# Nonhyperbolic Periodic Orbits of Vector Fields in the Plane Revisited

Authors: Denis de Carvalho Braga; Luis Fernando Mello; Antonio Carlos Zambroni de Souza
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The main goal of this paper is to present a theory of approximation of periodic orbits of vector fields in the plane. From the theory developed here, it is possible to obtain an approximation to the curve of nonhyperbolic periodic orbits in the bifurcation diagram of a family of differential equations that has a transversal Hopf point of codimension two. Applications of the developed theory are made in Liénard-type equations and in Bazykin’s predator-prey system.


---

## Body

## 1. Introduction

The existence of a curve of nonhyperbolic periodic orbits in the bifurcation diagram of a family of differential equations that has a transversal Hopf point of codimension two can be demonstrated with the theories presented in [1, 2]. However, these theories do not allow us to find or even approximate the curve of nonhyperbolic periodic orbits, except in very special cases as in [3]. On the other hand, good approximations to this curve are essential not only to mathematicians, but primarily for engineers, physicists, and other users of mathematics.In general, the curve of nonhyperbolic periodic orbits is obtained by numerical methods as in [4] or through specific softwares such as [5], for instance. An analytical alternative proposed in this paper is to generalize the theory of approximation of periodic orbits of [6], using some results and notations of [1, 2], in order to obtain an approximation to the curve of nonhyperbolic periodic orbits of a family of differential equations that has transversal Hopf bifurcations of codimension two. Furthermore, the theory developed here does not need normal forms of the vector field in the neighborhood of the Hopf points.Article [7], among other cases, treats also the generalized Hopf bifurcation in general as n-dimensional systems. In particular, it provides quadratic asymptotics for the bifurcation parameter values corresponding to the nonhyperbolic limit cycle, and for this cycle itself. Moreover, these asymptotics are implemented into the standard software MATCONT [5], allowing to automatically initialize the continuation of the cycle-saddle-node curve from the generalized Hopf point. However, the authors believe that the constructions presented here are independent and self-contained. More precisely, both articles give an approximation to the curve of nonhyperbolic periodic orbits of a family of differential equations that has transversal Hopf bifurcations of codimension two. Here we present this theory for 2-dimensional systems without the use of normal forms while in [7], the authors present n-dimensional systems using normal forms.This paper is organized as follows. In Section2, the theory of approximation of periodic orbits for vector fields in the plane is developed. The stability of the approximate periodic orbits is discussed in Section 3. In Section 4, applications of the theory in Liénard-type differential equations are made, while applications to the Bazykin's predator-prey system are made in Section 5. Concluding comments about the results obtained here are in Section 6.


