

Since the trajectories are symmetric about the $y$-axis, we have
$p_{1}(r)=-p(r)=-\sqrt{1-r}$. Melnikov functions for (5.11) are given by the
following proposition.

(a) $f(x)=0.01\sin x$

(b) $f(x)=x(x^{2}-1)(x^{2}-4)$

Figure 3: Saddle-center system with perturbation boundary $y=\varepsilon f(x)$

######  Proposition 5.3.

Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ is $C^{1}$ smooth function.
Then we have the following:

  1. 1.

The first order Melnikov function for the system (5.11) is given by

$M_{1}(r)=2f_{o}(a\sqrt{1-r}),$

where $f_{o}(x)=\dfrac{f(x)-f(-x)}{2}$ for $0<r<1$. Further, if
$M_{1}(r)\equiv 0$, then $M_{2}(r)\equiv 0$.

  2. 2.

The number of limit cycles bifurcated from period annulus around the origin
inside the homoclinic orbit containing the saddle point $(0,a)$, is the number
of isolated positive roots of $f_{o}(x)$. In particular, if $f$ is an even
function, then no limit cycles bifurcated from the period annulus.

###### Proof.

From (4.3), we have

$\displaystyle
M_{1}(r)=\dfrac{(-p(r))(-p_{1}(r))}{(-p(r))}\left(\dfrac{-1}{(-p_{1}(r))}-\dfrac{0}{(-p_{1}(r))}\right)(f(ap(r))-f(ap_{1}(r)))=f(ap(r))-f(-ap(r)).$

Therefore $M_{1}(r)=2f_{0}(ap(r))=2f_{0}(a\sqrt{1-r})$ for all $r\in[0,1]$.
Here, $H_{xx}^{\pm}=H_{yy}^{\pm}=H_{xy}^{\pm}=0$. Therefore from (4.2), we get
$M_{2}(r)=\dfrac{-f^{\prime}(a\sqrt{1-r})}{\sqrt{1-r}}M_{1}(r),r\in[0,1).$
Hence the proof of (1).

Proof of (2) follows from the expression for $M_{1}$. ∎

######  Remark 5.2.

From Proposition 3.1, it is clear that the limit cycle of (5.12) through the
point $(\sqrt{1-r},0)$ is stable if and only if
$\dfrac{dM_{1}}{dr}=\dfrac{-2a}{\sqrt{1-r}}{f_{e}}^{\prime}(\sqrt{1-r})<0$,
where $f_{e}(x)=\dfrac{f(x)+f(-x)}{2}.$

Similarly we can characterize all planar piecewise smooth differential systems
having saddle-center type as stated in the following proposition.

######  Proposition 5.4.

If the Filippov system

$\displaystyle\dot{X}=\begin{cases}(F^{+},G^{+}),\leavevmode\nobreak\ y>0\\\
(F^{-},G^{-}),\leavevmode\nobreak\ y<0\end{cases}$ (5.13)

has a period annulus around the origin inside the homoclinic orbit containing
the saddle point $(0,a),a>0$, then there is a homeomorphism which maps the
period annulus of (5.11) to the period annulus of (5.13) and maps switching
manifold to switching manifold.

###### Proof.

Let $U^{+}$ be an open region in the upper half plane lying inside the
homoclinic connection containing the saddle point $(0,a)$ of

$\displaystyle\dot{X}=(F^{+},G^{+}).$ (5.14)

Let $(b,0)$ and $(c,0)$ be the points of intersection of the homoclinic orbit
of (5.14) with the $x$-axis and $b<0<c$. We may assume that the periodic
orbits of (5.14) are convex (see [10]). Hence we use the polar co-ordinates
$(r,\theta)$ to transform the system (5.14) into

$\displaystyle\frac{dr}{d\theta}=P^{+}(r,\theta).$ (5.15)

Let $0<r_{0}<c$ and $0<\theta<\pi$. Consider the initial value problem
$\dfrac{dr}{d\theta}=P^{+}(r,\theta),r(0)=r_{0}$. Let $\xi=\xi(\theta,r_{0})$
be its solution. Now define the function
$\Phi^{+}:[0,\pi]\times[0,c]\rightarrow U^{+}$ by

$\displaystyle\Phi^{+}(\theta,r)=(\xi(\theta,r)\cos\theta,\xi(\theta,r)\sin\theta).$
(5.16)

Then $\Phi^{+}$ is a diffeomorphism and maps each horizontal line segment
$r=r_{0}$ to the trajectory $\xi=\xi(\theta,r_{0})$ of (5.14).

