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The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems

The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems 1IntroductionThis paper studies the non-autonomous planar Hamiltonian system(1.1){x′=-λ⁢α⁢(t)⁢f⁢(y),y′=λ⁢β⁢(t)⁢g⁢(x),\left\{\begin{aligned} \displaystyle{}x^{\prime}&\displaystyle=-\lambda\alpha(t)f(y),\\ \displaystyle y^{\prime}&\displaystyle=\lambda\beta(t)g(x),\end{aligned}\right.where λ>0\lambda>0 is regarded as a real parameter, and given a real number T>0T>0, 𝛼 and 𝛽 are nonnegative 𝑇-periodic continuous functions such thatA:=∫0Tα>0,B:=∫0Tβ>0,A:=\int_{0}^{T}\alpha>0,\quad B:=\int_{0}^{T}\beta>0,for which the set(1.2)Z:=supp⁡α∩supp⁡βZ:=\operatorname{supp}\alpha\cap\operatorname{supp}\betahas Lebesgue measure zero, |Z|=0\lvert Z\rvert=0. This is why model (1.1) is said to be degenerate.In (1.1), f,g∈C⁢(R)f,g\in\mathcal{C}({{\mathbb{R}}}) are locally Lipschitz continuous functions such that f,g∈C1⁢(-ρ,ρ)f,g\in\mathcal{C}^{1}(-\rho,\rho) for some ρ>0\rho>0 and(1.3){f⁢(0)=0,f⁢(y)⁢y>0for all⁢y≠0,g⁢(0)=0,g⁢(x)⁢x>0for all⁢x≠0,f′⁢(0)>0,g′⁢(0)>0.\left\{\begin{aligned} \displaystyle{}f(0)&\displaystyle=0,&\displaystyle f(y)y&\displaystyle>0&&\displaystyle\text{for all}\ y\neq 0,\\ \displaystyle g(0)&\displaystyle=0,&\displaystyle g(x)x&\displaystyle>0&&\displaystyle\text{for all}\ x\neq 0,\\ \displaystyle f^{\prime}(0)&\displaystyle>0,&\displaystyle g^{\prime}(0)&\displaystyle>0.\end{aligned}\right.Moreover, it is assumed that either 𝑓 or 𝑔 satisfies one of the following four conditions:(1.4)(f-)(f_{-})f⁢is bounded in⁢R-,f\ \text{is bounded in}\ \mathbb{R}^{-},(f+)(f_{+})f⁢is bounded in⁢R+,f\ \text{is bounded in}\ \mathbb{R}^{+},(g-)(g_{-})g⁢is bounded in⁢R-,g\ \text{is bounded in}\ \mathbb{R}^{-},(g+)(g_{+})g⁢is bounded in⁢R+.g\ \text{is bounded in}\ \mathbb{R}^{+}.This paper analyzes the existence of n⁢TnT-periodic solutions of model (1.1) for any integer n≥1n\geq 1. Those with n≥2n\geq 2 (and 𝑛 minimal) are often referred to as subharmonics of order 𝑛. Besides |Z|=0\lvert Z\rvert=0, through this paper, we assume that, given two positive integers k,ℓ≥1k,\ell\geq 1 such that |k-ℓ|≤1\lvert k-\ell\rvert\leq 1, there exist k+ℓk+\ell continuous functions in the interval [0,T][0,T], αi⪈0\alpha_{i}\gneq 0, 1≤i≤k1\leq i\leq k, and βj⪈0\beta_{j}\gneq 0, 1≤j≤ℓ1\leq j\leq\ell, such that α=α1+α2+⋯+αk\alpha=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{k}, β=β1+β2+⋯+βℓ\beta=\beta_{1}+\beta_{2}+\cdots+\beta_{\ell}, with(1.5)supp⁡αi⊆[t0i,t1i] and supp⁡βj⊆[t2j,t3j]\operatorname{supp}\alpha_{i}\subseteq[t_{0}^{i},t_{1}^{i}]\quad\text{and}\quad\operatorname{supp}\beta_{j}\subseteq[t_{2}^{j},t_{3}^{j}]for some partition of [0,T][0,T],0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0k<t1k≤t2k<t3k≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{k}<t_{1}^{k}\leq t_{2}^{k}<t_{3}^{k}\leq Tif⁢k=ℓ,or\text{if}\ k=\ell,\,\text{or}0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0k<t1k≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{k}<t_{1}^{k}\leq Tif⁢k=ℓ+1.