# Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

Authors: Yanqin Xiong; Maoan Han
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.


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## Body

## 1. Introduction and Main Results

As well known, Liénard systems describe the dynamics of systems of one degree of freedom under existence of a linear restoring fore and a nonlinear dumping. In the first half of the last century models based on the Liénard system were important for the development of radio and vacuum tube technology. Nowadays the system is widely used to describe oscillatory processes arising in various studies of mathematical models of physical, biological, chemical, epidemiological, physiological, economical, and many other phenomena (see Glade et al. [1], Llibre [2] and references therein). Further, quadratic systems and some other systems can be transformed into Liénard systems by suitable changes, see for instance Han et al. [3], Cherkas [4], Gasull [5], Giné, and Llibre [6]. As we have seen, the main concern on Liénard systems is the center and focus problem and the number of limit cycles, see [7–17], and references therein. Here, we briefly list some known results related to our study in this paper. Consider a Liénard system of the form
(1.1)x˙=y-F(x),y˙=-g(x),
where we suppose that the functions F and g satisfy two assumptions below.(I)
There exists a positive numberε0 such that F is continuous for |x|<ε0 with F(0)=0 and g is continuous for 0<|x|<ε0 with xg(x)>0, and limx→0+g(x),limx→0-g(x) existing.(II)
For|x|<ε0, |F(x)|<a1G(x), where 0<a1<8 and G(x)=∫0x‍g(u)du.  Obviously, if g(0)=0, system (1.1) has a singular point at the origin. By Han and Zhang [12], the origin is a generalized singular point if g(0)≠0. For example, the following system:
(1.2)(x˙,y˙)={(y,-1),x≥0,(y,1),x<0
has a generalized center at the origin.Under the condition (I), the equationG(x)=z has two solutions x=x1(z)>0 and x=x2(z)<0 for 0<z≪1. Let Fi(z)=F(xi(z)),i=1,2. Then in 1952, Filippov proved the following theorem (see Chapter 5 of Ye [15]).Theorem 1.1.
Let (I) and (II) be satisfied. Further supposeg in (1.1) is continuous at x=0. Then system (1.1) has a stable focus (resp., center, unstable focus) if
(1.3)F2(z)-F1(z)<0(resp.,≡0,>0)for0<z≪1.In 1985, Han [10] obtained the following.Theorem 1.2.
Consider the system(1.4)x˙=h(y)-F(x),y˙=-g(x),
where h,F, and g are continuous functions satisfying(i)
h(y)=(sgny)|y|m-+O(ym-+1) with m->0,(ii)
xg(x)>0 for 0<|x|≪1 and
(1.5)|F(x)|<c[G(x)]m-/(m-+1)for|x|≪1,
where 0<c<(m-+1)((m-+1)/m-)m-/(m-+1). Let α(x)=-x+O(x2) satisfy G(α(x))=G(x) for |x|≪1. Then system (1.4) has a stable focus (resp., center, unstable focus) if
(1.6)F(α(x))-F(x)<0(resp.,≡0,>0)
for |x|≪1.For center conditions, in 1976, Cherkas [7] proved the following.Theorem 1.3.
Let the functionsF and g in system (1.1) be polynomials in x with
(1.7)F(0)=g(0)=0,g'(0)>0.
Then the origin is a center of system (1.1) if and only if the equations
(1.8)F(x)=F(y),G(x)=G(y)
have a unique solution y=α(x)<0 for x>0 sufficiently small.From Cherkas [7], one can see that the above result is also true for analytic system (1.1). In 1998, Gasull and Torregrosa [9] studied the center problem for analytic systems of form (1.4) using the Cherkas’ method, generalized Theorem 1.3 to (1.4) and presented interesting applications to some polynomial systems. Then in 2006, Cherkas and Romanovski [8] gave a necessary and sufficient condition for a Liénard system with nonlinearities of degree six to have a center at the origin. About Theorem 1.2, we have the following two remarks.Remark 1.4.
By the variable change(1.9)u=(sgnx)2G(x)
and the scaling of the time dτ=(g(X(u))/u)dt, we can obtain from (1.4)
(1.10)u˙=h(y)-F(X(u)),y˙=-u,
where X(u) satisfies |u|=2G(X(u)), uX(u)>0. Thus, Theorem 1.2 is also true if g(x) is not continuous at x=0.Remark 1.5.
IfF and g are C∞ with g'(0)>0. Then F(α(x))-F(x) in (1.6) has the form
(1.11)F(α(x))-F(x)=∑j≥1‍Bjxj.
In this case, Han [11] proved that if Bj=0 for j=1,…,2k-1(k≥1), then B2k=0, and the origin is a focus of order k if B2k+1≠0 in addition. Further, if a vector parameter a appears in F, then Bj=Bj(a) and
(1.12)B2k=O(|B1,B3,…,B2k-1|)
for all a and any k≥1.In this paper, we consider the following discontinuous Liénard system:(1.13)x˙=h(y)-F(x,a),y˙=-g(x,b),
where (a,b)∈Rn1 with n1 being an integer, h(y) is a C∞ function satisfying h(y)=y+O(y2) and
(1.14)F(x,a)={F+(x,a),x≥0,F-(x,a),x<0,g(x)={g+(x,b),x≥0,g-(x,b),x<0,
where F+,g+ and F-,g- are C∞ on [0,x0], and [-x0,0], respectively, with x0>0. Further, suppose there exist integers m≥1, n≥1, k≥0, l≥0 such that
(1.15)F+(x,a)=∑j≥m‍Fj+(a)xj,g+(x,b)=∑j≥k‍gj+(b)xj,gk+(b)>0
for 0<x≪1 and
(1.16)F-(x,a)=∑j≥n‍Fj-(a)xj,g-(x,b)=gl-(b)(-x)l+∑j≥l+1‍gj-(b)xj,gl-(b)<0
for 0<-x≪1.Let(1.17)G(x,b)=∫0x‍g(u,b)du={G+(x,b),0≤x≤x0,G-(x,b),0<-x≤x0,
where
(1.18)G+(x,b)=gk+(b)k+1xk+1+∑j≥k+1‍gj+(b)j+1xj+1,G-(x,b)=-gl-(b)l+1(-x)l+1+∑j≥l+1‍gj-(b)j+1xj+1
for 0<x≪1 and 0<-x≪1, respectively.This paper is devoted to studying the local property of system (1.13) at the origin and the number of limit cycles bifurcated from the origin as (a,b) varies. The authors [13] considered system (1.13) for the case m=n=k=l=1 and studied the center focus problem. It is easy to prove that for system (1.13) under (1.15) and (1.16), the condition (ii) of Theorem 1.2 is equivalent to the following:
(1.19)(H1)m>((k+1)/2)orm=((k+1)/2),|Fm+(a)|<c((gk+(b))/(k+1))1/2,(H2)n>((l+1)/2)orn=((l+1)/2),|Fn-(a)|<c((-gl-(b))/(l+1))1/2,
where 0<c<22. For convenience, introduce
(1.20)r=k+1l+1=qp,
where p,q are relatively prime numbers, that is, (p,q)=1. Then it is easy to see that there must exist an integer η∈[0,p-1] such that (n+η)r is an integer, and our main results are the following.Theorem 1.6.
