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Limit cycles for perturbing Hamiltonian system inside piecewise smooth polynomial differential system
Advances in Difference Equations volumeÂ 2016, ArticleÂ number:Â 228 (2016)
Abstract
In this paper, we obtain the firstorder Melnikov function of piecewise smooth polynomial perturbation of a Hamiltonian system. As application, we consider the number of limit cycles for perturbing the global center and truncated pendulum inside a piecewise smooth cubic polynomial differential system. Our results show that a piecewise smooth differential system can bifurcate more limit cycles than the smooth one.
1 Introduction and statement of the main results
The second part of Hilbertâ€™s 16th problem and its weak version are two open problems in the qualitative theory of real planar differential systems; see [1â€“4]. Since both problems are difficult, mathematicians try to study particular and simple cases. For example, Smaleâ€™s 13th problem restricts Hilbertâ€™s 16th problem to the LiÃ©nard systems [5].
In recent years, stimulated by nonsmooth phenomena in the real world, piecewise smooth differential systems have attracted a good deal of attention; see, for instance, [6, 7]. There are several papers [8, 9] considering the limit cycles for piecewise smooth LiÃ©nard systems. The authors of [10] studied the Hopf bifurcation for a piecewise smooth planar Hamiltonian system. In the paper [11], the authors considered the PoincarÃ© bifurcation for piecewise smooth Hamiltonian systems and obtained the firstorder Melnikov function. Then, they applied the firstorder Melnikov function to study the number of limit cycles that bifurcate from the period annulus of the center and obtained some new results. Later, by introducing multiple parameters, in [12], some new formulas are obtained for piecewise smooth systems. In the paper [13], the authors considered the number of limit cycles that bifurcate from the period annulus of the center for a piecewise smooth quadratic isochronous center. Generally speaking, a piecewise smooth system can bifurcate more limit cycles than a smooth one. In a recent paper [9], the authors studied the limit cycles for mpiecewise smooth LiÃ©nard system, They conjecture that the number of limit cycles decreases as m increases.
In this paper, we consider the following piecewise smooth perturbed Hamiltonian system:
where \(f^{\pm}(x,y)\) and \(g^{\pm}(x,y)\) are polynomial functions with respect to x and y.
Note that the unperturbed system \(\mbox{(1)}_{\varepsilon=0}\) is a Hamiltonian system with the first integral \(H(x,y)\), and there is a family of periodic orbits \(L_{h}=L^{+}_{h}\cup L^{}_{h}\) surrounding the center, where \(L^{\pm}_{h}:H(x,y)=h,x>0\) (\(x<0\)), \(h\in(h_{1},h_{2})\). Without loss of generality, we suppose that \(L_{h}\) has the clockwise orientation.
In this paper, we try to study the firstorder Melnikov function for piecewise smooth system (1). Applying the firstorder Melnikov function, we consider the number of limit cycles that bifurcate from the period annulus of the center for unperturbed system \(\mbox{(1)}_{\varepsilon=0}\) under piecewise smooth polynomial perturbation.
Theorem 1
The firstorder Melnikov function of system (1) can be expressed as
where \(L^{\pm}(h):H(x,y)=h,x>0\) (\(x<0\)), \(h\in(h_{1},h_{2})\), and \(L_{h}=L^{+}_{h}\cup L^{}_{h}\) has the clockwise orientation.
Moreover, if \(M_{1}(h^{*})=0\) and \(M_{1}^{\prime}(h^{*})\neq0\) for some \(h^{*}\in(h_{1},h_{2})\), then for \(\varepsilon>0\) sufficiently small, system (1) has a unique limit cycle near \(L_{h^{*}}\).
Remark 2
If \(f^{+}(x,y)\equiv f^{}(x,y)\) and \(g^{+}(x,y)\equiv g^{}(x,y)\), then system (1) is a smooth nearHamiltonian system, and the firstorder Melnikov function is well known; see [3]. Let \(\operatorname{deg} H(x,y)=m\). There are many papers considering the number of limit cycles that bifurcate from the period annulus of the center. See, for instance, \(m=2\) (e.g. [14]), \(m=3\) (e.g. [15]), \(m=4\) (e.g. [16]), \(m=5\) (e.g. [17, 18]), \(m=6\) (e.g. [19]).