\text{if}\ k=\ell+1.Similarly, we also consider the case when, instead of (1.5),(1.6)supp⁡βj⊆[t0j,t1j] and supp⁡αi⊆[t2i,t3i]\operatorname{supp}\beta_{j}\subseteq[t_{0}^{j},t_{1}^{j}]\quad\text{and}\quad\operatorname{supp}\alpha_{i}\subseteq[t_{2}^{i},t_{3}^{i}]for some partition of [0,T][0,T],0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0ℓ<t1ℓ≤t2ℓ<t3ℓ≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{\ell}<t_{1}^{\ell}\leq t_{2}^{\ell}<t_{3}^{\ell}\leq Tif⁢ℓ=k,or\text{if}\ \ell=k,\,\text{or}0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0ℓ<t1ℓ≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{\ell}<t_{1}^{\ell}\leq Tif⁢ℓ=k+1.\text{if}\ \ell=k+1.We will refer to an 𝛼-interval (resp. 𝛽-interval) as the maximal interval 𝐼, where |supp⁡β|I|=0\lvert\operatorname{supp}\beta|_{I}\rvert=0 and |supp⁡α|I|>0\lvert\operatorname{supp}\alpha|_{I}\rvert>0 (resp. |supp⁡α|I|=0\lvert\operatorname{supp}\alpha|_{I}\rvert=0 and |supp⁡β|I|>0\lvert\operatorname{supp}\beta|_{I}\rvert>0). So the total number of 𝛼-intervals and 𝛽-intervals in [0,T][0,T] is k+ℓk+\ell. However, ascertaining the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] when n≥2n\geq 2 is slightly more subtle, as it depends on whether k=ℓk=\ell or |k-ℓ|=1\lvert k-\ell\rvert=1. If k=ℓk=\ell, it is apparent that the number of 𝛼-intervals is n⁢knk, whereas the number of 𝛽-intervals is n⁢ℓn\ell. Thus, the total number of 𝛼 and 𝛽-intervals in this case equals(1.7)n⁢(k+ℓ)=2⁢n⁢k.n(k+\ell)=2nk.Now, assume that |k-ℓ|=1\lvert k-\ell\rvert=1. Then, in case k=ℓ+1k=\ell+1, there are n⁢ℓ+1n\ell+1 𝛼-intervals and n⁢ℓn\ell 𝛽-intervals in [0,n⁢T][0,nT]. Indeed, as for every i∈{0,1,…,n-2}i\in\{0,1,\ldots,n-2\} the last 𝛼-interval of [i⁢T,(i+1)⁢T][iT,(i+1)T] and the first one of [(i+1)⁢T,(i+2)⁢T][(i+1)T,(i+2)T] produce a unique 𝛼-interval in [0,n⁢T][0,nT], the total number of 𝛼-intervals in [0,n⁢T][0,nT] is given by n⁢(ℓ+1)-(n-1)=n⁢ℓ+1n(\ell+1)-(n-1)=n\ell+1. Obviously, the number of 𝛽-intervals in [0,n⁢T][0,nT] is n⁢ℓn\ell. Thus, the total number of 𝛼 and 𝛽-intervals equals n⁢ℓ+1+n⁢ℓ=2⁢n⁢ℓ+1n\ell+1+n\ell=2n\ell+1. Similarly, in case ℓ=k+1\ell=k+1, the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] is n⁢k+1+n⁢k=2⁢n⁢k+1nk+1+nk=2nk+1. Therefore, setting m:=min⁡{k,ℓ}m:=\min\{k,\ell\}, the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] in case |k-ℓ|=1\lvert k-\ell\rvert=1 is(1.8)2⁢n⁢m+1.2nm+1.Figure 1 shows a series of examples satisfying the previous requirements. Note that the support of αi\alpha_{i} and βj\beta_{j} on each of the intervals [tri,tr+1i][t_{r}^{i},t_{r+1}^{i}], 1≤i≤k1\leq i\leq k, and [trj,tr+1j][t_{r}^{j},t_{r+1}^{j}], 1≤j≤ℓ1\leq j\leq\ell, might not be connected.Figure 1Some admissible distributions of 𝛼 and 𝛽.On each of the intervals [trs,tr+1s][t_{r}^{s},t_{r+1}^{s}], s∈{i,j}s\in\{i,j\}, r∈{0,2}r\in\{0,2\}, the structure of the support