Let (1.15), (1.16), (H1), and (H2) hold. Then(i)
F-(α(x),a)-F+(x,a) can be expressed as
(1.21)F-(α(x,b),a)-F+(x,a)=∑j≥1‍Bj(a,b)xkj,
where α(x,b)=-[((l+1)gk+(b))/(-(k+1)gl-(b))]1/(l+1)xr(1+o(1)) satisfies G-(α(x,b),b)=G+(x,b) for x>0 small and kτ∈{(n+i)r+j|0≤i≤p-1,i≠η,j≥0}∪{j|j≥min{m,(n+η)r}} for τ≥1 with min{nr,m}=k1<k2<⋯.(ii)
If there exist(a0,b0)∈Rn1 and s(≥1) such that Bs+1(a0,b0)≠0 and
(1.22)Bj(a0,b0)=0,j=1,…,s,rank∂(B1,B2,…,Bs)∂(a,b)(a0,b0)=s,
where n1≥s, then system (1.13) has cyclicity s near the origin for all (a,b) near (a0,b0).Theorem 1.7.
Let (1.15), (1.16), (H1), and (H2) hold. If there exists s≥1 such that
(1.23)F-(α(x,b),a)≡F+(x,a)asB1=B2=⋯=Bs=0
for x≥0 small and (1.22) is satisfied, then system (1.13) has cyclicity s-1 at the origin for |a-a0|+|b-b0| small.In many cases, the functionF in (1.13) is linear in a. Then the coefficients Bj(a,b)(j≥1) in (1.21) are linear in a. Let there exist a0∈Rn2,b0∈Rn3(n2+n3=n1) and integer s>0(s≤n2) such that
(1.24)Bj(a0,b0)=0,j=1,2,…,s,det∂(B1,B3,…,Bs)∂(a1,a2,…,as)(a0,b0)≠0
which implies that the linear equations Bj=0, j=1,2,…,s of a have a unique solution of the form
(1.25)(a1,…,as)=φ(as+1,…,an2,b)
for b near b0. Obviously, φ is linear in as+1,…,an2. Further, let
(1.26)Bs+j∣(a1,…,as)=φ(as+1,…,an2,b)=Lj(as+1,…,an2)Δj(b),j=1,…,σ
for some integer σ>0. Then, we can obtain the following.Theorem 1.8.
Consider system (1.13), where the function F is linear in a∈Rn2. Suppose (1.15), (1.16), (H1), and (H2) hold. Let there exist integers s>0,σ>0 and points a0=(a10,…,an2,0)∈Rn2 and b0∈Rn3 such that (1.24) and (1.26) hold with
(1.27)Lj(as+1,0,…,an2,0)≠0,j=1,…,σ,Δj(b0)=0,j=1,…,σ-1.(i)
IfΔσ(b0)≠0 and
(1.28)rank∂(Δ1…,Δσ-1)∂(b1,…,bσ-1)(b0)=σ-1,
then system (1.13) has cyclicity s+σ-1 limit cycles near the origin for all (a,b) near (a0,b0).(ii)
IfΔσ(b0)=0 and F-(α(x),a)≡F+(x,a) as (a1,…,as)=φ(as+1,…,an2,b), Δj(b)=0, j=1,…,σ with
(1.29)rank∂(Δ1,…,Δσ)∂(b1,…,bσ)(b0)=σ,
then system (1.13) has cyclicity s+σ-1 limit cycles at the origin for all (a,b) near (a0,b0).The conclusion (i) of Theorem1.8 can be proved in a similar manner to the proof of Theorem 2 of [16] and (ii) can be verified by Theorem 1.7. Thus, we do not verify it here. The proof of Theorems 1.6 and 1.7 is presented in Section 2. In Section 3, we give some applications.


## 2. Proof of Theorems1.6 and 1.7

In this section, we verify Theorems1.6 and 1.7. Before proving them, we need to introduce a bifurcation function d of system (1.13) and establish some preliminary lemmas, which will be used in our proof. First, we have the following lemma.Lemma 2.1.
Suppose (1.18) holds. Then the function α(x,b) defined in Theorem 1.6 has the form
(2.1)α(x,b)=∑i=1p‍∑j≥0‍αijxir+j
for 0<x≪1 with α10=-[((l+1)gk+(b))/(-(k+1)gl-(b))]1/(l+1).Proof.
Consider the equation(2.2)G-(u,b)=G+(x,b)
for 0<-u≪1 and 0<x≪1. By (1.18), we can obtain
(2.3)[G-(u,b)]1/(l+1)=(-gl-(b)l+1)1/(l+1)(-u)[1+∑j≥l+1‍(l+1)gj-(b)(-1)l+2(j+1)gl-(b)uj-l]1/(l+1),[G+(x,b)]1/(l+1)=(gk+(b)k+1)1/(l+1)x(k+1)/(l+1)[1+∑j≥k+1‍(k+1)gj+(b)(j+1)gk+(b)xj-k]1/(l+1)=(gk+(b)k+1)1/(l+1)x(k+1)/(l+1)∑j≥0‍v~jxj=∑j≥0‍vjxj+r≡v(x),
where v0=((gk+(b))/(k+1))1/(l+1). The implicit function theorem implies that the equation [G-(u,b)]1/(l+1)=v has a unique solution
(2.4)u=u*(v)=∑i≥1‍uivi
for v>0 small with u1=((l+1)/(-gl-(b)))1/(l+1). Let α(x,b)=u*(v(x)). Then, u=α(x,b) for x>0 sufficiently small is the solution of (2.2). Combining (2.3) and (2.4), the above formula can be represented as
(2.5)α(x,b)=∑i≥1‍ui(∑j≥0‍vjxj+r)i=∑i≥1‍∑j≥0‍α~ijxir+j=∑i≥1‍∑j≥0‍α~ijx(qi/p)+j=∑i=1p‍∑j≥0‍αijxir+j
with α10=v0u1. Thus, the proof is ended.Following the idea of Han [11], we have the following lemma.Lemma 2.2.