Similarly, if $V^{+}$ denote the region in the upper half plane occupied by
the periodic orbits of

$\displaystyle\dot{X}=(y-1,x),$ (5.17)

then we have the diffeomorphism
$\Psi^{+}(\theta,r):[0,\pi]\times[0,1]\rightarrow V^{+}$. Let
$\chi^{+}:[0,\pi]\times[0,c]\rightarrow[0,\pi]\times[0,1]$ be the map given by
$\chi^{+}(x,y)=(x,y/c)$. Then $\chi^{+}$ is also a diffeomorphism.

The composition
$\mathcal{H}^{+}:=\Psi^{+}\circ\chi^{+}\circ(\Phi^{+})^{-1}:U^{+}\rightarrow
V^{+}$ is a diffeomorphism.

Next, let $U^{-}$ and $V^{-}$ denote the open regions in lower half plane
consisting of orbits of the sysetms

$\displaystyle\dot{X}=(F^{-},G^{-})$ (5.18)

and

$\displaystyle\dot{X}=(-y,x),$ (5.19)

respectively. Then we can construct a diffeomorphism
$\mathcal{H}^{-}:U^{-}\rightarrow V^{-}$ which maps orbits of (5.18) to that
of (5.19).

Since the system (5.13) is Filippov, every point on the switching manifold
$y=0$ is a singularity of order one. Hence, $\displaystyle\lim_{y\rightarrow
0^{+}}\mathcal{H}^{+}(x,y)=\lim_{y\rightarrow 0^{-}}\mathcal{H}^{-}(x,y)$.

Now we define the map

$\mathcal{H}:U^{+}\cup U^{-}\cup\\{(x,0):b<x<0\leavevmode\nobreak\
\mbox{or}\leavevmode\nobreak\ 0<x<c\\}\rightarrow V^{+}\cup
V^{-}\cup\\{(x,0):0<|x|<1\\}$

by

$\displaystyle\mathcal{H}(x,y)=\begin{cases}\mathcal{H}^{+}(x,y),(x,y)\in
U^{+}\\\ \mathcal{H}^{-}(x,y),(x,y)\in U^{-}\\\
\displaystyle{\lim_{y\rightarrow
0^{+}}\mathcal{H}^{+}(x,y)}=\lim_{y\rightarrow
0^{-}}\mathcal{H}^{-}(x,y),y=0\\\ \end{cases}.$ (5.20)

Note that due to the Filippov convension, $\displaystyle\lim_{y\rightarrow
0^{+}}F^{+}(x,y)=\lim_{y\rightarrow 0^{-}}F^{-}(x,y)$ and
$\displaystyle\lim_{y\rightarrow 0^{+}}G^{+}(x,y)=\lim_{y\rightarrow
0^{-}}G^{-}(x,y)$, so that $\mathcal{H}$ is continuously differentiable on the
switching manifold $y=0$. Therefore $\mathcal{H}$ is a diffeomorphism. ∎

From Proposition (5.4) we conclude that the Proposition (5.3) holds for the
system (5.13).

##  6 Concluding Remark

In this article we found expressions for first order as well as second order
Melnikov functions for perturbed planar piecewise smooth Hamiltonian systems.
Using Melnikov functions we study limit cycle bifurcations of piecewise smooth
Hamiltonian systems due to the perturbation of the switching manifold.

This idea could be extended to study limit bifurcation of any piecewise smooth
planar differential system.

##

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