## 2. Approximation of Periodic Orbits

Consider a family of the differential equations(1)x′=f(x,ξ),
where f:W×U→ℝ2, W⊂ℝ2 is an open set in ℝ2, f∈𝒞∞(W×U,ℝ2), and ξ=(μ,ν)∈U⊂ℝ2 is the parameter vector. Let (x0(ξ),ξ)∈W×U be an equilibrium point of (1); that is, f(x0(ξ),ξ)=0 for ξ∈U. Suppose the following assumption:  (H1)
the linear part of the vector fieldf:W×U→ℝ2, evaluated at (x0(ξ),ξ) and denoted by A(ξ)=Df(x0(ξ),ξ), has eigenvalues λ and λ¯, with λ(ξ)=γ(ξ)+iη(ξ). For ξ0=(μ0,ν)∈U, γ(ξ0)=0, ∂μγ(ξ0)≠0, and η(ξ0)=ω0(ν)>0, where
(2)∂μγ(ξ0)=∂∂μγ(ξ)|ξ=ξ0.There is no loss of generality in considering thatx0(ξ)=0 for all ξ∈U, (0,0)∈U and μ0=0. Just make a translation of the equilibrium point and of the critical parameter to their origins and adjust in a convenient way the sets W⊂ℝ2 and U⊂ℝ2. By doing this, (1) can be rewritten as
(3)x′=A(ξ)x+G(x,ξ),
where (x,ξ)↦G(x,ξ) is a smooth vector field with Taylor expansion around x=0, starting with second-order terms at least, as follows:
(4)G(x,ξ)=12B(x,x,ξ)+16C(x,x,x,ξ)+124D(x,x,x,x,ξ)+1120E(x,x,x,x,x,ξ)+OG(∥x∥6,ξ),
where
(5)Bi(x,y,ξ)=∑j,k=12‍∂2∂ηj∂ηkGi(η,ξ)|η=0xjyk,Ci(x,y,u,ξ)=∑j,k,l=12‍∂3∂ηj∂ηk∂ηlGi(η,ξ)|η=0xjykul,Di(x,y,u,v,ξ)=∑j,k,l,r=12‍∂4∂ηj∂ηk∂ηl∂ηrGi(η,ξ)|η=0xjykulvr,Ei(x,y,u,v,w,ξ)=∑j,k,l,r,p=12‍∂5∂ηj∂ηk∂ηl∂ηr∂ηpGi(η,ξ)|η=0xjykulvrwp
are the components of symmetric multilinear functions B, C, D, and E.Letq(ξ)∈ℂ2 be an eigenvector corresponding to the eigenvalue λ(ξ), and let p(ξ)∈ℂ2 be an adjoint eigenvector corresponding to the eigenvalue λ¯(ξ) satisfying
(6)A(ξ)q(ξ)=λ(ξ)q(ξ),(7)A(ξ)Tp(ξ)=λ¯(ξ)p(ξ),
and the normalization
(8)〈p(ξ),q(ξ)〉=∑i=12‍p¯i(ξ)qi(ξ)=1,
where 〈·,·〉:ℂ2×ℂ2→ℂ is the standard inner product in ℂ2 and A(ξ)T is the transpose of the matrix A(ξ). The set {q(ξ),q¯(ξ)} is a basis of ℂ2 and the subspace of ℂ2 defined by
(9)ℝ02={(x,y)∈ℂ2:x,y∈ℝ2,y=0}
is isomorphic to the vector space ℝ2. Taking into account the isomorphism between ℝ2 and ℝ02, if (x,0)∈ℝ02, then the notation used is x∈ℝ2. Thus, every vector x∈ℝ2 can be uniquely represented as a linear combination of elements of {q(ξ),q¯(ξ)}; that is, there is z∈ℂ such that
(10)x=zq(ξ)+z¯q¯(ξ).It is easy to show that〈p(ξ),q¯(ξ)〉=0 and z=〈p(ξ),x〉. So (1) can be written as a complex family of differential equations as follows:
(11)z′=g(z,z¯,ξ),
for ∥ξ-ξ0∥ sufficiently small, where g∈𝒞∞(ℂ×ℂ×U,ℂ) and
(12)g(z,z¯,ξ)=λ(ξ)z+〈p(ξ),G(zq(ξ)+z¯q¯(ξ),ξ)〉.
The function (z,z¯,ξ)↦g(z,z¯,ξ) has formal Taylor series
(13)g(z,z¯,ξ)=λ(ξ)z+∑k=2∞‍∑j=0k‍1(k-j)!j!gk-j,j(ξ)zk-jz¯j,
where
(14)gk-j,j(ξ)=∂k∂zk-j∂z¯j〈p(ξ),G(zq(ξ)+z¯q¯(ξ),ξ)〉|z=0,
for k=2,3,… and j=0,…,k.The coefficientsgk-j,j(ξ) for k=2,3,… and j=0,…,k play an important role in the method of approximation of a family of periodic orbits of (1). A simple way to calculate these coefficients, alternative to (14), is through the symmetric multilinear functions. From the symmetric bilinear function (x,y,ξ)↦B(x,y,ξ) and (10), it follows that
(15)B(zq(ξ)+z¯q¯(ξ),zq(ξ)+z¯q¯(ξ),ξ)=B(q(ξ),q(ξ),ξ)z2+2B(q(ξ),q¯(ξ),ξ)zz¯+B(q¯(ξ),q¯(ξ),ξ)z¯2,
and, therefore,
(16)g2,0(ξ)=〈p(ξ),B(q(ξ),q(ξ),ξ)〉,g1,1(ξ)=〈p(ξ),B(q(ξ),q¯(ξ),ξ)〉,g0,2(ξ)=〈p(ξ),B(q¯(ξ),q¯(ξ),ξ)〉.
Similarly, for the symmetric trilinear function (x,y,u,ξ)↦C(x,y,u,ξ),
(17)g3,0(ξ)=〈p(ξ),C(q(ξ),q(ξ),q(ξ),ξ)〉,g2,1(ξ)=〈p(ξ),C(q(ξ),q(ξ),q¯(ξ),ξ)〉,g1,2(ξ)=〈p(ξ),C(q(ξ),q¯(ξ),q¯(ξ),ξ)〉,g0,3(ξ)=〈p(ξ),C(q¯(ξ),q¯(ξ),q¯(ξ),ξ)〉,
and so on for other symmetric multilinear functions.The aim of the theory of approximation of periodic orbits in [6] is to build an approximation for a periodic orbit of the complex differential equation (11), from the solution of the linear differential equation
(18)z′=λ(ξ)z
for ξ=ξ0. This linear differential equation has the solution
(19)z(t)=z0eλ(ξ)t,
where z0∈ℂ. For ξ=ξ0, it follows that
(20)z(t)=z0eiω0(ν)t
and making the change in time s=ω0(ν)t, this solution is periodic of period 2π in the variable s. To formalize the method, consider the functions (ϵ,ν)↦μ=ϕ(ϵ,ν), (ϵ,ν)↦ω(ϵ,ν) and the change of coordinates and time
(21)z(t)=w(s,ϵ,ν),s=ω(ϵ,ν)t,ω(ξ0)=ω0(ν),
where
(22)ϵ=12π∫02π‍e-isw(s,ϵ,ν)ds.Note that the parameterϵ, as defined in (22), is a complex number or, more precisely, a complex function whose independent variable is ν. However, it is possible, through a change of variables, to consider the parameter ϵ as a real number. In fact, as
(23)ϵ=εeiϕ=12π∫02π‍e-isw(s,ϵ,ν)ds,
it follows that
(24)ε=|ϵ|=e-iϕ2π∫02π‍e-isw(s,ϵ,ν)ds=12π∫02π‍e-i(s+ϕ)w(s,ϵ,ν)ds.
Thus, making the change of variable u=s+ϕ in (24) and setting (u,ϵ,ν)↦w~(u,ϵ,ν)=w(u-ϕ,ϵ,ν),
(25)ε=|ϵ|=12π∫ϕ2π+ϕ‍e-iuw(u-ϕ,ϵ,ν)du=12π∫02π‍e-iuw~(u,ϵ,ν)du,
since the function (u,ϵ,ν)↦e-iuw~(u,ϵ,ν) is periodic of period 2π in the variable s. Therefore, by (25), the parameter ϵ as defined in (22) will be considered a real parameter.The generalization of the theory of approximation of periodic orbits introduced in [6] consists in achieving an approximation to the two-parameter family of periodic orbits
(26){(s,ϵ,ν)∈ℝ×Uϵ↦w(s,ϵ,ν)∈ℂ:(ϵ,ν)∈Uϵ},
where Uϵ={(ϵ,ν)∈ℝ2:(ϕ(ϵ,ν),ν)∈U}.The change in times=ω(ϵ,ν)t is essential, since the period of the family of periodic orbits (26) is unknown and, therefore, the change in time is used only to provide an approximation of the known period 2π for the family of periodic orbits (26). If (ϵ,ν)↦T(ϵ,ν) denotes the period of the family of periodic orbits, then
(27)ω(ϵ,ν)=2πT(ϵ,ν).
In other words, the knowledge of the function (ϵ,ν)↦ω(ϵ,ν) completely determines the period of the family of periodic orbits of (26).By changing the coordinates and time (21) and applying the chain rule, the complex differential equation (11) is rewritten as
(28)ω(ϵ,ν)ddsw(s,ϵ,ν)=g(w(s,ϵ,ν),w¯(s,ϵ,ν),ϕ(ϵ,ν),ν).Approximations to the functions(s,ϵ,ν)↦w(s,ϵ,ν), (ϵ,ν)↦μ=ϕ(ϵ,ν) and (ϵ,ν)↦ω(ϵ,ν) are obtained through (28) and the formal power series