Let (1.15) and (1.16) hold. Then system (1.13) is equivalent to
(2.6)u˙=v-K(v)F*(u,a,b),v˙=-u,
where K(v)=(1/2)+O(v)∈C∞, and
(2.7)F*(u,a,b)={∑j≥m‍fj+u2j/(k+1),0≤u≪1,∑j≥n‍fj-|u|2j/(l+1),0<-u≪1
with fm+=Fm+((k+1)/(2gk+(b)))m/(k+1) and fn-=(-1)nFn-((l+1)/(-2gl-(b)))n/(l+1).Proof.
LetH(y)=∫0y‍h(u)du and make the transformation
(2.8)u=G(x,b)(sgnx),v=H(y)(sgny),
together with the scaling of the time dτ=(g(X(u))h(Y(v))/2uv)dt to system (1.13) so that (1.13) can be changed into
(2.9)u˙=v-K(v)F*(u,a,b),v˙=-u,
where K(v)=v/h(Y(v))=(1/2)+O(v)∈C∞, F*(u,a,b)=F(X(u),a) and X(u),Y(v) denote the inverse of the transformation (2.8). In fact, by (1.18) and similar to the proof of Lemma 2.1, we can obtain
(2.10)X(u)={X+(u)X-(u)={∑j≥1‍xj+u2j/(k+1),x1+=(k+12gk+(b))1/(k+1),0≤u≪1,∑j≥1‍xj-|u|2j/(l+1),x1-=-(l+1-2gl-(b))1/(l+1),0<-u≪1.
For 0≤u≪1, by (1.15) and (2.10), we have
(2.11)F*(u,a,b)=F+(X+(u),a)=∑j≥m‍Fj+(a)(∑i≥1‍xi+u2i/(k+1))j=∑j≥m‍fj+u2j/(k+1),
where fm+=Fm+((k+1)/2gk+(b))m/(k+1). Similarly, for 0<-u≪1, by (1.16) and (2.10), we can obtain
(2.12)F*(u,a,b)=F-(X-(u),a)=∑j≥n‍fj-|u|2j/(l+1),
where fn-=(-1)nFn-((l+1)/-2gl-(b))n/(l+1). This ends the proof.For a relation between the functionF(x,a) in (1.13) and the function F*(u,a,b) in (2.6) we haveLemma 2.3.
Suppose (1.15) and (1.16) hold. Then we have(i)
F-(α(x,b),a)-F+(x,a) has the form (1.21) for 0≤x≪1;(ii)
letF0(u,a,b)=F*(-u,a,b)-F*(u,a,b). Then F0 can be expressed as
(2.13)F0(u,a,b)=∑j≥1‍Aj(a,b)u2kj/(k+1),for0≤u≪1,
where
(2.14)A1(a,b)=B1(a,b)N1(x1+),Aj(a,b)=Bj(a,b)Nj(x1+)+O(|B1,B2,…,Bj-1|),j≥2
with Nj(j≥1) are positive C∞ functions in x1+, and kj,j≥1, are as appeared in (1.21).Proof.
By (1.15), (1.16), and Lemma 2.1, we can have
(2.15)F-(α(x,b),a)-F+(x,a)=∑s≥n‍Fs-(a)[∑i=1p‍xir∑j≥0‍αijxj]s-∑s≥m‍Fs+(a)xs=∑s≥n‍Fs-(a)x(q/p)s[∑i=0p-1‍xqi/p∑j≥0‍αi+1,jxj]s-∑s≥m‍Fs+(a)xs=∑i=0p-1‍x(n+i)r∑j≥0‍B-ij(a,b)xj-∑s≥m‍Fs+(a)xs.
Since r(n+η) is an integer, the above formula can be written as
(2.16)F-(α(x,b),a)-F+(x,a)=∑i=0,i≠ηp-1‍∑j≥0‍B-ij(a,b)x(n+i)r+j+∑j≥T‍B~j(a,b)xj,
where T=min{m,r(n+η)}. Then we express (2.16) with respect to the power of x in ascending order, yielding the form of (1.21). In addition, for kτ defined in Theorem 1.6 there must exist some integers iτ and jτ such that kτ=(n+iτ)r+jτ. Hence, we can obtain
(2.17)Bτ(a,b)=B-iτjτifiτ≠η,Bτ(a,b)=B~kτifiτ=η.
This completes the proof of the conclusion (i).
Now we prove the conclusion (ii). Foru≥0 sufficiently small, (2.8) and (2.10) imply that
(2.18)G+(X+(u),b)=G-(X-(-u),b)=u2.
Then noting that G-(α(x,b),b)=G+(x,b), we can get
(2.19)X-(-u)=α(X+(u)).
By (2.11), (2.12) and the above formula, we have for u≥0 sufficiently small
(2.20)F0(u,a,b)=F*(-u,a,b)-F*(u,a,b)=[F-(X-(-u),a)-F+(X+(u),a)]=[F-(α(x,b),a)-F+(x,a)]∣x=X+(u).
Let
(2.21)φ(μ)=X+(μ(k+1)/2).
Combining (2.10) and (2.21) gives that
(2.22)φ(μ)=∑j≥1‍xj+μj=x1+μ(1+O(μ)).
Substituting the above formula into (1.21) or (2.16) yields that
(2.23)F-(α(x,b),a)-F+(x,a)∣x=φ(μ)=∑j≥1‍Bj(a,b)(x1+)kjμkj(1+O(μ))=∑i=0,i≠ηp-1‍∑j≥0‍B-ij(a,b)(∑s≥1‍xs+μs)(n+i)r+j+∑j≥T‍B~j(a,b)(∑s≥1‍xs+μs)j.