As applications, we study the number of limit cycles for the piecewise smooth perturbation of the global center
and the truncated pendulum
where
Note that for \(\varepsilon=0\), systems (3) and (4) are Hamiltonian systems. These systems occur in oscillating chemical reactor models and have been studied in several papers; see, for instance, [14, 20].
Applying the firstorder Melnikov function given by (2), we consider the number of limit cycles that can bifurcate from the period annuls surrounding the origin of systems (3) and (4) under piecewise smooth cubic polynomial perturbation. Our result is the following theorem.
Theorem 3
There are at least five limit cycles that can bifurcate from the period annulus surrounding the origin of (3) (resp. (4)) by the firstorder Melnikov function.
Remark 4
If \(a_{i}=c_{i},b_{i}=d_{i},i=1,2,\ldots,9\), then systems (3) and (4) become smooth nearHamiltonian systems and have been studied in the papers [14, 20], where the authors obtained that there are at most two limit cycles that bifurcate from the period annulus of the origin for the smooth systems (3) and (4). Our result shows that planar piecewise smooth differential systems (3) and (4) can bifurcate three more limit cycles than the smooth one.
2 Proof of Theorem 1
We need the following lemma derived from [11] to prove TheoremÂ 1.
Lemma 5
Consider the perturbed piecewise smooth Hamiltonian system
where \(f^{\pm}(x,y)\) and \(g^{\pm}(x,y)\) are analytic functions with respect to \(x,y\). Assume that:

(I)
There exist an interval \(J=(\alpha,\beta)\) and two points \(A(h)=(0,\alpha(h))\) and \(A_{1}(h)=(0,\alpha_{1}(h))\), where \(\alpha(h)\neq\alpha_{1}(h)\), such that, for \(h\in J\), we have
$$ \begin{aligned} &H^{+} \bigl(A(h) \bigr)=H^{+} \bigl(A_{1}(h) \bigr)=h, \\ &H^{} \bigl(A(h) \bigr)=H^{} \bigl(A_{1}(h) \bigr). \end{aligned} $$(7) 
(II)
The system has an orbital arc \(L^{+}_{h}\) starting from \(A(h)\) and ending at \(A_{1}(h)\) defined by \(H^{+}(x,y)=h,x>0\). The system has an orbital arc \(L^{}_{h}\) starting from \(A_{1}(h)\) and ending at \(A(h)\) defined by \(H^{}(x,y)=h,x\leqslant0\).
Under assumptions (I) and (II), system \(\mbox{(6)}_{\varepsilon=0}\) has a family of periodic orbits \(L_{h} = L^{+}_{h}\cup L^{}_{h}\) for \(h\in J\). Each of the closed curves \(L_{h}\) is piecewise smooth in general. Further, without loss of generality, suppose that \(L_{h}\) has a clockwise orientation.
Then the firstorder Melnikov function of system (6) can be expressed as
Further, if \(M_{1}(h^{*})=0\) and \(M_{1}^{\prime}(h^{*})\neq0\) for some \(h^{*}\in J\), then for \(\varepsilon>0\) sufficiently small, system (6) has a unique limit cycle near \(L_{h^{*}}\).
Proof of TheoremÂ 1
Since the unperturbed system \(\mbox{(1)}_{\varepsilon=0}\) is a Hamiltonian system with first integral \(H(x,y)\), it is obvious that assumptions (I) and (II) are satisfied. Note that \(H^{+}(x,y)\equiv H^{}(x,y)=H(x,y)\) for system (1). Then
where
Applying Greenâ€™s formula two times to the integrals (11), we have
where \(L^{+}_{h}: H(x,y)=h,x>0\).
Similarly, we have
where \(L^{}_{h}: H(x,y)=h,x<0\).
Replacing (12) and (13) by (10), we obtain the firstorder Melnikov function (2). According to LemmaÂ 5, every simple zero of the firstorder Melnikov function (2) provides a limit cycle of system (1). This completes the proof.â€ƒâ–¡
3 Proof of Theorem 3
In order to estimate the number of the zeros of the firstorder Melnikov function, we need the following lemma.