(29)(w(s,ϵ,ν)ϕ(ϵ,ν)ω(ϵ,ν)-ω0(ν))=∑k=1∞‍1k!(wk(s,ν)μk(ν)ωk(ν))ϵk.A property of the terms of the sequence{wk(s,ν)}k∈ℕ, widely used in this theory of approximation of periodic orbits of vector fields in ℝ2, is obtained in Proposition 1.Proposition 1.
Each term of the sequence{wk(s,ν)}k∈ℕ satisfies
(30)[wk]=12π∫02π‍e-iswk(s,ν)ds={1,k=10,k=2,3,….Proof.
Setting𝒲={w:ℝ×Uϵ→ℂ:w∈𝒞∞(ℝ×Uϵ,ℂ)}, the proof is an immediate consequence of the definition of linear map
(31)[]:𝒲⟶ℝw↦[w]=12π∫02π‍e-isw(s,ϵ,ν)ds
and the formal power series in the variable ϵ of the function (s,ϵ,ν)↦w(s,ϵ,ν), because
(32)ϵ=12π∫02π‍e-is(∑k=1∞‍1k!wk(s,ν)ϵk)ds=∑k=1∞‍1k!(12π∫02π‍e-iswk(s,ν)ds)ϵk=∑k=1∞‍1k![wk]ϵk.The terms of the sequences{wk(s,ν)}k∈ℕ, {μk(ν)}k∈ℕ and {ωk(ν)}k∈ℕ are determined through a process that involves analysis of the powers in ϵ, obtained by replacing (29) into the differential equation (28). Note that, for k=2,3,… and j=0,…,k, the coefficients of powers in ϵ are determined by expanding the composition (ϵ,ν)↦gk-j,j(ϕ(ϵ,ν),ν) in the Taylor series around ϵ=0. Such an expansion, up to the fifth-order terms, is of the following form:
(33)gk-j,j(ϕ(ϵ,ν),ν)=gk-j,j(ξ0)+μ1(ν)∂μgk-j,j(ξ0)ϵ+12(μ2(ν)∂μgk-j,j(ξ0)+16(μ3(ν)∂μgk-j,j(ξ0)+μ1(ν)2∂μ2gk-j,j(ξ0))ϵ2+3μ1(ν)μ2(ν)∂μ2gk-j,j(ξ0)+μ1(ν)3∂μ3gk-j,j(ξ0))ϵ3+124(μ4(ν)∂μgk-j,j(ξ0)+3μ2(ν)2∂μ2gk-j,j(ξ0)+4μ1(ν)μ2(ν)∂μgk-j,j(ξ0)+6μ1(ν)2μ2(ν)×∂μ3gk-j,j(ξ0)+μ1(ν)4∂μ4gk-j,j(ξ0))ϵ4+1120(μ5(ν)∂μgk-j,j(ξ0)+(10μ2(ν)μ3(ν)+5μ1(ν)μ4(ν))∂μ2gk-j,j(ξ0)+(15μ1(ν)μ2(ν)2+10μ1(ν)2μ3(ν))×∂μ3gk-j,j(ξ0)+10μ1(ν)3μ2(ν)∂μ4gk-j,j(ξ0)+μ1(ν)5∂μ5gk-j,j(ξ0))ϵ5+Og(ϵ6,ν),
with the same being valid for the composition (ϵ,ν)↦λ(ϕ(ϵ,ν),ν)=γ(ϕ(ϵ,ν),ν)+iη(ϕ(ϵ,ν),ν).The coefficient of the term inϵ leads to the following boundary value problem:
(34)w1′(s,ν)-iw1(s,ν)=0,w1(s,ν)=w1(s+2π,ν).
The solution of the differential equation in (34) is
(35)w1(s,ν)=C1eis,
and as by Proposition 1, [w1]=1, it follows that
(36)1=[w1]=[C1eis]=C1[eis]=C1.
Thus,
(37)w1(s,ν)=eis,
which is a periodic function of period 2π in the variable s. In fact, the terms of the sequence {wk(s,ν)}k∈ℕ are solutions of certain boundary value problems which appear when (29) is substituted into the differential equation (28). For each k=1,2,…, the boundary value problem is of the following form:
(38)wk+1′(s,ν)-iwk+1(s,ν)=Hk+1(s,μk(ν),ωk(ν)),wk+1(s,ν)=wk+1(s+2π,ν),
where Hk+1(s,μk(ν),ωk(ν))=Hk+1(s+2π,μk(ν),ωk(ν)).The following theorem guarantees the existence of the solutions of the boundary value problem (38).Theorem 2.
For eachk=1,2,…, the boundary value problem (38) admits solution if and only if
(39)[Hk+1]=0.Proof.
For fixedk=1,2,…, suppose that (s,ν)↦φk+1(s,ν) is the solution of (38). Thus,
(40)12π∫02π‍e-is(φk+1′(s,ν)-iφk+1(s,ν))ds=12π∫02π‍e-isHk+1(s,μk(ν),ωk(ν))ds
and by integrating by parts the left member of (40), it follows that [Hk+1]=0. Now suppose that [Hk+1]=0 for a fixed k=1,2,…. The general solution (s,ν)↦φk+1(s,ν) of the differential equation in (38) is of the following form:
(41)φk+1(s,ν)=eisφ0k+1+eis∫0s‍e-iζHk+1(ζ,μk(ν),ωk(ν))dζ,
where φ0k+1=φk+1(0,ν). This solution will be periodic of period 2π if φk+1(0,ν)=φ0k+1=φk+1(2π,ν); that is, if
(42)φk+1(0,ν)=φk+1(2π,ν)=φ0k+1+∫02π‍e-iζHk+1(ζ,μk(ν),ωk(ν))dζ=φ0k+1+2π[Hk+1].
Thus, using the hypothesis [Hk+1]=0, it follows that φk+1(s,ν)=φk+1(s+2π,ν), and, therefore, for each fixed k=1,2,…, the function (s,ν)↦φk+1(s,ν) is the solution of the boundary value problem (38).The previous theorem shows that, fork=1,2,…, the solution of (38) is obtained by solving the differential equation in (38) with conditions [Hk+1]=0 and [wk+1]=0.Continuing the process and using the result (37), the coefficient of the term in ϵ2 provides the boundary value problem
(43)w2′(s,ν)-iw2(s,ν)=H2(s,μ1(ν),ω1(ν)),w2(s,ν)=w2(s+2π,ν),
where
(44)H2(s,μ1(ν),ω1(ν))=1ω0(ν)(e2isg2,0(ξ0)+2g1,1(ξ0)+e-2isg0,2(ξ0)+2eis(μ1(ν)∂μλ(ξ0)-iω1(ν))).
By applying Theorem 2 to the function (s,μ1(ν),ω1(ν))↦H2(s,μ1(ν),ω1(ν)), it follows that
(45)[H2]=μ1(ν)(∂μγ(ξ0)+i∂μη(ξ0))-iω1(ν)=0,
and by separating the real and imaginary parts of (45), we have μ1(ν)=0 and ω1(ν)=0. Under these conditions, Theorem 2 guarantees the existence of the solution of the boundary value problem (43), which is given by
(46)w2(s,ν)=13iω0(ν)(3e2isg2,0(ξ0)-6g1,1(ξ0)-e-2isg0,2(ξ0)).For the coefficient of the term inϵ3, we have the following boundary value problem:
(47)w3′(s,ν)-iw3(s,ν)=H3(s,μ2(ν),ω2(ν)),w3(s,ν)=w3(s+2π,ν),
with
(48)H3(s,μ2(ν),ω2(ν))=3ω0(ν)(H33(ξ0)e3is+H31(ξ0)eis+H3-1(ξ0)e-is+H3-3(ξ0)e-3is),
where
(49)H33(ξ0)=13g3,0(ξ0)-ig2,0(ξ0)2ω0(ν)-ig1,1(ξ0)g¯0,2(ξ0)ω0(ν),H31(ξ0)=μ2(ν)∂μλ(ξ0)-iω2(ν)+G2,1(ξ0),H3-1(ξ0)=2ig1,1(ξ0)2ω0(ν)+ig¯2,0(ξ0)g1,1(ξ0)ω0(ν)+g1,2(ξ0)+ig2,0(ξ0)g0,2(ξ0)3ω0(ν)-2ig0,2(ξ0)g¯1,1(ξ0)ω0(ν),H3-3(ξ0)=13g0,3(ξ0)+ig1,1(ξ0)g0,2(ξ0)3ω0(ν)+ig0,2(ξ0)g¯2,0(ξ0)ω0(ν)
and the coefficient G2,1(ξ0) is defined as
(50)G2,1(ξ0)=ig2,0(ξ0)g1,1(ξ0)+ω0(ν)g2,1(ξ0)ω0(ν)-2i|g1,1(ξ0)|2ω0(ν)-i|g0,2(ξ0)|23ω0(ν).
Expression (50) is identical to the one given in [1].Continuing the process and calculating[H3], it follows that
(51)[H3]=μ2(ν)(∂μγ(ξ0)+i∂μη(ξ0))-iω2(ν)+G2,1(ξ0)=0.
And by separating the real and imaginary parts,
(52)μ2(ν)=-Re(G2,1(ξ0))∂μγ(ξ0),(53)ω2(ν)=Im(G2,1(ξ0))+μ2(ν)∂μη(ξ0).Once the coefficientsμ2(ν) and ω2(ν) are determined, the solution of the boundary value problem (47) has the following form:
(54)w3(s,ν)=14ω0(ν)2(w33(ξ0)e3is+w3-1(ξ0)e-is+w3-3(ξ0)e-3is),
where
(55)w33(ξ0)=-6g2,0(ξ0)2-2iω0(ν)g3,0(ξ0)-2g1,1(ξ0)g¯0,2(ξ0),w3-1(ξ0)=-12g1,1(ξ0)2-6g¯2,0(ξ0)g1,1(ξ0)+6iω0(ν)g1,2(ξ0)-2g0,2(ξ0)g2,0(ξ0)+12g0,2(ξ0)g¯1,1(ξ0),w3-3(ξ0)=iω0(ν)g0,3(ξ0)-g0,2(ξ0)g1,1(ξ0)-3g0,2(ξ0)g¯2,0(ξ0).Definition 3.