Note that
(2.24)∑i=0,i≠ηp-1‍∑j≥0‍B-ij(a,b)(∑s≥1‍xs+μs)(n+i)r+j=∑i=0,i≠ηp-1‍∑j≥0‍B-ij(a,b)(x1+)(n+i)r+jμ(n+i)r+j(1+∑s≥1‍xs+1+x1+us)(n+i)r+j=∑i=0,i≠ηp-1‍∑j≥0‍B-ij(a,b)(x1+)(n+i)r+jμ(n+i)r+j(1+∑s≥1‍x-sμs)=∑i=0,i≠ηp-1‍∑j≥0‍Bij*(a,b)μ(n+i)r+j,
where
(2.25)Bi0*=(x1+)(n+i)rB-i0,Bij*=(x1+)(n+i)r+jB-ij+O(|B-i0,B-i1,…,B-i,j-1|),j≥1
with i=0,…,p-1 and i≠η. Similarly,
(2.26)∑j≥T‍B~j(a,b)(∑s≥1‍xs+μs)j=∑j≥T‍Bj*(a,b)μj,
where
(2.27)BT*=(x1+)TB-T,Bj*=(x1+)jB-j+O(|B-T,B-T+1,…,B-j-1|),j≥T+1.
Combining (2.23), (2.24), and (2.26), we can obtain
(2.28)F-(α(x,b),a)-F+(x,a)∣x=φ(μ)=∑i=0,i≠ηp-1‍∑j≥0‍Bij*(a,b)μ(n+i)r+j+∑j≥T‍Bj*(a,b)μj.
Similar to (1.21) and (2.16), the above formula can be rewritten as
(2.29)F-(α(x,b),a)-F+(x,a)∣x=φ(μ)=∑j≥1‍Aj(a,b)μkj,
where from (2.17)
(2.30)Aτ(a,b)=Biτjτ*ifiτ≠η,Aτ(a,b)=Bkτ*ifiτ=η.
From (2.25), (2.27), and the above equation, Aj in (2.29) can be rewritten as the form of (2.14). Further, combining (2.17), (2.20), and (2.29) yields (2.13). Thus, the proof is ended.Next, we establish the bifurcation functiond of system (1.13). For the purpose, let u=rcosθ,v=rsinθ. Then (2.6) can be translated into
(2.31)drdθ=cosθK(rsinθ)F*(rcosθ,a,b)1-sinθK(rsinθ)F*(rcosθ,a,b)/r≡R*(θ,r,a,b).
In a similar way to the proof of Lemma 2.3 of [13], it is easy to prove that
(2.32)1-sinθK(rsinθ)F*(rcosθ,a,b)r>0
for r>0 small under (H1) and (H2), which shows that R*(θ,r,a,b) in (2.31) is well defined. Introduce
(2.33)β=(k+1)(l+1),ρ=r1/β.
Then by (2.31) and (2.33), we can obtain
(2.34)dρdθ=ρ1-ββ×cosθK(ρβsinθ)F*(ρβcosθ,a,b)1-sinθK(ρβsinθ)F*(ρβcosθ,a,b)/ρβ=ρβ×cosθK(ρβsinθ)F1(θ,ρ,a,b)1-sinθK(ρβsinθ)F1(θ,ρ,a,b)≡R(θ,ρ,a,b),
where F1(θ,ρ,a,b)=ρ-βF*(ρβcosθ,a,b). Clearly, (2.34) is a 2π-periodic equation in θ. By (2.7), we have
(2.35)F1(θ,ρ,a,b)={∑j≥m‍fj+(cosθ)2j/(k+1)ρ(2j/(k+1))β-β,cosθ≥0,∑j≥n‍fj-|cosθ|2j/(l+1)ρ(2j/(l+1))β-β,cosθ<0.
From (2.33) and (2.35), it is easy to see that F1 is C∞ in ρ for any given θ. Let ρ(θ,ρ0,a,b) denote the solution of (2.34) satisfying ρ(π/2,ρ0,a,b)=ρ0. For convenience, let
(2.36)ρ(-π2,ρ0,a,b)=ρ1,ρ(3π2,ρ0,a,b)=ρ2.
Define a function as follows:
(2.37)d(ρ0,a,b)=ρ(-π2,ρ0,a,b)-ρ(3π2,ρ0,a,b)=ρ1-ρ2.
It is obvious that system (1.13), (2.6), (2.31), or (2.34) has a periodic orbit near the origin if and only if d(ρ0,a,b) in (2.37) has a zero in ρ0 for ρ0>0 sufficiently small. See Figure 1.Figure 1Hence, the functiond in (2.37) is called a displacement function or bifurcation function of system (1.13), (2.6), (2.31), or (2.34). About the bifurcation function d, we have.Lemma 2.4.
Let (1.15), (1.16), (H1), and (H2) hold. Then the bifurcation function d in (2.37) has the form
(2.38)d(ρ0,a,b)=∑j≥1‍Cj(a,b)ρ0rj(1+dj(ρ0)),
where rj=2(l+1)kj-kl-k-l are positive integers satisfying 1≤r1<r2<⋯,
(2.39)C1(a,b)=B1(a,b)N1*(fn-,fm+),Cj(a,b)=Bj(a,b)Nj*(fn-,fm+)+O(|B1,B2,…,Bj-1|),j≥2
with Nj*∈C∞, Nj(0,0)>0 for j≥1 and dj(j≥1) are C∞ functions in ρ0 satisfying di(0)=0.Proof.
Denoteρ- by ρ-(θ,ρ0,a,b)=ρ(π-θ,ρ0,a,b). Then
(2.40)ρ-(π2,ρ0,a,b)=ρ(π2,ρ0,a,b),ρ-(-π2,ρ0,a,b)=ρ(3π2,ρ0,a,b)=ρ2
and it satisfies
(2.41)dρ-dθ=-R(π-θ,ρ,a,b).
Let R-(θ,ρ,a,b)=-R(π-θ,ρ,a,b). Then we can obtain that ρ-(θ,ρ0,a,b) is a solution of the equation
(2.42)dρdθ=R-(θ,ρ,a,b)
satisfying ρ-(π/2,ρ0,a,b)=ρ0. Let θ∈[-π/2,π/2]. Then π-θ∈[π/2,3π/2] and
(2.43)R(θ,ρ,a,b)=R1(θ,a,b)ρ((2m/(k+1))-1)β+1(1+O(ρ)),R-(θ,ρ,a,b)=R-1(θ,a,b)ρ((2n/(l+1))-1)β+1(1+O(ρ)),
where
(2.44)R1(θ,a,b)={K(0)fm+β(cosθ)(2m/(k+1))+1,m>k+12,K(0)fm+cos2θβ(1-sinθcosθK(0)fm+),m=k+12,R-1(θ,a,b)={-K(0)fn-β(cosθ)(2m/(l+1))+1,n>l+12,-K(0)fn-cos2θβ(1-sinθcosθK(0)fn-),n=l+12.