of αi\alpha_{i} or βj\beta_{j} might be rather involved topologically, as illustrated by Figure 2, where we have plotted a sketch of the graph of a function αi\alpha_{i} or βj\beta_{j}, vanishing on the tertiary Cantor set of the interval [trs,tr+1s][t_{r}^{s},t_{r+1}^{s}] and being positive on the interior of its complement.Figure 2The internal complexity of the weights on each of their support intervals.Note that the cases when k+ℓ≠2k+\ell\neq 2, or k+ℓ=2k+\ell=2 under condition (1.6), were not dealt with in any previous references. In particular, they stay outside the general scope of [16, 13, 14]. The main result of this paper can be stated as follows. (Subsequently, for every r∈Rr\in{\mathbb{R}}, we are denoting by [r][r] the integer part of 𝑟.)Theorem 1Assume n⁢m≥3nm\geq 3 for some integer n=3⁢h+in=3h+i, with i∈{0,1,2}i\in\{0,1,2\}, where m:=min⁡{k,ℓ}m:=\min\{k,\ell\}. Then there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, (1.1) possesses at leastσ⁢(n):=2⁢(h⁢m+[i⁢m3])\sigma(n):=2\biggl{(}hm+\biggl{[}\frac{im}{3}\biggr{]}\biggr{)}periodic solutions with period n⁢TnT. Moreover, settingγ⁢(n):=min⁡{γ≥0:gcd⁡(n,σ⁢(n)2-γ)=1},\gamma(n):=\min\biggl{\{}\gamma\geq 0:\gcd\biggl{(}n,\frac{\sigma(n)}{2}-\gamma\biggr{)}=1\biggr{\}},it turns out that, for every λ>λn\lambda>\lambda_{n}, (1.1) has at least σ⁢(n)-2⁢γ⁢(n)\sigma(n)-2\gamma(n) periodic solutions with minimal period n⁢TnT.The main technical device to prove Theorem 1 is the Poincaré–Birkhoff twist theorem collected in Theorem 2. Theorem 1 deals with a degenerate case in the context of Hamiltonian systems not previously studied in the literature, because neither the monotonicity of α⁢(t)⁢f⁢(y)\alpha(t)f(y) or β⁢(t)⁢g⁢(x)\beta(t)g(x)for all 𝑡, nor the non-degeneration of α⁢(t)\alpha(t) and β⁢(t)\beta(t) are required (see [12, 5, 18, 10], and [15, § 1]).In Section 2, we state the version of the Poincaré–Birkhoff theorem invoked in the proof of Theorem 1 and make sure that it can be applied to deal with the degenerate model (1.1). Then, in Section 3, the proof of Theorem 1.1 is delivered. We refer to [9, 3] for a general discussion about the applications of the Poincaré–Birkhoff theorem to non-autonomous equations.2The Poincaré–Birkhoff Theorem in a Degenerate SettingIn this section, we adapt the Poincaré–Birkhoff theorem to deal with problem (1.1) in the degenerate case when |Z|=0\lvert Z\rvert=0 (see (1.2), if necessary). The Poincaré–Birkhoff theorem has been applied, very successfully, to study some non-degenerate Volterra predator-prey models of type (1.1) (see, e.g., [12, 4, 5, 18, 1, 10, 7] and the recent paper by the authors [15]).According to [15, Theorem 2.2], as soon as |Z|>0\lvert Z\rvert>0, system (1.1) possesses at least two n⁢TnT-periodic solutions for every integer n≥1n\geq 1 and sufficiently large λ>0\lambda>0. The main result of this section, Theorem 3, provides with some general sufficient conditions on the coefficients α⁢(t)\alpha(t) and β⁢(t)\beta(t) for the validity of the same result in the case when |Z|=0\lvert Z\rvert=0.Subsequently, for any given nontrivial solution (x⁢(t),y⁢(t))(x(t),y(t)) with initial data z0:=(x⁢(0),y⁢(0))≠(0,0)z_{0}:=(x(0),y(0))\neq(0,0), we denote by θ⁢(t)\theta(t) the angular polar coordinate so that, on any interval [0,n⁢T][0,nT], the rotation number of the solution can be defined throughrot⁡(z0;[0,n⁢T]):=θ⁢(n⁢T)-θ⁢(0)2⁢π.