The real number(56)l1(ξ0)=12Re(G2,1(ξ0))=12ω0(ν)Re(ig2,0(ξ0)g1,1(ξ0)+ω0(ν)g2,1(ξ0))
is called the first Lyapunov coefficient.Remark 4.
A Hopf point of codimension one for (1) is an equilibrium point (0,ξ0)∈W×U, with ξ0=(0,ν), such that A(ξ0)=Df(0,ξ0) has eigenvalues λ and λ¯, with λ(ξ0)=γ(ξ0)+iη(ξ0), γ(ξ0)=0, η(ξ0)=ω0(ν)>0, and the first Lyapunov coefficient, l1(ξ0)∈ℝ, is different from zero. A transversal Hopf point of codimension one is a Hopf point of codimension one such that
(57)∂μγ(ξ0)≠0,
for ξ0∈U. In a neighborhood of a transversal Hopf point of codimension one (0,ξ0)∈W×U, with l1(ξ0)≠0, the dynamic behavior of differential equation (1) is orbitally topologically equivalent to the following complex normal form:
(58)w′=(α+i)w+sw|w|2,
where s=sign(l1(ξ0)). The sign of the first Lyapunov coefficient determines the stability of the family of periodic orbits that appears (or disappears) from (0,ξ0)∈W×U as will be seen later.Whenl1(ξ1)=0, for ξ1=(0,0)∈U, there is the possibility of Hopf bifurcations of codimension two. In this case, it is necessary to obtain an expression for G3,2(ξ1).Applying Theorem2 to the boundary value problem for k=3, it follows that μ3(ν)=0, ω3(ν)=0 and
(59)w4(s,ν)=145ω0(ν)3(w44(ξ0)e4is+w42(ξ0)e2is+w40(ξ0)+w4-2(ξ0)e-2is+w4-4(ξ0)e-4is),
where
(60)w44(ξ0)=135ig2,0(ξ0)3-120ω0(ν)g3,0(ξ0)g2,0(ξ0)+105ig1,1(ξ0)g¯0,2(ξ0)g2,0(ξ0)+5ig0,2(ξ0)g¯0,2(ξ0)2-15iω0(ν)2g4,0(ξ0)-30ω0(ν)g2,1(ξ0)g¯0,2(ξ0)-15ω0(ν)g1,1(ξ0)g¯0,3(ξ0)+15ig1,1(ξ0)g¯0,2(ξ0)g¯1,1(ξ0),w42(ξ0)=-180ig3,1(ξ0)ω0(ν)2-270iμ2(ν)∂μg2,0(ξ0)ω0(ν)2+450g1,1(ξ0)g3,0(ξ0)ω0(ν)-180g1,2(ξ0)g¯0,2(ξ0)ω0(ν)-45g0,2(ξ0)g¯0,3(ξ0)ω0(ν)-540g2,1(ξ0)g¯1,1(ξ0)ω0(ν)-270g1,1(ξ0)g¯1,2(ξ0)ω0(ν)+270μ2(ν)g2,0(ξ0)∂μγ(ξ0)ω0(ν)+270iμ2(ν)g2,0(ξ0)∂μη(ξ0)ω0(ν)+270ig1,1(ξ0)g2,0(ξ0)2+540ig1,1(ξ0)g¯1,1(ξ0)2-630ig1,1(ξ0)2g¯0,2(ξ0)-45ig0,2(ξ0)g2,0(ξ0)g¯0,2(ξ0)-270ig1,1(ξ0)g2,0(ξ0)g¯1,1(ξ0)+225ig0,2(ξ0)g¯0,2(ξ0)g¯1,1(ξ0)+90ig1,1(ξ0)g¯0,2(ξ0)g¯2,0(ξ0),w40(ξ0)=270ig2,2(ξ0)ω0(ν)2+540iμ2(ν)∂μg1,1(ξ0)ω0(ν)2-1080g1,1(ξ0)g2,1(ξ0)ω0(ν)-90g0,2(ξ0)g3,0(ξ0)ω0(ν)+90g0,3(ξ0)g¯0,2(ξ0)ω0(ν)+1080g1,2(ξ0)g¯1,1(ξ0)ω0(ν)+270g0,2(ξ0)g¯1,2(ξ0)ω0(ν)-270g2,1(ξ0)g¯2,0(ξ0)ω0(ν)-540μ2(ν)g1,1(ξ0)∂μγ(ξ0)ω0(ν)-540iμ2(ν)g1,1(ξ0)∂μη(ξ0)ω0(ν)-1080ig0,2(ξ0)g¯1,1(ξ0)2-1080ig1,1(ξ0)2g2,0(ξ0)+570ig0,2(ξ0)g1,1(ξ0)g¯0,2(ξ0)+1080ig1,1(ξ0)2g¯1,1(ξ0)+270ig0,2(ξ0)g2,0(ξ0)g¯1,1(ξ0),w4-2(ξ0)=-180ig1,1(ξ0)3-270ig¯2,0(ξ0)g1,1(ξ0)2-90ig¯2,0(ξ0)2g1,1(ξ0)-270ω0(ν)g1,2(ξ0)g1,1(ξ0)-165ig0,2(ξ0)g2,0(ξ0)g1,1(ξ0)+330ig0,2(ξ0)g¯1,1(ξ0)g1,1(ξ0)-30ω0(ν)g¯3,0(ξ0)g1,1(ξ0)+60iω0(ν)2g1,3(ξ0)-15ω0(ν)g0,3(ξ0)g2,0(ξ0)-120ω0(ν)g0,2(ξ0)g2,1(ξ0)+20ig0,2(ξ0)2g¯0,2(ξ0)+180ω0(ν)g0,3(ξ0)g¯1,1(ξ0)-180ω0(ν)g1,2(ξ0)g¯2,0(ξ0)-45ig0,2(ξ0)g2,0(ξ0)g¯2,0(ξ0)+180ig0,2(ξ0)g¯1,1(ξ0)g¯2,0(ξ0)-90μ2(ν)ω0(ν)g0,2(ξ0)∂μγ(ξ0)-90iμ2(ν)ω0(ν)g0,2(ξ0)∂μη(ξ0)+90iμ2(ν)ω0(ν)2∂μg0,2(ξ0),w4-4(ξ0)=9ig0,4(ξ0)ω0(ν)2-9g0,3(ξ0)g1,1(ξ0)ω0(ν)-18g0,2(ξ0)g1,2(ξ0)ω0(ν)-9ig0,2(ξ0)g1,1(ξ0)2-54g0,3(ξ0)g¯2,0(ξ0)ω0(ν)-18g0,2(ξ0)g¯3,0(ξ0)ω0-81ig0,2(ξ0)g¯2,0(ξ0)2-3ig0,2(ξ0)2g2,0(ξ0)-18ig0,2(ξ0)2g¯1,1(ξ0)-45ig0,2(ξ0)g1,1(ξ0)g¯2,0(ξ0).From the boundary value problem fork=5, it follows that
(61)μ4(ν)=-(2Re(G3,2(ξ0))+6μ2(ν)Re(∂μG2,1(ξ0))+3μ2(ν)2∂μ2γ(ξ0))×(∂μγ(ξ0))-1,(62)ω4(ν)=2Im(G3,2(ξ0))+μ4(ν)∂μη(ξ0)+6μ2(ν)Im(∂μG2,1(ξ0))+3μ2(ν)2∂μ2η(ξ0),
where
(63)∂μG2,1(ξ0)=3g1,1(ξ0)g2,0(ξ0)∂μγ(ξ0)ω0(ν)2+g0,2(ξ0)g¯0,2(ξ0)∂μγ(ξ0)9ω0(ν)2+2g1,1(ξ0)g¯1,1(ξ0)∂μγ(ξ0)ω0(ν)2-ig1,1(ξ0)g2,0(ξ0)∂μη(ξ0)ω0(ν)2+ig0,2(ξ0)g¯0,2(ξ0)∂μη(ξ0)3ω0(ν)2+2ig1,1(ξ0)g¯1,1(ξ0)∂μη(ξ0)ω0(ν)2-ig¯0,2(ξ0)∂μg0,2(ξ0)3ω0(ν)+ig2,0(ξ0)∂μg1,1(ξ0)ω0(ν)-2ig¯1,1(ξ0)∂μg1,1(ξ0)ω0(ν)+ig1,1(ξ0)∂μg2,0(ξ0)ω0(ν)+∂μg2,1(ξ0)-ig0,2(ξ0)∂μg¯0,2(ξ0)3ω0(ν)-2ig1,1(ξ0)∂μg¯1,1(ξ0)ω0(ν),(64)G3,2(ξ0)=12ig¯0,2(ξ0)g1,1(ξ0)3ω0(ν)3-12ig¯1,1(ξ0)2g1,1(ξ0)2ω0(ν)3-4g3,0(ξ0)g1,1(ξ0)2ω0(ν)2+12ig2,0(ξ0)g¯1,1(ξ0)g1,1(ξ0)2ω0(ν)3+12g¯1,2(ξ0)g1,1(ξ0)2ω0(ν)2-3ig¯0,2(ξ0)g¯2,0(ξ0)g1,1(ξ0)2ω0(ν)3+4ig3,1(ξ0)g1,1(ξ0)ω0(ν)+9g1,2(ξ0)g¯0,2(ξ0)g1,1(ξ0)ω0(ν)2+17ig0,2(ξ0)g2,0(ξ0)g¯0,2(ξ0)g1,1(ξ0)4ω0(ν)3+31g0,2(ξ0)g¯0,3(ξ0)g1,1(ξ0)12ω0(ν)2-175ig0,2(ξ0)g¯0,2(ξ0)g¯1,1(ξ0)g1,1(ξ0)12ω0(ν)3+12g2,1(ξ0)g¯1,1(ξ0)g1,1(ξ0)ω0(ν)2+6ig¯1,1(ξ0)2g¯2,0(ξ0)g1,1(ξ0)ω0(ν)3-g3,0(ξ0)g¯2,0(ξ0)g1,1(ξ0)ω0(v)2-6ig2,0(ξ0)g¯1,1(ξ0)g¯2,0(ξ0)g1,1(ξ0)ω0(ν)3+3g2,0(ξ0)g¯2,1(ξ0)g1,1(ξ0)ω0(ν)2-3ig¯2,2(ξ0)g1,1(ξ0)ω0(ν)-6g¯1,1(ξ0)g¯2,1(ξ0)g1,1(ξ0)ω0(ν)2-g¯0,2(ξ0)g¯3,0(ξ0)g1,1(ξ0)ω0(ν)2+8ig0,2(ξ0)g¯1,1(ξ0)3ω0(ν)3-2ig0,2(ξ0)2g¯0,2(ξ0)29ω0(ν)3-12g1,2(ξ0)g¯1,1(ξ0)2ω0(ν)2-6ig0,2(ξ0)g2,0(ξ0)g¯1,1(ξ0)2ω0(ν)3+ig1,2(ξ0)g3,0(ξ0)ω0(ν)-g0,2(ξ0)g2,0(ξ0)g3,0(ξ0)3ω0(ν)2+g3,2(ξ0)+ig0,2(ξ0)g4,0(ξ0)3ω0(ν)-ig1,3(ξ0)g¯0,2(ξ0)ω0(ν)+g0,3(ξ0)g2,0(ξ0)g¯0,2(ξ0)4ω0(ν)2+g0,2(ξ0)g2,1(ξ0)g¯0,2(ξ0)ω0(ν)2-ig0,3(ξ0)g¯0,3(ξ0)4ω0(ν)+ig0,2(ξ0)g2,0(ξ0)2g¯1,1(ξ0)ω0(ν)3+3g1,2(ξ0)g2,0(ξ0)g¯1,1(ξ0)ω0(ν)2-6ig2,2(ξ0)g¯1,1(ξ0)ω0(ν)+8g0,2(ξ0)g3,0(ξ0)g¯1,1(ξ0)3ω0(ν)2-9g0,3(ξ0)g¯0,2(ξ0)g¯1,1(ξ0)4ω0(ν)2-3ig1,2(ξ0)g¯1,2(ξ0)ω0(ν)+g0,2(ξ0)g2,0(ξ0)g¯1,2(ξ0)ω0(ν)2-6g0,2(ξ0)g¯1,1(ξ0)g¯1,2(ξ0)ω0(ν)2-2ig0,2(ξ0)g¯1,3(ξ0)3ω0(ν)+ig3,1(ξ0)g¯2,0(ξ0)ω0(ν)-ig0,2(ξ0)g2,0(ξ0)g¯0,2(ξ0)g¯2,0(ξ0)4ω0(ν)3+g0,2(ξ0)g¯0,3(ξ0)g¯2,0(ξ0)12ω0(ν)2+23ig0,2(ξ0)g¯0,2(ξ0)g¯1,1(ξ0)g¯2,0(ξ0)12ω0(ν)3-2g0,2(ξ0)g¯0,2(ξ0)g¯2,1(ξ0)3ω0(ν)2.Rewriting the coefficientG3,2(ξ0) in a convenient way, expression (64) is exactly the one that appears in [1].Definition 5.
The real number(65)l2(ξ0)=112Re(G3,2(ξ0)),
where G3,2(ξ0) is given in (64), is called the second Lyapunov coefficient.Remark 6.
A Hopf point of codimension two for (1) is an equilibrium point (0,ξ1)∈W×U, where ξ1=(0,0), that satisfies the definition of a point Hopf of codimension one, except that l1(ξ1)=0. Moreover, it satisfies an additional condition; the second Lyapunov coefficient l2(ξ1) is nonzero. A Hopf point of codimension two is transversal if
(66)∂μγ(ξ1)Re(∂νG2,1(ξ1))≠0.