Thus, by (2.43), (2.44) we have easily
(2.45)ρ(θ,ρ0,a,b)={ρ0+O(ρ0((2m/(k+1))-1)β+1),m>k+12,ρ0exp∫π/2θ‍R1(u,a,b)du+O(ρ02),m=k+12,(2.46)R-(θ,ρ(θ,ρ0,a,b),a,b)-R-(θ,ρ-(θ,ρ0,a,b),a,b)=[ρ(θ,ρ0,a,b)-ρ-(θ,ρ0,a,b)]Δ(θ,ρ0)
by the mean value theorem, where Δ(θ,ρ0)=O(ρ0((2n/(l+1))-1)β). Further by (2.34) and (2.42), we can obtain
(2.47)R(θ,ρ,a,b)-R-(θ,ρ,a,b)=-cosθK(ρβsinθ)ρ1-β[F*(-ρβcosθ,a,b)-F*(ρβcosθ,a,b)]β[1-sinθK(ρβsinθ)ρ-βF*(ρβcosθ,a,b)][1-sinθK(ρβsinθ)ρ-βF*(-ρβcosθ,a,b)]
which follows that by Lemma 2.3(2.48)R(θ,ρ,a,b)-R-(θ,ρ,a,b)=-cosθρ1-βF0(ρβcosθ,a,b)S(θ,ρ),
where S(θ,ρ) is a C∞ function in ρ with
(2.49)S(θ,0)={K(0)β,m>k+12,n>l+12,K(0)β(1-sinθcosθK(0)fm+),m=k+12,n>l+12,K(0)β(1-sinθcosθK(0)fn-),m>k+12,n=l+12,K(0)β(1-sinθcosθK(0)fm+)(1-sinθcosθK(0)fn-),m=k+12,n=l+12,
which implies S(θ,0)>0 by the above discussion. Then it follows from (2.13), (2.45), and (2.48) that
(2.50)R(θ,ρ(θ,ρ0,a,b),a,b)-R-(θ,ρ(θ,ρ0,a,b),a,b)=-∑j≥1‍Aj(a,b)(cosθ)((2kj/(k+1))+1)ρ0(2kj/(k+1))β+1-βqj(θ,ρ0),
where qj(j≥1) are C∞ functions in ρ0 with
(2.51)qj(θ,0)={S(θ,0),m>k+12,S(θ,0)exp(2kjk+1β∫π/2θ‍R1(u,a,b)du),m=k+12,
which shows that qj(θ,0)>0 for all j≥1. Since ρ-(θ,ρ,a,b) and ρ(θ,ρ,a,b) satisfy (2.42) and (2.34), respectively, using (2.46) we have
(2.52)ddθ(ρ-ρ-)=(ρ-ρ-)Δ(θ,ρ0)+R(θ,ρ,a,b)-R-(θ,ρ,a,b),
where ρ-=ρ-(θ,ρ0,a,b) and ρ=ρ(θ,ρ0,a,b). Then noting (2.40), applying the formula of variation of constants to the above equation yields
(2.53)ρ(θ,ρ0,a,b)-ρ-(θ,ρ0,a,b)=∫π/2θ‍(R(δ,ρ(δ,ρ,a,b),a,b)-R-(δ,ρ(δ,ρ0,a,b),a,b))×exp(∫δθ‍Δ(τ,ρ0)dτ)dδ.
Inserting (2.50) into the above formula and taking θ=-π/2 give by (2.37) that
(2.54)d(ρ0,a,b)=∑j≥1‍Aj(a,b)ρ0(2kj/(k+1))β+1-β∫-π/2π/2‍(cosθ)(2kj/(k+1))+1qj(θ,0)×exp(∫δθ‍Δ(τ,0)dτ)(1+O(ρ0))dθ,
which implies (2.38) by (2.33). Hence, The proof is completed.In the following part, we verify Theorems1.6 and 1.7 using the above lemmas.Proof of Theorem1.6.
One can see that the conclusion (i) is true by Lemmas2.1 and 2.3. Now, we prove the conclusion (ii). Obviously, it suffices to prove the following two points.(1)
There are at mosts limit cycles near the origin for all (a,b) near (a0,b0).(2)
There can appears limit cycles in any given neighborhood of the origin for some (a,b) arbitrarily close to (a0,b0).
In fact, noting that(2.55)Cs+1(a0,b0)=Bs+1(a0,b0)Ns+1*≠0.
By our assumption, we have from (2.38) and (2.55)
(2.56)d(ρ0,a,b)=∑j=1s+1‍Cj(a,b)ρ0rj(1+dj(ρ0))+ρ0rs+2Q(ρ0,a,b)=∑j=1s+1‍Cj(a,b)ρ0rj(1+d-j(ρ0))≡Ds,
where Q(ρ0,a,b)=∑j≥s+2Cj(a,b)ρ0rj-rs+1,
(2.57)d-j=dj,j=1,…,s,d-s+1=ds+1+Q(ρ0,a,b)Cs+1(a,b)ρ0rs+2-rs+1,
which implies d-j in (2.56) are C∞ in ρ0 satisfying d-j(0)=0 for (a,b) sufficiently close to (a0,b0).
We claim thatDs in (2.56) has at most s zeros in ρ0>0 small for |a-a0|+|b-b0|≪1. We can proceed with the proof by induction on s.
First, whens=1, from (2.56), we can obtain
(2.58)D1=(1+d-1(ρ0))ρ0r1[C1(a,b)+C2(a,b)ρ0r2-r1(1+d~2(ρ0))]=(1+d-1(ρ0))ρ0r1d1(ρ0,a,b).
Note that
(2.59)dd1(ρ0,a,b)dρ0=C2(a,b)ρ0r2-r1-1[(r2-r1)(1+d~2(ρ0))+ρ0d~2′(ρ0)],
which implies
(2.60)dd1(ρ0,a,b)dρ0≠0
for 0<ρ0≪1 and (a,b) near (a0,b0) since C2(a0,b0)=B2(a0,b0)N2*≠0 by (2.55). Hence, by Rolle’s theorem, the conclusion (1) is true for s=1.