\operatorname{rot}(z_{0};[0,nT]):=\frac{\theta(nT)-\theta(0)}{2\pi}.To obtain the main result of this section, we need the following version of the Poincaré–Birkhoff twist theorem. It is, essentially, an application of Ding’s version of the twist theorem for planar annuli (see [6]), as presented in [17, Theorem A] (see also [2] for another application).Theorem 2Assume that, for some 0<r0<R00<r_{0}<R_{0} and an integer ω≥1\omega\geq 1, the next twist condition holds:(2.1)rot⁡(z0;[0,n⁢T])>ω⁢if⁢∥z0∥=r0 and rot⁡(z0;[0,n⁢T])<ω⁢if⁢∥z0∥=R0.\operatorname{rot}(z_{0};[0,nT])>\omega\ \text{if\/}\ \lVert z_{0}\rVert=r_{0}\quad\text{and}\quad\operatorname{rot}(z_{0};[0,nT])<\omega\ \text{if\/}\ \lVert z_{0}\rVert=R_{0}.Then system (1.1) has at least 2 nontrivial n⁢TnT-periodic solutions belonging to different periodicity classes with rotation number 𝜔.As observed in [5, § 3], given any n⁢TnT-periodic solution (x,y)(x,y) with n≥2n\geq 2, for every j∈{1,2,…,n-1}j\in\{1,2,\ldots,n-1\}, also xj⁢(t):=x⁢(t+j⁢T)x_{j}(t):=x(t+jT), yj⁢(t):=y⁢(t+j⁢T)y_{j}(t):=y(t+jT) is an n⁢TnT-periodic solution. In Theorem 2, all these solutions are considered to be equivalent, and it is said that they belong to the same periodicity class.Remark 1The information on the rotation number provided by Theorem 2 is very relevant. First, because the solutions with different rotation numbers are essentially different since, as paths in R2∖{(0,0)}{\mathbb{R}}^{2}\setminus\{(0,0)\}, they have a different fundamental group. Moreover, because Theorem 2, as stated, does not guarantee the minimality of the period n⁢TnT, except in the special case when gcd⁡(n,ω)=1\gcd(n,\omega)=1. Indeed, if (x⁢(t),y⁢(t))(x(t),y(t)) is ℓ⁢T\ell T-periodic for some integer ℓ<n\ell<n, then the rotation number in the interval [0,ℓ⁢T][0,\ell T] must be an integer, say ω1≥1\omega_{1}\geq 1. Thus, by the additivity property of the rotation numbers, it becomes apparent that rot⁡(z0;[0,n⁢ℓ⁢T])=ℓ⁢ω=ω1⁢n\operatorname{rot}(z_{0};[0,n\ell T])=\ell\omega=\omega_{1}n, where z0=(x⁢(0),y⁢(0))z_{0}=(x(0),y(0)), which contradicts the fact that gcd⁡(n,ω)=1\gcd(n,\omega)=1. Consequently, Theorem 2 is providing us with solutions of minimal period n⁢TnT if ω=1\omega=1.Remark 2By the continuous dependence of the solutions of (1.1) with respect to the initial conditions, for every ε>0\varepsilon>0, λ>0\lambda>0 and any integer n≥1n\geq 1, there exists δ≡δ⁢(n,λ,ε)>0\delta\equiv\delta(n,\lambda,\varepsilon)>0 such that the unique solution of (1.1), (x⁢(t),y⁢(t))(x(t),y(t)), satisfies (x⁢(t),y⁢(t))∈Dε(x(t),y(t))\in