In a neighborhood of a transversal Hopf point of codimension two (0,ξ1)∈W×U, with l2(ξ1)≠0, the dynamic behavior of differential equation (1) is orbitally topologically equivalent to the following complex normal form:
(67)w′=(α+i)w+βw|w|2+sw|w|4,
where s=sign(l2(ξ1)). In the bifurcation diagram of (67), there exists a curve of nonhyperbolic periodic orbits that has the exact representations
(68)Γ(ϵ)=(sϵ4,-2sϵ2),
as a curve parameterized by ϵ or as a graph of the function
(69)α=Λ(β)=14sβ2,
for β≥0.The function(s,ν)↦w5(s,ν) will not be shown here because it is a long expression and it is not necessary in this work. In many results in this section and, particularly in (63), the following expressions ∂μγ(ξ0), ∂μη(ξ0), ∂μ2γ(ξ0), ∂μ2η(ξ0), ∂μg2,0(ξ0), ∂μg1,1(ξ0), ∂μg0,2(ξ0), and ∂μg2,1(ξ0) appear. These expressions are calculated according to Propositions 7 and 8.Proposition 7.
Consider the differential equation (1) with an equilibrium point (0,ξ)∈W×U, such that the linear part of the map (x,ξ)↦f(x,ξ), evaluated at (0,ξ0), A(ξ0)=Df(0,ξ0), has eigenvalues λ and λ¯, where λ(ξ0)=γ(ξ0)+iη(ξ0), γ(ξ0)=0 and η(ξ0)=ω0(ν)>0. Let also q(ξ)∈ℂ2 be an eigenvector corresponding to the eigenvalue λ(ξ), and let p(ξ)∈ℂ2 be an adjoint eigenvector corresponding to the eigenvalue λ¯(ξ), satisfying (6), (7), and (8). The following statements hold. (a)
The vector∂μq(ξ0)∈ℂ2is the solution of the following nonsingular 3-dimensional system:(70)(iω0(ν)I2-A(ξ0)q(ξ0)p¯(ξ0)0)(∂μq(ξ0)s)=(R2(ξ0)0),with the condition〈p(ξ0),∂μq(ξ0)〉=0, where(71)R2(ξ0)=(∂μA(ξ0)-∂μλ(ξ0)I2)q(ξ0).(b)
The vector∂μp(ξ0)∈ℂ2is the solution of the following nonsingular 3-dimensional system:(72)(-(iω0(ν)I2+AT(ξ0))p(ξ0)q¯(ξ0)0)(∂μp(ξ0)s)=(R¯2(ξ0)0),with the condition〈q(ξ0),∂μp(ξ0)〉=0, where(73)R¯2(ξ0)=(∂μAT(ξ0)-∂μλ¯(ξ0)I2)p(ξ0).(c)
The partial derivative with respect toμof the real part of the eigenvalueλ(ξ), evaluated atξ=ξ0, is given by(74)∂μγ(ξ0)=Re(〈p(ξ0),∂μA(ξ0)q(ξ0)〉).(d)
The partial derivative with respect toμof the imaginary part of the eigenvalueλ(ξ), evaluated atξ=ξ0, is given by(75)∂μη(ξ0)=Im(〈p(ξ0),∂μA(ξ0)q(ξ0)〉).(e)
The second-order partial derivative with respect toμof the real part of the eigenvalueλ(ξ), evaluated atξ=ξ0, is given by(76)∂μ2γ(ξ0)=Re(〈p(ξ0),∂μ2A(ξ0)q(ξ0)+2(∂μA(ξ0)-∂μλ(ξ0)I2)∂μq(ξ0)〉).(f)
The second-order partial derivative with respect toμof the imaginary part of the eigenvalueλ(ξ), evaluated atξ=ξ0, is given by(77)∂μ2η(ξ0)=Im(〈p(ξ0),∂μ2A(ξ0)q(ξ0)+2(∂μA(ξ0)-∂μλ(ξ0)I2)∂μq(ξ0)〉).Proof.
Differentiating (6) with respect to the parameter μ and evaluating at ξ=ξ0, we have
(78)∂μA(ξ0)q(ξ0)+A(ξ0)∂μq(ξ0)=∂μλ(ξ0)q(ξ0)+λ(ξ0)∂μq(ξ0).
Using the hypotheses, the previous equation is rewritten as
(79)(iω0(ν)I2-A(ξ0))∂μq(ξ0)=(∂μA(ξ0)-∂μλ(ξ0))q(ξ0).
Taking the inner product of p(ξ0)∈ℂ2 on both sides of the above equation and using (8), it follows that
(80)0=〈p(ξ0),(iω0(ν)I2-A(ξ0))∂μq(ξ0)〉=〈p(ξ0),∂μA(ξ0)q(ξ0)〉-∂μλ(ξ0).
Items (a), (c), and (d) follow from the above equation, the Fredholm alternative (see [1]), and the results of [8]. The proof of part (b) is equal to the previous proof; that is, it is sufficient to differentiate (7) with respect to the parameter μ and to evaluate at ξ=ξ0. The proofs of items (e) and (f) consist of calculating the second-order partial derivative of (6) with respect to the parameter μ, evaluated at ξ=ξ0, and to use the Fredholm alternative.Proposition 8.
Consider the coefficients of the formal Taylor series of the map(z,z¯,ξ)↦g(z,z¯,ξ),
(81)g2,0(ξ)=〈p(ξ),B(q(ξ),q(ξ),ξ)〉,g1,1(ξ)=〈p(ξ),B(q(ξ),q¯(ξ),ξ)〉,g0,2(ξ)=〈p(ξ),B(q¯(ξ),q¯(ξ),ξ)〉,g2,1(ξ)=〈p(ξ),C(q(ξ),q(ξ),q¯(ξ),ξ)〉.
The following statements hold.  (a)
The partial derivative with respect toμof the coefficientg2,0(ξ), evaluated atξ=ξ0, is(82)∂μg2,0(ξ0)=〈∂μp(ξ0),B(q(ξ0),q(ξ0),ξ0)〉+〈p(ξ0),2B(q(ξ0),∂μq(ξ0),ξ0)+∂μB(q(ξ0),q(ξ0),ξ0)〉.(b)
The partial derivative with respect toμof the coefficientg1,1(ξ), evaluated atξ=ξ0, is given by(83)∂μg1,1(ξ0)=〈∂μp(ξ0),B(q(ξ0),q¯(ξ0),ξ0)〉+〈p(ξ0),B(∂μq(ξ0),q¯(ξ0),ξ0)+B(q(ξ0),∂μq¯(ξ0),ξ0)+∂μB(q(ξ0),q¯(ξ0),ξ0)〉.(c)
The partial derivative with respect toμof the coefficientg0,2(ξ), evaluated atξ=ξ0, is obtained as(84)∂μg0,2(ξ0)=〈∂μp(ξ0),B(q¯(ξ0),q¯(ξ0),ξ0)〉+〈p(ξ0),2B(q¯(ξ0),∂μq¯(ξ0),ξ0)+∂μB(q¯(ξ0),q¯(ξ0),ξ0)〉.(d)
The partial derivative with respect toμof the coefficientg2,1(ξ), evaluated atξ=ξ0, is calculated as(85)∂μg2,1(ξ0)=〈∂μp(ξ0),C(q(ξ0),q(ξ0),q¯(ξ0),ξ0)〉+〈p(ξ0),2C(q(ξ0),∂μq(ξ0),q¯(ξ0),ξ0)+C(q(ξ0),q(ξ0),∂μq¯(ξ0),ξ0)+∂μC(q(ξ0),q(ξ0),q¯(ξ0),ξ0)〉.Proof.
Observing how the symmetric multilinear functions are defined, the proofs of items (a) to (d) consist in differentiating each expression in (81) with respect to the parameter μ and evaluating at ξ=ξ0.The theory built up to this point approximates a family of periodic orbits of the complex differential equation (11). In the hypotheses of the Hopf bifurcation, if (ω(ϵ,ν)t,ϵ,ν)↦w(ω(ϵ,ν)t,ϵ,ν) is a family of periodic orbits of (11), then (ω(ϵ,ν)t,ϵ,ν)↦u(ω(ϵ,ν)t,ϵ,ν) is a family of periodic orbits associated with the differential equation (1), where
(86)u(ω(ϵ,ν)t,ϵ,ν)=w(ω(ϵ,ν)t,ϵ,ν)q(ϕ(ϵ,ν),ν)+w¯(ω(ϵ,ν)t,ϵ,ν)q¯(ϕ(ϵ,ν),ν),
or, in a more simple way,
(87)u(s,ϵ,ν)=w(s,ϵ,ν)q(ϕ(ϵ,ν),ν)+w¯(s,ϵ,ν)q¯(ϕ(ϵ,ν),ν).The family of periodic orbits(s,ϵ,ν)↦u(s,ϵ,ν) has formal Taylor series around ϵ=0 of the following form:
(88)u(s,ϵ,ν)=∑k=1∞‍1k!uk(s,ν)ϵk
and the theory developed previously and the Taylor expansion of (87), around ϵ=0, show that
(89)u1(s,ν)=q(ξ0)w1(s,ν)+q¯(ξ0)w¯1(s,ν),u2(s,ν)=q(ξ0)w2(s,ν)+q¯(ξ0)w¯2(s,ν),u3(s,ν)=q(ξ0)w3(s,ν)+q¯(ξ0)w¯3(s,ν)+3μ2(ν)(w1(s,ν)∂μq(ξ0)+w¯1(s,ν)∂μq¯(ξ0)),u4(s,ν)=q(ξ0)w4(s,ν)+q¯(ξ0)w¯4(s,ν)+6μ2(ν)(w2(s,ν)∂μq(ξ0)+w¯2(s,ν)∂μq¯(ξ0)).The stability of the approximate family of periodic orbits is studied in the next section by means of the Floquet exponent.