Assume that the conclusion (1) is true fors=i. That is by (2.56)
(2.61)Di=∑j=1i+1‍Cj(a,b)ρ0rj(1+d-j(ρ0))
has at most i zeros for ρ0>0 and |a-a0|+|b-b0| sufficiently small with Ci+1(a0,b0)≠0. Let us prove that the conclusion (1) is also true for s=i+1. Then from (2.56), we can obtain
(2.62)Di+1=∑j=1i+2‍Cj(a,b)ρ0rj(1+d-j(ρ0))=ρ0r1(1+d-1(ρ0))∑j=1i+2‍Cj(a,b)ρ0rj-r1dj*≡ρ0r1(1+d-1(ρ0))D-i+1,
where d1*=1, dj*=(1+d-j(ρ0))/(1+d-1(ρ0))=1+O(ρ0), j=2,3,…,i+2. Then,
(2.63)dD-i+1dρ0=∑j=2i+2‍(rj-r1)Cj(a,b)ρ0rj-r1-1(dj*+ρ0rj-r1ddj*dρ0)=∑j=1i+1‍C-j(a,b)ρ0r~j(1+d^j(ρ0)),
where C-j(a,b)=(rj+1-r1)Cj+1(a,b), r~j=rj+1-r1-1, j=1,…,i+1 satisfying 1≤r~1<⋯<r~i+1 and d^j(ρ0) is C∞ in ρ0 with d^j(0)=0, which implies that dD-i+1/dρ0 has at most i zeros in ρ0>0 small for (a,b) near (a0,b0) by induction assumption since C-i+1(a0,b0)=(ri+2-r1)Ci+2(a0,b0)≠0. Hence, by Rolle’s theorem, it follows that Di+1 in (2.62) has at most i+1 zeros in 0<ρ0≪1 for (a,b) near (a0,b0). This ends the proof of conclusion (1).
Now, we verify the conclusion (2). For simplicity, we can assume(a,b)=a-=(a-1,a-2,…,a-n1). Further, by (1.22), without loss of generality, suppose
(2.64)det∂(B1,…,Bs)∂(a-1,…,a-s)≠0.
Then we can fix a-j=a-j0 for j=s+1,…,n1 since n1≥s. Consider the equations
(2.65)Bj=Bj(a-),j=1,…,s.
By the implicit function theorem, the equations have a unique set of solutions
(2.66)a-j=a-j*(B1,…,Bs,a-s+1,…,a-n1),j=1,2,…,s
for a- near a-0, which means Bj(j=1,…,s) can be taken as free parameters. Hence, by changing the sign of Bs,Bs-1,…,B1 in turn such that
(2.67)Bj-1Bj<0,j=s+1,…,1,0<|B1|≪|B2|≪⋯=|Bs|≪1,
which follows that Cj in (2.56) satisfy
(2.68)Cj-1Cj<0,j=s+1,…,1,0<|C1|≪|C2|≪⋯≪|Cs|≪1.
This ensures that the bifurcation function d has s zeros in ρ0 for some (a,b) near (a0,b0), which implies the conclusion (2). This ends the proof.Proof of Theorem1.7.
From the proof of Theorem1.6, it is easy to see Bj(j=1,…,s) can be taken as free parameters by (1.22), which implies Cj(j=1,…,s) can also be taken as free parameters by the definition of Cj in Lemma 2.4. From (1.23) and Lemma 2.4, we have
(2.69)d(ρ0,a,b)=0whenCj=0,j=1,…,s
for |a-a0|+|b-b0| sufficiently small since Bj=0 if and only if Cj=0, which means that the coefficients Cj(j≥s+1) in (2.38) satisfy
(2.70)Cj(a,b)=0,j=s+1,s+2,…aslongasC1=⋯=Cs=0.
Since Cj can be taken as free parameters, we can write
(2.71)Cj(a,b)=∑i=1s‍CiQij(a,b),
where Qij∈C∞,j≥s+1. Inserting (2.71) into (2.38) gives that
(2.72)d(ρ0,a,b)=∑j=1s‍Cjρ0rj(1+ej(ρ0)),
where
(2.73)ej(ρ0)=1+dj(ρ0)+∑i≥s+1‍Qjiρ0ri-rj(1+di(ρ0))forj=1,…,s.
We can easily prove d in (2.72) has at most s-1 zeros for (a,b) near (a0,b0) by induction s in a similar way to the above proof. Further, we can choose Cj satisfying
(2.74)Cj-1Cj<0,j=s,…,1,0<|C1|≪|C2|≪⋯≪|Cs|≪1,
which ensures that d in (2.72) has s-1 zeros for (a,b) near (a0,b0). The proof is finished.From the proof of Theorems1.6 and 1.7, we have immediately the following corollaries.Corollary 2.5.
Let (1.15), (1.16), (H1), and (H2) hold. If there exist a point (a0,b0) and an integer s≥1such that
(2.75)Bj(a0,b0)=0,j=1,2,…,s,Bs+1(a0,b0)<0(resp.,>0),
then the origin is a stable (resp., unstable) focus of system (1.13) for (a,b)=(a0,b0) and there has at most s limit cycles at the origin for all (a,b) near (a0,b0).Corollary 2.6.
Let (1.15), (1.16), (H1), and (H2) hold. If there exists s≥1 such that
(2.76)Bj=O(|B1,B2,…,Bs|)
for j≥s+1, then for any given N>0 there exists a neighborhood U of the origin such that for all |Bj|<N, j=1,…,s system (1.13) has at most s-1 limits cycles in U.Further, about system (1.13), we have the below two remarks in order.Remark 2.7.
If the functionh(y) in (1.13) has the form h(y)=h1y+O(y2) with h1>0, we also can obtain the same conclusions since this system is equivalent to the following:
(2.77)x˙=h(y)h1-F(x)h1,y˙=-g(x)h1.Remark 2.8.
When vector parametera or b is constant in (1.13), Theorems 1.6 and 1.7 remain true in this case.


## 3. Applications

In this section, we present some applications of our main results to systems of the form(3.1)x˙=y-F(x,a),y˙=-g(x,b),
where F,g satisfy (1.15) and (1.16).Example 3.1.