D_{\varepsilon} for all t∈[0,n⁢T]t\in[0,nT] if (x⁢(0),y⁢(0))∈Dδ(x(0),y(0))\in D_{\delta} (see [15, Proposition 2.1]). For every R>0R>0, we are denoting by DRD_{R} the disk of radius R>0R>0 centered at the origin.According to (1.7) and (1.8) and recalling that m=min⁡{k,ℓ}m=\min\{k,\ell\}, throughout the rest of this paper, we will assume that 2⁢n⁢m≥62nm\geq 6 if k=ℓk=\ell, and 2⁢n⁢m+1≥72nm+1\geq 7 if |k-ℓ|=1\lvert k-\ell\rvert=1. Thus, unifying both conditions, throughout the rest of this paper, we will actually assume that(2.2)n⁢m≥3.nm\geq 3.This condition entails, essentially, at least five alternations between the components of the supports of α⁢(t)\alpha(t) and β⁢(t)\beta(t), as illustrated in Figure 3.Figure 3An example of five alternations between the supports of 𝛼 and 𝛽.The main result of this paper reads as follows.Theorem 3Assume n⁢m≥3nm\geq 3. Then there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, the twist condition (2.1) in model (1.1) holds for ω≥1\omega\geq 1.This theorem analyzes a degenerate case for Hamiltonian systems of the form of (1.1), through the Poincaré–Birkhoff twist theorem, which had not been previously studied in this context, for as neither the monotonicity of α⁢(t)⁢f⁢(y)\alpha(t)f(y) or β⁢(t)⁢g⁢(x)\beta(t)g(x)for all 𝑡, nor the non-degeneration of α⁢(t)\alpha(t) and β⁢(t)\beta(t) are required (see [12, 5, 18, 10] and [15, § 1]).The technical details of the proof will be given in the special case when 𝛼 and 𝛽 satisfy (1.5), as the case when (1.6) holds follows similarly. By (1.3),min⁡{f′⁢(0),g′⁢(0)}>η\min\{f^{\prime}(0),g^{\prime}(0)\}>\etafor some constant η>0\eta>0. Thus, for sufficiently small |ζ|≤ε\lvert\zeta\rvert\leq\varepsilon,(2.3)f⁢(ζ)⁢ζ≥η⁢ζ2,g⁢(ζ)⁢ζ≥η⁢ζ2.f(\zeta)\zeta\geq\eta\zeta^{2},\quad g(\zeta)\zeta\geq\eta\zeta^{2}.Hence, since up to an additive constant,θ⁢(t)=arctan⁡y⁢(t)x⁢(t),\theta(t)=\arctan\frac{y(t)}{x(t)},differentiating with respect to 𝑡 and using the fact that (x⁢(t),y⁢(t))(x(t),y(t)) solves (1.1) yieldsθ′⁢(t)=y′⁢(t)⁢x⁢(t)-x′⁢(t)⁢y⁢(t)x2⁢(t)+y2⁢(t)=λ⁢β⁢(t)⁢g⁢(x⁢(t))⁢x⁢(t)+λ⁢α⁢(t)⁢f⁢(y⁢(t))⁢y⁢(t)x2⁢(t)+y2⁢(t)\theta^{\prime}(t)=\frac{y^{\prime}(t)x(t)-x^{\prime}(t)y(t)}{x^{2}(t)+y^{2}(t)}=\frac{\lambda\beta(t)g(x(t))x(t)+\lambda\alpha(t)f(y(t))y(t)}{x^{2}(t)+y^{2}(t)}for every t∈[0,n⁢T]t\in[0,nT]. So, owing to (2.3), we have that(2.4)θ′⁢(t)≥λ⁢η⁢[β⁢(t)⁢x2⁢(t)x2⁢(t)+y2⁢(t)+α⁢(t)⁢y2⁢(t)x2⁢(t)+y2⁢(t)]=λ⁢η⁢[β⁢(t)⁢cos2⁡θ⁢(t)+α⁢(t)⁢sin2⁡θ⁢(t)]≥0\theta^{\prime}(t)\geq\lambda\eta\biggl{[}\beta(t)\frac{x^{2}(t)}{x^{2}(t)+y^{2}(t)}+\alpha(t)\frac{y^{2}(t)}{x^{2}(t)+y^{2}(t)}\biggr{]}=\lambda\eta[\beta(t)\cos^{2}\theta(t)+\alpha(t)\sin^{2}\theta(t)]\geq 0for every t∈[0,n⁢T]t\in[0,nT]. In particular, θ⁢(t)\theta(t) is non-decreasing. Moreover, by Remark 2, for every integer n≥1n\geq 1, there exists δ>0\delta>0 such that (x⁢(t),y⁢(t))∈Dε(x(t),y(t))\in D_{\varepsilon} for all t∈[0,n⁢T]t\in[0,nT] if (x0,y0):=(x⁢(0),y⁢(0))∈Dδ(x_{0},y_{0}):=(x(0),y(0))\in D_{\delta}. This condition