Lemma 6
[21]
Consider \(p+1\) linearly independent analytical functions \(f_{i}: U \rightarrow\mathbb{R}\), \(i=0,1,\ldots,p\), where \(U \in\mathbb{R}\) is an interval. Suppose that there exists \(j \in \{0,1,\ldots,p\}\) such that \(f_{j}\) has constant sign. Then there exist \(p+1\) constants \(C_{i}, i=0,1,\ldots,p\), such that \(f(r)=\sum^{p}_{i=0}C_{i}f_{i}(r)\) has at least p simple zeros in U.
Proof of TheoremÂ 3
First, we consider system (3). From
we obtain that
and
According to TheoremÂ 1, the firstorder Melnikov function is
where \(f^{\pm}(x,y)\) and \(g^{\pm}(x,y)\) are given by (5).â€ƒâ–¡
For simplicity, we define the following functions:
Lemma 7
The following equalities hold: 5

(i)
\(I_{0,1}(h)=I_{1,1}(h)=I_{2,1}(h)=I_{0,3}(h)=0\);

(ii)
\(I_{0,0}(h),I_{1,0}(h),I_{2,0}(h),I_{0,2}(h),I_{3,0}(h),I_{1,2}(h)\) are linearly independent functions.
Proof
(i) Note that \(L^{+}_{h}:x=\sqrt{2hy^{2}\frac{y^{4}}{2}}\) and \(\alpha _{1}(h)=\alpha(h)\). By symmetry we have
The equalities \(I_{1,1}(h)=I_{2,1}(h)=I_{0,3}(h)=0\) can be proved similarly.
(ii) In order to prove that \(I_{0,0}(h),I_{1,0}(h),I_{2,0}(h),I_{0,2}(h),I_{3,0}(h),I_{1,2}(h)\) are linearly independent functions, for these functions, we make the following Taylor expansions in the variable h around \(h=0\):
Suppose that
In the following, we need to prove that \(k_{i}=0\), \(i=1,2,\ldots,6\).
From \(F^{\prime}(0)=k_{1}=0\) we have \(k_{1}=0\). Substituting \(k_{1}=0\) into (21), from \(\lim_{h\rightarrow0^{+}}\frac{F(h)}{h^{\frac {3}{2}}}=\sqrt{2}k_{2}=0\) we have \(k_{2}=0\). Substituting \(k_{1}=k_{2}=0\) into (21), from \(F^{\prime\prime}(0)=4k_{3}=0\) we have \(k_{3}=0\). Similarly, we can obtain that \(k_{4}=k_{5}=k_{6}=0\).â€ƒâ–¡
Substituting statement (i) of LemmaÂ 7 into (17), we have
By statement (ii) of LemmaÂ 7 the firstorder Melnikov function \(M_{1}(h)\) given by (21) is a linear combination of six linearly independent functions \(I_{0,0}(h),I_{1,0}(h),I_{2,0}(h)\), \(I_{0,2}(h),I_{3,0}(h), I_{1,2}(h)\) with arbitrary coefficients. From LemmaÂ 6 we obtain that \(M_{1}(h)\) has at least five simple zeros in \((0,+\infty)\). According to TheoremÂ 1, we can deduce that there are at least five limit cycles that can bifurcate from the period annulus surrounding the origin of (3) by the firstorder Melnikov function.
The proof of TheoremÂ 3 for system (4) is similar. The main difference is that
and
so we omit it.
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Acknowledgements
The authors would like to thank the editor and the anonymous reviewers for their useful and valuable suggestions. This work is supported by the National Natural Science Foundation of China (Nos. 11301105, 11401111) and Science Technology fund of Foundation Guizhou Province (No. [2015]2036), Guizhou Science Foundation for Distinguished Young Scholars (201313), and Science and Technical Innovative Personnel Foundation of Guizhou Education Department (2012085).
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Wu, K., Li, S. Limit cycles for perturbing Hamiltonian system inside piecewise smooth polynomial differential system. Adv Differ Equ 2016, 228 (2016). https://doi.org/10.1186/s1366201609575
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DOI: https://doi.org/10.1186/s1366201609575