First consider system (3.1) with
(3.2)g(x,b)={∑j=0k‍bjxj,x≥0,-1,x<0,F(x,a)={∑j=1n‍aj+xj,x≥0,∑j=1m‍aj-xj,x<0,
where b0>0 and a=(a1+,…,an+,a1-,…,am-), b=(b0,b1,…,bk). By Lemma 2.1, it is clear that
(3.3)α(x,b)=-∑j=0k‍bjj+1xj+1,
for x≥0 small. For any given N>0, denote L(k,n,m) the cyclicity of system (3.1) near the origin for |a|+|b|≤N. We study the cyclicity of system (3.1) under (3.2) near the origin with the following three cases.Case 1.
m
=
1, k,m,n≥1. In this case, we claim that L(k,n,1)=max{k+1,n}-1. In fact, by (3.2) and (3.3), we have
(3.4)F-(α(x,b),a)-F+(x,a)=-∑j=0k‍a1-bjj+1xj+1-∑j=1n‍aj+xj=∑j=1max{k+1,n}‍Bj(a,b)xj
for 0≤x≪1. From (3.4) it is easy to see that F-(α(x,b),a)≡F+(x,a) for 0≤x≪1 if and only if Bj=0 for j=1,…,max{k+1,n}, which shows that L(k,n,1)≤max{n,k+1}-1 by Corollary 2.6. Thus we only need to prove that L(k,n,1)≥max{n,k+1}-1. For n<k+1, by (3.4), we can obtain
(3.5)Bj(a,b)=a1-bj-1j-aj+,j=1,…,n,Bj(a,b)=a1-bj-1j,j=n+1,…,k+1.
Take (a,b)=(a0,b~0) such that aj0+=(a10-bj-1,0)/j, j=1,…,n, bj0=0, j=n,…,k and a10-≠0. Then by (3.5), we have
(3.6)Bj(a0,b0)=0,j=1,…,k+1,rank∂(B1,…,Bn,Bn+1,…,Bk+1)∂(a1+,…,an+,bn,…,bk)|(a0,b~0)=k+1
which implies k limit cycles can appear near the origin for (a,b) near (a0,b~0). For n≥k+1, we can discuss similarly.Case 2.
k
=
2, m=2, n≥1. We will see that L(2,2,1)=3, L(2,2,2)=L(2,2,3)=4, L(2,2,4)=L(2,2,5)=5, and L(2,2,n)=n-1 for n≥6.
In fact, combining (3.2) and (3.3) gives that
(3.7)F-(α(x,b),a)-F+(x,a)=a1-(-b0x-b12x2-b23x3)+a2-(-b0x-b12x2-b23x3)2-∑j=1n‍aj+xj=∑j=1max{6,n}‍Bj(a,b)xj.
As before, it is easy to get that L(2,2,n)=n-1 for n≥6. For n≤5, we can discuss system (3.1) case by case with respect to n.
Forn=1, by (3.7) we can obtain
(3.8)B1(a,b)=-a1-b0-a1+,B2(a,b)=-12a1-b1+a2-b02,B3(a,b)=-13a1-b2+a2-b0b1,B4(a,b)=a2-(23b0b2+14b12),B5(a,b)=13a2-b1b2,B6(a,b)=19a2-b22.
From (3.8), it is easy to see that B5=O(|B2,B3,B4|), B6=O(|B2,B3,B4|). By Corollary 2.6, one can see that L(2,2,1)≤3. Further, from (3.8) we can have a0=(a10+,a10-,a20-) with
(3.9)a10+=-a10-b0,a20-=b12b02a10-,a10-≠0
such that
(3.10)B1(a0,b)=B2(a0,b)=0,B3(a0,b)=a10-Δ1(b),B4(a0,b)=a10-Δ2(b),B5(a0,b)=a10-Δ3(b),B6(a0,b)=a10-Δ4(b),
where
(3.11)Δ1(b)=-b23+b122b0,Δ2(b)=b1b02(13b0b2+18b12),Δ3(b)=b12b26b02,Δ4(b)=b1b2218b02.
Take b=b~0=(b00,b10,(3b102/2b00)) with b00≠0, b10≠0. Then by (3.11), we have
(3.12)Δ1(b0)=0,Δ2(b0)=5b1038b002≠0.
Note that
(3.13)det∂(B1,B2)∂(a1+,a2-)=|-100b02|=-b02≠0,∂Δ1∂b2=-13.
Then by Theorem 1.8, we know that system (3.1) has cyclicity 3 for (a,b) sufficiently close to (a0,b~0). In other words, L(2,2,1)=3. Similarly, we have L(2,2,2)=L(2,2,3)=4, L(2,2,4)=L(2,2,5)=5.Case 3.
k
=
m
=
n, n=1,2,3. We claim that L(n,n,n)=3n-2 for n=1,2,3. In fact, for k=m=n, we have
(3.14)F-(α(x,b),a)-F+(x,a)=∑j=1n‍aj-(-∑i=0n‍bii+1xi+1)j-∑j=1n‍aj+xj=∑j=1n(n+1)‍Bj(a,b)xj.
From Cases 1 and 2, it is easy to see that L(1,1,1)=1, L(2,2,2)=4. For k=m=n=3, Bj(a,b) in (3.14) can be written as
(3.15)B1(a,b)=-a1-b0-a1+,B2(a,b)=-12a1-b1+a2-b02-a2+,B3(a,b)=-13a1-b2+a2-b0b1-a3-b03-a3+,B4(a,b)=-14a1-b3+a2-(23b0b2+14b12)-32a3-b02b1,B5(a,b)=a2-(12b0b3+13b1b2)-a3-(b02b2+34b0b12),B6(a,b)=a2-(14b1b3+19b22)-a3-(b0b1b2+34b02b3+18b13),B7(a,b)=16a2-b2b3-a3-(34b0b1b3+13b0b22+14b12b2),B8(a,b)=116a2-b32-a3-(12b0b2b3+316b12b3+16b1b22),B9(a,b)=-a3-(14b1b2b3+316b0b32+127b23),B10(a,b)=-a3-(332b1b32+112b22b3),B11(a,b)=-115a3-b2b32,B12=-164a3-b33.
From (3.15), it is easy to prove that Bi=O(|B4,…,B8|), i=9,…,12. Then by Corollary 2.6, we have L(3,3,3)≤7. In order to prove L(3,3,3)=7, let for convenience
(3.16)Z(b)=36b02b1b3-12b0b12b2-32b02b22-9b14.