## 3. Stability of the Family of Periodic Orbits

According to the Floquet theory (see [9]), the stability of a periodic orbit can be determined through the characteristic exponent that, in this context and for differential equations in ℝ2, is a function (ϵ,ν)↦χ(ϵ,ν) such that
(90)χ(ϵ,ν)=1T(ϵ,ν)∫0T(ϵ,ν)‍Tr(ℳ(ω(ϵ,ν)t,ϵ,ν))dt,
where ℳ(ω(ϵ,ν)t,ϵ,ν)=Df(u(ω(ϵ,ν)t,ϵ,ν),ϕ(ϵ,ν),ν). The next proposition provides a simple way to compute (90) in terms of the map (z,z¯,ξ)↦g(z,z¯,ξ).Proposition 9.
Through a change in times=ω(ϵ,ν)t, the characteristic exponent associated with the differential equation z′=g(z,z¯,ξ) is of the following form:
(91)χ(ϵ,ν)=12π∫02π‍ℋ(s,ϵ,ν)ds,
where
(92)ℋ(s,ϵ,ν)=∂∂wg(w(s,ϵ,ν),w¯(s,ϵ,ν),ϕ(ϵ,ν),ν)+∂∂w¯g¯(w(s,ϵ,ν),w¯(s,ϵ,ν),ϕ(ϵ,ν),ν).Proof.
The differential equation (1) can be written as (11), where (z,z¯,ξ)↦g(z,z¯,ξ)=g1(z1,z2,ξ)+ig2(z1,z2,ξ),
(93)g1(w1,w2,ξ)=12(g(w,w¯,ξ)+g¯(w,w¯,ξ)),g2(w1,w2,ξ)=-i2(g(w,w¯,ξ)-g¯(w,w¯,ξ)),
and z=z1+iz2. Thus, through the changes z(t)=w(s,ϵ,ν) and s=ω(ϵ,ν)t, the characteristic exponent (90) can be rewritten as
(94)χ(ϵ,ν)=12π∫02π‍(∂∂w1g1(w1,w2,ξ)+∂∂w2g2(w1,w2,ξ))ds,
where
(95)∂∂w1g1(w1,w2,ξ)=12(∂∂wg(w,w¯,ξ)∂w∂w1+∂∂w¯g(w,w¯,ξ)∂w¯∂w1)+12(∂∂wg¯(w,w¯,ξ)∂w∂w1+∂∂w¯g¯(w,w¯,ξ)∂w¯∂w1),(96)∂∂w2g2(w1,w2,ξ)=-i2(∂∂wg(w,w¯,ξ)∂w∂w2+∂∂w¯g(w,w¯,ξ)∂w¯∂w2)+i2(∂∂wg¯(w,w¯,ξ)∂w∂w2+∂∂w¯g¯(w,w¯,ξ)∂w¯∂w2).
Adding equations (95) and (96) and taking into account that w=w1+iw2, it follows that
(97)∂∂w1g1(w1,w2,ξ)+∂∂w2g2(w1,w2,ξ)=∂∂wg(w,w¯,ξ)+∂∂w¯g¯(w,w¯,ξ).
Therefore, ℋ(s,ϵ,ν)=K(w,w¯,ξ), with
(98)K(w,w¯,ξ)=∂∂wg(w,w¯,ξ)+∂∂w¯g¯(w,w¯,ξ).By the formal Taylor series in the variableϵ of the function (ϵ,ν)↦χ(ϵ,ν),
(99)χ(ϵ,ν)=∑k=1∞‍1k!χk(ν)ϵk,
the theory of approximation of a family of periodic orbits developed in the previous section and Proposition 9 allow us to obtain the terms of the sequence {χk(ν)}k∈ℕ. For k=1,…,4, the next theorem provides these terms.Theorem 10.
Let(100)χ(ϵ,ν)=∑k=1∞‍1k!χk(ν)ϵk
be the formal Taylor series of the characteristic exponent (ϵ,ν)↦χ(ϵ,ν) associated with the differential equation z′=g(z,z¯,ξ). Then,
(101)χ1(ν)=0,χ2(ν)=2Re(G2,1(ξ0)),χ3(ν)=0,χ4(ν)=8Re(G3,2(ξ0))+12μ2(ν)Re(∂μG2,1(ξ0)),χ5(ν)=0,
where G2,1(ξ0), ∂μG2,1(ξ0), and G3,2(ξ0) are given by (50), (63), and (64), respectively.Proof.
From (13) and (14), we have
(102)∂∂wg(w,w¯,ξ)=λ(ξ)+∑k=2∞‍∑j=0k‍k-j(k-j)!j!gk-j,j(ξ)wk-j-1w¯j,∂∂w¯g¯(w,w¯,ξ)=λ¯(ξ)+∑k=2∞‍∑j=0k‍k-j(k-j)!j!g¯k-j,j(ξ)w¯k-j-1wj.
Thus, formally, the map (w,w¯,ξ)↦K(w,w¯,ξ) has the Taylor series
(103)K(w,w¯,ξ)=λ(ξ)+λ¯(ξ)+∑k=2∞‍∑j=0k‍k-j(k-j)!j!×(gk-j,j(ξ)+g¯k-j,j(ξ))wk-j-1w¯j.
Doing the fourth-order Taylor expansion of the map
(104)(s,ϵ,ν)↦ℋ(s,ϵ,ν)=K(w(s,ϵ,ν),w¯(s,ϵ,ν),ϕ(ϵ,ν),ν),
around ϵ=0, and taking into account that μ1(ν)=μ3(ν)=0, it results that
(105)ℋ(s,ϵ,ν)=ℋ1(s,ν)ϵ+12ℋ2(s,ν)ϵ2+16ℋ3(s,ν)ϵ3+124ℋ4(s,ν)ϵ4+Oℋ(s,ϵ5,|ν|),
with
(106)ℋ1(s,ν)=w¯1(s,ν)g1,1(ξ0)+w1(s,ν)g2,0(ξ0)+w1(s,ν)g¯1,1(ξ0)+w¯1(s,ν)g¯2,0(ξ0),ℋ2(s,ν)=w1(s,ν)2g3,0(ξ0)+w1(s,ν)2g¯1,2(ξ0)+2w1(s,ν)w¯1(s,ν)g2,1(ξ0)+2w1(s,ν)w¯1(s,ν)g¯2,1(ξ0)+w¯2(s,ν)g1,1(ξ0)+w¯1(s,ν)2g1,2(ξ0)+w2(s,ν)g2,0(ξ0)+w2(s,ν)g¯1,1(ξ0)+w¯2(s,ν)g¯2,0(ξ0)+w¯1(s,ν)2g¯3,0(ξ0)+2μ2(ν)∂μγ(ξ0),ℋ3(s,ν)=w1(s,ν)3g4,0(ξ0)+w1(s,ν)3g¯1,3(ξ0)+3w1(s,ν)2w¯1(s,ν)g3,1(ξ0)+3w1(s,ν)2w¯1(s,ν)g¯2,2(ξ0)+3w1(s,ν)w¯2(s,ν)g2,1(ξ0)+3w1(s,ν)w¯1(s,ν)2g2,2(ξ0)+3w1(s,ν)w2(s,ν)g3,0(ξ0)+3w1(s,ν)w2(s,ν)g¯1,2(ξ0)+3w1(s,ν)w¯2(s,ν)g¯2,1(ξ0)+3w1(s,ν)w¯1(s,ν)2g¯3,1(ξ0)+3μ2(ν)w1(s,ν)∂μg2,0(ξ0)+3μ2(ν)∂μg¯1,1(ξ0)w1(s,ν)+w¯3(s,ν)g1,1(ξ0)+3w¯1(s,ν)w¯2(s,ν)g1,2(ξ0)+w¯1(s,ν)3g1,3(ξ0)+w3(s,ν)g2,0(ξ0)+3w2(s,ν)w¯1(s,ν)g2,1(ξ0)+w3(s,ν)g¯1,1(ξ0)+w¯3(s,ν)g¯2,0(ξ0)+3w2(s,ν)w¯1(s,ν)g¯2,1(ξ0)+3w¯1(s,ν)w¯2(s,ν)g¯3,0(ξ0)+w¯1(s,ν)3g¯4,0(ξ0)+3μ2(ν)w¯1(s,ν)∂μg1,1(ξ0)+3μ2(ν)w¯1(s,ν)∂μg¯2,0(ξ0),ℋ4(s,ν)=w1(s,ν)4g5,0(ξ0)+w1(s,ν)4g¯1,4(ξ0)+4w1(s,ν)3w¯1(s,ν)g4,1(ξ0)+4w1(s,ν)3w¯1(s,ν)g¯2,3(ξ0)+6w1(s,ν)2w¯2(s,ν)g3,1(ξ0)+6w1(s,ν)2w¯1(s,ν)2g3,2(ξ0)+6w1(s,ν)2w2(s,ν)g4,0(ξ0)+6w1(s,ν)2w2(s,ν)g¯1,3(ξ0)+6w1(s,ν)2w¯2(s,ν)g¯2,2(ξ0)+6w1(s,ν)2w¯1(s,ν)2g¯3,2(ξ0)+6μ2(ν)w1(s,ν)2∂μg3,0(ξ0)+6μ2(ν)w1(s,ν)2∂μg¯1,2(ξ0)+4w1(s,ν)w¯3(s,ν)g2,1(ξ0)+12w1(s,ν)w¯1(s,ν)w¯2(s,ν)g2,2(ξ0)+4w1(s,ν)w¯1(s,ν)3g2,3(ξ0)+12w1(s,ν)w¯1(s,ν)w2(s,ν)g3,1(ξ0)+4w1(s,ν)w3(s,ν)g3,0(ξ0)+4w1(s,ν)w3(s,ν)g¯1,2(ξ0)+4w1(s,ν)w¯3(s,ν)g¯2,1(ξ0)+12w1(s,ν)w¯1(s,ν)w2(s,ν)g¯2,2(ξ0)+12w1(s,ν)w¯1(s,ν)w¯2(s,ν)g¯3,1(ξ0)+4w1(s,ν)w¯1(s,ν)3g¯4,1(ξ0)+12μ2(ν)w1(s,ν)w¯1(s,ν)∂μg2,1(ξ0)+12μ2(ν)w1(s,ν)w¯1(s,ν)∂μg¯2,1(ξ0)+w¯4(s,ν)g1,1(ξ0)+3w¯2(s,ν)2g1,2(ξ0)+4w¯1(s,ν)w¯3(s,ν)g1,2(ξ0)+6w¯1(s,ν)2w¯2(s,ν)g1,3(ξ0)+w¯1(s,ν)4g1,4(ξ0)+w4(s,ν)g2,0(ξ0)+4w¯1(s,ν)w3(s,ν)g2,1(ξ0)+6w2(s,ν)w¯2(s,ν)g2,1(ξ0)+6w¯1(s,ν)2w2(s,ν)g2,2(ξ0)+3w2(s,ν)2g3,0(ξ0)+w4(s,ν)g¯1,1(ξ0)+3w2(s,ν)2g¯1,2(ξ0)+w¯4(s,ν)g¯2,0(ξ0)+4w¯1(s,ν)w3(s,ν)g¯2,1(ξ0)+6w2(s,ν)w¯2(s,ν)g¯2,1(ξ0)+3w¯2(s,ν)2g¯3,0(ξ0)+4w¯1(s,ν)w¯3(s,ν)g¯3,0(ξ0)+6w¯1(s,ν)2w2(s,ν)g¯3,1(ξ0)+6w¯1(s,ν)2w¯2(s,ν)g¯4,0(ξ0)+w¯1(s,ν)4g¯5,0(ξ0)+6μ2(ν)w¯2(s,ν)∂μg1,1(ξ0)+2μ4(ν)∂μγ(ξ0)+6μ2(ν)w¯1(s,ν)2∂μg1,2(ξ0)+6μ2(ν)w2(s,ν)∂μg2,0(ξ0)+6μ2(ν)w2(s,ν)∂μg¯1,1(ξ0)+6μ2(ν)w¯2(s,ν)∂μg¯2,0(ξ0)+6μ2(ν)w¯1(s,ν)2∂μg¯3,0(ξ0)+6μ2(ν)2∂μ2γ(ξ0),
where for k=1,2,3,4, the functions (s,ν)↦wk(s,ν) are such as in (37), (46), (54), and (59) and the expressions μ2(ν) and μ4(ν) are given by (52), and (61), respectively. Thus, from (99) and (105) and by Proposition 9,
(107)∑k=14‍1k!(χk(ϵ,ν)-12π∫02π‍ℋk(s,ν)ds)ϵk+OℋI(ϵ5,|ν|)=0.
Therefore,
(108)12π∫02π‍ℋ1(s,ν)ds=12π∫02π‍ℋ3(s,ν)ds=0,12π∫02π‍ℋ2(s,ν)ds=2Re(G2,1(ξ0)),12π∫02π‍ℋ4(s,ν)ds=8Re(G3,2(ξ0))+12μ2(ν)Re(∂μG2,1(ξ0)),
which proves the theorem.It follows from Theorem10 a corollary that deals with the stability of a family of periodic orbits of the differential equation (1) which exists due to a Hopf bifurcation.Corollary 11.
Let(109)χ(ϵ,ν)=12χ2(ν)ϵ2+124χ4(ν)ϵ4+Oχ(ϵ5,|ν|)
be the fourth-order Taylor expansion around ϵ=0 of the characteristic exponent (ϵ,ν)↦χ(ϵ,ν) associated with the differential equation z′=g(z,z¯,ξ), and let
(110)μ=ϕ(ϵ,ν)=12μ2(ν)ϵ2+124μ4(ν)ϵ4+Oχ(ϵ5,|ν|),
be the fourth-order Taylor expansion, around ϵ=0, of the function (ϵ,ν)↦μ=ϕ(ϵ,ν). The following statements hold.  (a)
For a fixed(ϵ,ν)∈Uϵ, ϵ∈ℝ sufficiently small, and Re(G2,1(ξ0))≠0, the stability of the periodic orbit of  the differential equation (1) is given by the sign of Re(G2,1(ξ0)). When Re(G2,1(ξ0))<0 for ξ0∈U, the periodic orbit in the phase portrait of differential equation (1) is stable. As for ϵ∈ℝ, sufficiently small,