Then we can find (a,b)=(a0,b~0) with Z(b~0)≠0 and
(3.17)a10+=-a10-b0,a20+=12a10-b1-a20-b02,a30+=-13a10-b2+a2-b0b1-a30-b03,a10-≠0,a20-=3a10-b3(4b0b2+3b12)Z(b),a30-=2a10-b3(3b0b3+2b1b2)b0Z(b)
for b=b~0 such that
(3.18)B1(a0,b~0)=⋯=B5(a0,b~0)=0,B6(a0,b~0)=a10-Δ1(b~0),B7(a0,b~0)=a10-Δ2(b~0),B8(a0,b~0)=a10-Δ3(b~0),B9(a0,b~0)=a10-Δ4(b~0),B10(a0,b~0)=a10-Δ5(b~0),B11(a0,b~0)=a10-Δ6(b~0),B12(a0,b~0)=a10-Δ7(b~0),
where
(3.19)Δ1(b)=Z1(-92b03b32-6b02b1b2b3+43b02b23+32b0b12b3-3b0b12b22-12b14b2),Δ2(b)=Z1(-92b02b1b32-3b0b12b2b32-43b0b1b23-b13b22),Δ3(b)=Z1(-94b02b2b32-916b0b12b32-3b0b1b22b3-34b12b2b3-23b12b23),Δ4(b)=Z1(-98b02b32-94b0b1b2b32-29b0b23b3-b12b22b3-427b1b24),Δ5(b)=Z1(-916b0b1b32-12b0b22b32-38b12b2b32-13b1b23b3),Δ6(b)=Z1(-38b2b33-14b1b22b32),Δ7(b)=Z1(-332b0b34-116b1b2b32),
with Z1=b3/b0Z. Let’s choose b~0=(b00,0,(3/2)b001/3b302/3,b30) with b00>0, b30≠0. Obviously, Z(b~0)≠0. Then by (3.19), we have
(3.20)Δ1(b~0)=Δ2(b~0)=0,Δ3(b~0)=3b307/364b004/3≠0.
Note that
(3.21)det∂(B1,B2,B3,B4,B5)∂(a1+,a2+,a3+,a2-,a3-)∣(a0,b~0)=98b008/3b304/3≠0,det∂(Δ1,Δ2)∂(b1,b2)∣b~0=812b0014/3b3010/3≠0.
Then by Theorem 1.8, it is clear that 7 limit cycles can appear near the origin for (a,b) sufficiently close (a0,b~0). Then we have L(3,3,3)=7.Example 3.2.
Consider system (3.1) with
(3.22)g(x,b)={x2,x≥0,-1+b1x+b2x2,x<0,F(x,a)={∑j=2n‍aj+xj,x≥0,∑j=1m‍aj-xj,x<0.
Clearly, the origin is a center or focus by Theorem 1.2. Denote H(m,n) the cyclicity of system (3.1) near the origin. By (3.22), we can obtain
(3.23)G(x,b)={x33,x≥0,-x+b12x2+b23x3,x<0.
Using Lemma 2.1 and (3.23), for x≥0 small, we can solve from the equation G-(α(x,b),b)=G+(x,b)(3.24)α(x,b)=∑j≥1‍αjx3j=-13x3+118b1x6-(b1254+b281)x9+(5648b13+5486b1b2)x12-(71944b14+7972b12b2+1729b22)x15+(73888b15+71458b13b2+146561b1b22)x18+O(x21).
Take m=4, n=5. Combining (3.22) and (3.24), we can obtain
(3.25)F-(α(x,b),a)-F+(x,a)=∑j=15‍Bj(a,b)xj+1+∑j≥6‍Bj(a,b)x3(j-3),
where
(3.26)B1(a,b)=-a2+,B2(a,b)=-a3+-13a1-,B3(a,b)=-a4+,B4=-a5+,B5(a,b)=118a1-b1+19a2-,B6(a,b)=a1-(-154b12-181b2)-127a2-b1-127a3-,B7(a,b)=a1-(5648b13+5486b1b2)+a2-(5324b12+2243b2)+154a3-b1+181a4-,B8(a,b)=a1-(-11944b14-1972b12b2-1729b22)+a2-(-7972b13-2243b1b2)-a3-(1108b12+1243b2)-2243a4-b1,B9(a,b)=a1-(73888b15+71458b13b2+146561b1b22)+a2-(1648b14+52187b12b2+76561b22)+a3-(71458b13+71458b1b2)+a4-(71458b12+42187b2),
which implies that Bj=O(|B1,…,B9|), j≥10. Then we can obtain H(4,5)≤8 by Corollary 2.6. Now we prove that H(4,5)=8. Suppose a=(a2+,…,a5+,a1-,…,a4-). Then we can find
(3.27)a0=(a-20+,…,a-50+,a-10-,…,a-40-),
where
(3.28)a20+=a40+=a50+=a40-=0,a30+=-13a10-,a20-=-12a10-b1,a30-=-13a10-b2,a10-≠0,
such that
(3.29)B1(a0,b)=B2(a0,b)=B3(a0,b)=B4(a0,b)=B5(a0,b)=B6(a0,b)=B7(a0,b)=0,B8(a0,b)=1162a1-Δ0,B9(a0,b)=a1-162Δ1,
where
(3.30)Δ0=b12(12b12+b2),Δ1=b13(518531096b12+13b2).
Solving the equation Δ0=0 gives b=(b10,-(1/2)b102)≡b0, b10≠0. Inserting b0 into (3.30) gives
(3.31)Δ1(b0)=5185186576b105≠0.
Further, by (3.26) and (3.30), we have
(3.32)∂(B1,B2,B3,B4,B5,B6,B7)∂(a2+,a3+,a4+,a5+,a2-,a3-,a4-)=(-10000000-10000000-10000000-10000000-19000000-b127-127000005b12324+2b2243b154181),∂Δ0∂b2(b0)=b102≠0.
Hence, by the conclusion (i) of Theorem 1.8, it is easy to see that there exists cyclicity 8 when (a,b) near (a0,b0), which implies H(4,5)=8.Remark 3.3.
In a similar way, for1≤m≤4, 2≤n≤5, we can obtain H(m,n)=m+n-1 for n≠2, H(m,n)=m+n-2 for n=2, m≠2 and H(2,2)=3.



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*Source: 690453-2012-09-20.xml*