(111)μ=ϕ(ϵ,ν)=-12Re(G2,1(ξ0))∂μγ(ξ0)ϵ2+Oμ(ϵ4,|ν|),
if ∂μγ(ξ0)>0, the periodic orbit in the phase portrait of (1) exists for μ>0, and if ∂μγ(ξ0)<0, the periodic orbit in the phase portrait exists for μ<0. If Re(G2,1(ξ0))>0, the periodic orbit in the phase portrait of the differential equation (1) is unstable.(b)
Suppose that forξ1=(0,0), Re(G2,1(ξ1))=0 and Re(G3,2(ξ1))≠0. Then, for ϵ∈ℝ sufficiently small, the stability is given by the sign of Re(G3,2(ξ1)). When Re(G3,2(ξ1))<0, the periodic orbit in the phase portrait of the differential equation (1) is stable. As, in this case,
(112)μ=ϕ(ϵ,ν)=-112Re(G3,2(ξ1))∂μγ(ξ1)ϵ4+Oμ(ϵ5,|ν|),
if ∂μγ(ξ1)>0, the periodic orbit in the phase portrait of the differential equation (1) exists for μ>0, and if ∂μγ(ξ1)<0, the periodic orbit in the phase portrait of the differential equation (1) exists for μ<0. If Re(G3,2(ξ1))>0, the periodic orbit in the phase portrait of the differential equation (1) is unstable.Proof.
As the sign of the Floquet exponent provides the stability of a periodic orbit, by (109), and (110) the proof is immediate.Corollary11 does not deal with the case where χ(ϵ,ν)=0 for a set of points (ϵ,ν)∈Uϵ. The theory developed up to this point enables us to study the curve of nonhyperbolic periodic orbits CNH in the parameter plane (μ,ν)∈ℝ2, associated with a transversal Hopf point of codimension two. This curve is the set
(113)χ-1(0)={(ϵ,ν)∈Uϵ:χ(ϵ,ν)=0}.From the setχ-1(0) and the Implicit Function Theorem, the parameter ν can be obtained as a function of the parameter ϵ. Therefore, the curve CNH follows from functions ϵ↦ν=ψ(ϵ) and (ϵ,ν)↦μ=ϕ(ϵ,ν); that is, the curve CNH can be locally represented as a curve parameterized by ϵ(114)Γ(ϵ)=(ϕ(ϵ,ψ(ϵ)),ψ(ϵ)),
or can be locally represented as the graph of a function
(115)μ=Λ(ν).In fact, the Taylor expansion aroundϵ=0 of the exponent characteristic is such as in (99), and, therefore,
(116)χ(ϵ,ν)=ϵ2Ψ(ϵ,ν),
where the third-order Taylor expansion around ϵ=0 of the function (ϵ,ν)↦Ψ(ϵ,ν) is of the following form:
(117)Ψ(ϵ,ν)=12χ2(ν)+124χ4(ν)ϵ2+Oχ(ϵ3,|ν|).
It is easy to see that Ψ-1(0)={(ϵ,ν)∈Uϵ:Ψ(ϵ,ν)=0}⊂χ-1(0). Thus, the study of the curve of nonhyperbolic periodic orbits in the parameter plane (μ,ν)∈ℝ2, associated with the differential equation (1), and in the hypotheses of a transversal Hopf bifurcation of codimension two is reduced to the study of the set Ψ-1(0).The next lemma, whose proof is given in [3], guarantees the existence of the function ϵ↦ν=ψ(ϵ).Lemma 12.
Let(118)D:ℝ×ℝ4⟶ℝ(x,y)↦D(x,y)
be a smooth function, where y=(y0,y1,y2,y3). Suppose that for (0,y0)∈ℝ×ℝ4, y0=(y0,0,y2,y3), the function in (118) satisfies the following assumptions:  (A1)
D(0,y0)=0;(A2)
∂xD(0,y0)=0;(A3)
∂y1D(0,y0)≠0;(A4)
∂x2D(0,y0)≠0.Then, there exists a unique smooth function
(119)(x,y0)↦y1=ϕ(x,y0),
such that y=Φ(x,y0)=y0+(0,ϕ(x,y0),0,0) and D(x,Φ(x,y0))≡0. Moreover, the function (x,y0)↦y1=ϕ(x,y0) has the following representation:
(120)ϕ(x,y0)=12!ϕ2(y0)x2+Oϕ(|x|3,∥y0∥),
where
(121)ϕ2(y0)=-∂x2D(0,y0)∂y1D(0,y0).The following theorem can be stated now.Theorem 13.
Let(0,ξ1)∈W×U be a transversal Hopf point of codimension two of (1). Then, the curve of nonhyperbolic periodic orbits CNH, in the parameter plane (μ,ν)∈ℝ2, associated with the differential equation (1), has the following local representations:
(122)Γ(ϵ)=(Re(G3,2(ξ1))12∂μγ(ξ1)ϵ4,-Re(G3,2(ξ1))3Re(∂νG2,1(ξ1))ϵ2)+OΓ(ϵ),(123)μ=Λ(ν)=μ2(ν)ψ2ν+16μ4(ν)ψ22ν2+OΛ(|ν|),
where
(124)ψ2=-∂ϵ2Ψ(0,0)∂νΨ(0,0)=-2Re(G3,2(ξ1))3∂νRe(G2,1(ξ1)).Proof.
As(125)Ψ(0,0)=0,∂ϵΨ(0,0)=0,∂ϵ2Ψ(0,0)=112χ4(0)=23Re(G3,2(ξ1))≠0,∂νΨ(0,0)=12χ2′(0)=∂νRe(G2,1(ξ1))≠0.
Lemma 12 guarantees the existence of a smooth function ϵ↦ν=ψ(ϵ) such that Ψ(ϵ,ψ(ϵ))≡0, or even, χ(ϵ,ψ(ϵ))≡0. Moreover, the function ϵ↦ν=ψ(ϵ) has the second-order Taylor expansion around ϵ=0 of the following form:
(126)ν=ψ(ϵ)=12!ψ2ϵ2+Oψ(ϵ3),
where
(127)ψ2=-∂ϵ2Ψ(0,0)∂vΨ(0,0)=-2Re(G3,2(ξ1))3∂νRe(G2,1(ξ1)).
Thus,
(128)ν=ψ(ϵ)=-Re(G3,2(ξ1))3∂νRe(G2,1(ξ1))ϵ2+Oψ(ϵ3),
and substituting (128) into the function (ϵ,ν)↦μ=ϕ(ϵ,ν) results in the following Taylor expansion:
(129)μ=ϕ(ϵ,ψ(ϵ))=Re(G3,2(ξ1))12∂μγ(ξ1)ϵ4+Oϕ(ϵ5).
So, there is a curve in the parameter plane, ϵ↦Γ(ϵ)=(ϕ(ϵ,ψ(ϵ)),ψ(ϵ)), that can be parameterized by ϵ and represented as in (122). Another representation for this curve is obtained when the Implicit Function Theorem is applied to the following function:
(130)ϵ2=2ψ2ν+Oϵ(|ν|2).
By substituting (130) into (110), the curve ϵ↦Γ(ϵ) can also be represented locally as
(131)μ=Λ(ν)=12μ2(ν)(2ψ2ν)+124μ4(ν)(2ψ2ν)2+OΛ(|ν|)=μ2(ν)ψ2ν+16μ4(ν)ψ22ν2+OΛ(|ν|).
Therefore, there exists a curve Γ in the parameter plane that locally has the representation (122) or (123). By the hypotheses of the transversal Hopf bifurcation of codimension two, Re(G3,2(ξ1))≠0 and equation z′=g(z,z¯,ξ) are locally topologically equivalent, around z=0, to the complex differential equation (67). Therefore, the curve of nonhyperbolic periodic orbits has the representation (122) or (123).Example 14.
For the complex differential equation (67), we have
(132)γ(α,β)=α,η(α,β)=1,G2,1(α,β)=2β,G3,2(α,β)=12s.
So, by Theorem 13, the curve of nonhyperbolic periodic orbits has the following representations:
(133)Γ(ϵ)=(sϵ4,-2sϵ2)+OΓ(ϵ),α=Λ(β)=14sβ2+OΛ(β),
which agree with (68) and (69), respectively.The local representations (122) and (123) in Theorem 13 are valid when the Hopf curve is the set {(μ,ν):μ=0}. If the Hopf curve is the set {(μ,ν):μ=φ(ν)} and the transversal Hopf bifurcation of codimension two occurs for ξ1=(μ1,ν1)≠(0,0), it is easy to show that the local representations are given by
(134)Γ(ϵ)=(μ1+12φ2ϵ2+Re(G3,2(ξ1))12∂μγ(ξ1)ϵ4,ν1-Re(G3,2(ξ1))3Re(∂νG2,1(ξ1))ϵ2)+OΓ(ϵ),μ=Λ(ν)=φ(ν)+μ2(ν)ψ2(ν-ν1)+16μ4(ν)ψ22(ν-ν1)2+OΛ(|ν|),
for ν≤ν1, where
(135)μ1=φ(ν1),φ2=ψ2∂νφ(ν1).The next two sections present applications of the theory developed here in an extension of the van der Pol equation known as the Liénard equation and in Bazykin's predator-prey system and show how local representations of the the curveCNH are obtained.


## 4. Liénard Equation

One of the pioneers in nonlinear electrical circuits was, undoubtedly, Balthasar van der Pol, through studies with triodes (vacuum tubes). Balthasar van der Pol showed that in circuits with triodes, the electrical quantities can exhibit nonlinear oscillations under certain conditions. Nowadays, it is known that the model of this circuit with triode presents a Hopf bifurcation. In a simple and theoretical way, the electric circuit of van der Pol consists of a triode, a capacitor of capacitanceC, and an inductor of inductance L, according to the diagram of Figure 1.Figure 1
van der Pol circuit diagram.LetvC, iC and vL, iL be the models of voltage and current in the capacitor and inductor, respectively. The triode of van der Pol, by the hypothesis, satisfies the generalized Ohm's law iR↦vR=X(iR), where vR and iR are the models of voltage and current of the triode of van der Pol, respectively. Applying Kirchhoff's laws to the van der Pol electrical circuit model and using the capacitor and inductor equations, it follows that
(136)iR=iL=-iC,vr+vL-vC=0,vL=LddtiL,iC=CddtvC.
Therefore, the van der Pol circuit model is of the following form:
(137)LddtiL=vc-X(iL),Cddtvc=-iL.The study of differential equation (137) is simplified by the change of coordinates and time
(138)x=LLCiL,y=vC,τ=1LCt,
which leads to the differential equation
(139)x′=ddτx=y-𝒳(x),y′=ddτy=-x,
where 𝒳(x)=X((C/LC)x). Suppose that
(140)x↦𝒳(x)=-μx+νx3+15x5.
In the literature, the differential equation (139) satisfying (140) is known as the Liénard-type equation.The Liénard equation has a unique equilibrium point(0,ξ)∈ℝ2×ℝ2, with ξ=(μ,ν), and the linear part of the vector field, evaluated at (0,ξ),
(141)A(ξ)=(μ1-10),
has eigenvalues λ and λ¯, with
(142)λ(ξ)=γ(ξ)+iη(ξ)=12μ+i124-μ2,
for μ∈(-2,2). When ξ=ξ0=(0,ν), γ(ξ0)=0, and η(ξ0)=1, which indicates the occurrence of Hopf bifurcations. The eigenvectors q(ξ0)∈ℂ2 and p(ξ0)∈ℂ2, where q(ξ0) is normalized with respect to p(ξ0) according to (8), are chosen as
(143)q(ξ0)=(-i2,12),p(ξ0)=(-i,1).In the case of the Liénard equation, the symmetric multilinear functions are given by(144)B(x,y,ξ)=(0,0),C(x,y,u,ξ)=(6νx1y1u1,0),D(x,y,u,v,ξ)=(0,0),E(x,y,u,v,w,ξ)=(24x1y1u1v1w1,0).
Thus, for k=2,3,… and j=0,1,…,k, the only nonzero coefficients gk-j,j(ξ) are
(145)g3,0(ξ0)=g1,2(ξ0)=-g2,1(ξ0)=-g0,3(ξ0)=34ν,g4,1(ξ0)=g2,3(ξ0)=g0,5(ξ0)=-g5,0(ξ0)=-g3,2(ξ0)=-g1,4(ξ0)=34.The eigenvectors∂μq(ξ0) and ∂μp(ξ0) and the coefficients ∂μg2,0(ξ0), ∂μg1,1(ξ0), ∂μg0,2(ξ0), and ∂μg2,1(ξ0), computed by Propositions 7 and 8, are such that
(146)∂μq(ξ0)=(-18,i8),∂μp(ξ0)=(14,-i4),∂μg2,0(ξ0)=∂μg1,2(ξ0)=∂μg0,2(ξ0)=0,∂μg2,1(ξ0)=38iν.Thus, from the previous results and by (50), (63), and (64), it follows that
(147)G2,1(ξ0)=-34ν,∂μG2,1(ξ0)=38iν,G3,2(ξ0)=-34-8164iν2.Therefore, the first Lyapunov coefficient is given by(148)l1(ξ0)=Re(G2,1(ξ0))=-34ν,
and since
(149)∂μγ(ξ0)=12,
the Liénard equation presents a transversal Hopf bifurcation of codimension one for μ=0 and ν≠0. From Corollary 11,
(150)μ=(34ν)ϵ2+Oμ(ϵ4,|ν|),χ(ϵ,ν)=(-34ν)ϵ2+Oχ(ϵ3,|ν|),
and if ν<0, then there exists a unique unstable periodic orbit in the phase portrait of the Liénard equation when μ<0, and if ν>0, the periodic orbit is stable and there exists for μ>0.Forν=0, the Liénard equation has a transversal Hopf point of codimension two, and the second Lyapunov coefficient is given by
(151)l2(ξ1)=Re(G3,2(ξ1))=-116,
where ξ1=(0,0). Since
(152)∂νRe(G2,1(ξ1))∂μγ(ξ1)=(-34)(12)=-38≠0,
by Corollary 11 and Theorem 13, the Liénard equation has a bifurcation diagram as shown in Figure 2.Figure 2
Bifurcation diagram of the Liénard equation (139).The curve of nonhyperbolic periodic orbits has the following local representations:(153)Γ(ϵ)=(μ(ϵ,ν(ϵ)),ν(ϵ))=(-18ϵ4,-13ϵ2)+OΓ(ϵ),
as a curve parameterized by ϵ or as the graph of the function
(154)μ=Λ(ν)=-98ν2+Oν(ν3),
for ν≤0.Figure3 emphasizes the comparison between the curve of nonhyperbolic periodic orbits CNH of (139) obtained numerically with the software MATCONT (see [5]) and the quadratic approximation (154).Figure 3
Comparison between the curve of nonhyperbolic periodic orbitsCNH of (139): the dotted curve was obtained numerically with the software MATCONT and the continuous curve from the representation (154).


## 5. Bazykin's Predator-Prey System

Consider the dynamics of a predator-prey ecosystem, whose model is(155)x′=x2(1-x)μ+x-xy,y′=-γy(ν-x),
where γ>0 (fixed), μ≥0, and 0≤ν<1 are parameters. Model (155) is known in the literature as Bazykin's predator-prey system. See [1] or [10].Takingγ=1, the equilibrium point of interest is
(156)(x0(ξ),ξ)=((ν,ν(1-ν)μ+ν),ξ).
For
(157)μ=φ(ν)=ν21-2ν,
the linear part of the vector field, evaluated at (x0(ξ),ξ),
(158)A(ξ)=(ν(μ(1-2ν)-ν2)(μ+ν)2-νν(1-ν)μ+ν0),
has eigenvalues λ and λ¯, where λ(ξ0)=iω0(ν), ω0(ν)=ν(1-2ν) and ξ0=(φ(ν),ν).The eigenvectorsq(ξ0)∈ℂ2 and p(ξ0)∈ℂ2 are chosen as
(159)q(ξ0)=(iν2ω0(ν),12),p(ξ0)=(iω0(ν)ν,1),
and by Proposition 7,
(160)∂μγ(ξ0)=(1-2ν)32ν(1-ν)2,∂μη(ξ0)=-(1-2ν)22(1-ν)ω0(ν),∂μ2γ(ξ0)=-2(1-2ν)4ν2(1-ν)3,∂μq(ξ0)=(1-(1-ν)(4ν-iω0(ν))8ν(1-ν)2,ω0(ν)2(iω0(ν)3-ν2(1-ν))8ν4(1-ν)2),∂μp(ξ0)=(iω0(ν)3(iω0(ν)3-ν2(1-ν))4ν5(1-ν)2,-ω0(ν)2(iω0(ν)3-ν2(1-ν))4ν4(1-ν)2).The symmetric multilinear functions are given by(161)B(x,y,ξ)=(-x2y1-2(ν3+3μν2+μ2(3ν-1))(μ+ν)3x1y1-x1y2,x2y1+x1y22(ν3+3μν2+μ2(3ν-1))(μ+ν)3),∂μB(x,y,ξ)=(2μ(2ν+μ(3ν-1))(μ+ν)4x1y1,0),C(x,y,u,ξ)=(-6μ2(μ+1)(μ+ν)4x1y1z1,0),∂μC(x,y,u,ξ)=(6μ(μ2+(2-3ν)μ-2ν)(μ+ν)5x1y1z1,0),D(x,y,u,v,ξ)=(24μ2(μ+1)(μ+ν)5x1y1z1u1,0),E(x,y,u,v,w,ξ)=(-120μ2(μ+1)(μ+ν)6x1y1z1u1v1,0).Therefore, from the previous results,(162)Re(G2,1(ξ0))=1-4ν4(1-ν)2,∂μRe(G2,1(ξ0))=-5(1-2ν)(40ν3-67ν2+27ν-2)72ν2(1-ν)4,∂νRe(G2,1(ξ0))=-2ν+12(1-ν)3,Re(G3,2(ξ0))=292ν4-485ν3+286ν2-68ν+524ν(1-2ν)2(1-ν)4.Whenν=ν1=1/4, μ1=φ(ν1)=1/8. Thus, for ξ1=(μ1,ν1)=(1/8,1/4), Bazykin's system (155) has a transversal Hopf point of codimension two, since Re(G2,1(ξ1))=0, Re(G3,2(ξ1))=-32/27, and
(163)∂νRe(G2,1(ξ1))∂μγ(ξ1)=(-169)(49)=-6481≠0.Using (134), the curve of nonhyperbolic periodic orbits has the following local representations:
(164)Γ(ϵ)=(18-ϵ23+14ϵ481,14-2ϵ29)+OΓ(ϵ),
as a curve parameterized by ϵ or as a graph of the function
(165)μ=Λ(ν)=ν21-2ν+9ν(1-4ν)232(2ν-1)3+27(4ν-1)2ν(120ν3-98ν2+7ν+1)2048(ν-1)2(2ν-1)5+Oν(ν3),
for ν≤ν1=1/4.Figure4 emphasizes the comparison between the curve of nonhyperbolic periodic orbits CNH of (155) obtained numerically with the software MATCONT and the quadratic approximation (164).Figure 4
Comparison between the curve of nonhyperbolic periodic orbitsCNH of (155): the dotted curve was obtained numerically with the software MATCONT and the continuous curve from the representation (164). The dot-dashed curve is the Hopf curve.The comparison between the curve of nonhyperbolic periodic orbitsCNH of (155) obtained numerically with the software MATCONT and the approximation (165) is shown in Figure 5.Figure 5
Comparison between the curve of nonhyperbolic periodic orbitsCNH of (155): the dotted curve was obtained numerically with the software MATCONT and the continuous curve from the representation (165). The dot-dashed curve is the Hopf curve.


## 6.   Concluding Comments

This paper shows how to obtain approximations of periodic orbits of a family of differential equations in the plane that has a transversal Hopf point. Moreover, if the family of differential equations has a transversal Hopf point of codimension two, then it is also possible to build an approximation to the curve of nonhyperbolic periodic orbits in the bifurcation diagram. These results are summarized in Corollary11 and Theorem 13. Example 14, the study of the Liénard equation (139) in Section 4, and Bazykin's predator-prey system in Section 5 demonstrate the applicability of the theory. See also Figures 3, 4, and 5.Although the theory is formulated for a family of differential equations in the plane, it can be applied to any family of differential equations inℝn that presents a transversal Hopf bifurcation of codimension two. For this, it is necessary to use the Center Manifold Theorem, or more precisely, to apply the proposed theory to the family of differential equations in ℝn restricted to the center manifold.



---

*Source: 212340-2013-08-27.xml*