will be kept throughout the next lemmas and the proof of Theorem 3 in order to guarantee that the solution cannot escape from DεD_{\varepsilon}. Naturally, the bigger 𝜆 is, the smaller is 𝛿.Based on (2.4) and Remark 2, the next result holds. Essentially, it establishes that each of the components of the support of αi\alpha_{i} pushes the solutions of (1.1) from the first quadrant towards the second one, as well as from the third towards the fourth.Lemma 1Assume that there exists (ρ0,ρ1)⊊supp⁡α(\rho_{0},\rho_{1})\subsetneq\operatorname{supp}\alpha such that θ⁢(ρ0)∈[ω0,π-ω0]\theta(\rho_{0})\in[\omega_{0},\pi-\omega_{0}] for some ω0∈(0,π2)\omega_{0}\in(0,\frac{\pi}{2}). Then there exists λ1>0\lambda_{1}>0 such that θ⁢(ρ1)>π-ω0\theta(\rho_{1})>\pi-\omega_{0} for all λ>λ1\lambda>\lambda_{1}. Similarly, if θ⁢(ρ0)∈[π+ζ0,2⁢π-ζ0]\theta(\rho_{0})\in[\pi+\zeta_{0},2\pi-\zeta_{0}] for some ζ0∈(0,π2)\zeta_{0}\in(0,\frac{\pi}{2}), then θ⁢(ρ1)>2⁢π-ζ0\theta(\rho_{1})>2\pi-\zeta_{0} for sufficiently large 𝜆.ProofSince we are dealing with small solutions, it is apparent from (2.4) thatθ⁢(ρ1)=θ⁢(ρ0)+∫ρ0ρ1θ′⁢(s)⁢ds≥θ⁢(ρ0)+λ⁢η⁢∫ρ0ρ1α⁢(s)⁢y2⁢(ρ0)x2⁢(s)+y2⁢(ρ0)⁢ds\theta(\rho_{1})=\theta(\rho_{0})+\int_{\rho_{0}}^{\rho_{1}}\theta^{\prime}(s)\,ds\geq\theta(\rho_{0})+\lambda\eta\int_{\rho_{0}}^{\rho_{1}}\alpha(s)\frac{y^{2}(\rho_{0})}{x^{2}(s)+y^{2}(\rho_{0})}\,dsbecause β=0\beta=0 on (ρ0,ρ1)(\rho_{0},\rho_{1}) and, hence, y⁢(s)≡y⁢(ρ0)y(s)\equiv y(\rho_{0}) therein. Thus, settingν:=η⁢y2⁢(ρ0)ε2⁢∫ρ0ρ1α⁢(s)⁢ds\nu:=\frac{\eta y^{2}(\rho_{0})}{\varepsilon^{2}}\int_{\rho_{0}}^{\rho_{1}}\alpha(s)\,dsand taking into account that (x⁢(s),y⁢(s))=(x⁢(s),y⁢(ρ0))∈Dε(x(s),y(s))=(x(s),y(\rho_{0}))\in D_{\varepsilon} for all s∈(ρ0,ρ1)s\in(\rho_{0},\rho_{1}), it follows thatθ(ρ1)≥θ(ρ0)+λν≥ω0+λν>π-ω0 provided λ>π-2⁢ω0ν=:λ1.\theta(\rho_{1})\geq\theta(\rho_{0})+\lambda\nu\geq\omega_{0}+\lambda\nu>\pi-\omega_{0}\quad\text{provided}\quad\lambda>\frac{\pi-2\omega_{0}}{\nu}=:\lambda_{1}.Note that the bound λ1\lambda_{1} remains invariant if either θ⁢(ρ0)\theta(\rho_{0}) or y2⁢(ρ0)y^{2}(\rho_{0}) increases. Moreover, θ⁢(ρ1)\theta(\rho_{1}) increases with 𝜆 for any given (fixed) θ⁢(ρ0)\theta(\rho_{0}) and y2⁢(ρ0)y^{2}(\rho_{0}). Similarly, we have θ⁢(ρ1)>2⁢π-ζ0\theta(\rho_{1})>2\pi-\zeta_{0} for sufficiently large 𝜆 if θ⁢(ρ0)∈[π+ζ0,2⁢π-ζ0]\theta(\rho_{0})\in[\pi+\zeta_{0},2\pi-\zeta_{0}] for some ζ0∈(0,π2)\zeta_{0}\in(0,\frac{\pi}{2}). This ends the proof. ∎Analogously, the next result establishes that each of the components of the support of βi\beta_{i} pushes the solutions of (1.1) from the fourth quadrant towards the first, while it moves them from the second towards the third one.Lemma 2Assume that there exists (σ0,σ1)⊊supp⁡β(\sigma_{0},\sigma_{1})\subsetneq\operatorname{supp}\beta such that θ⁢(σ0)∈[-π2+τ0,π2-τ0]\theta(\sigma_{0})\in[-\frac{\pi}{2}+\tau_{0},\frac{\pi}{2}-\tau_{0}] for some τ0∈(0,π2)\tau_{0}\in(0,\frac{\pi}{2}). Then there exists μ1\mu_{1} such that θ⁢(σ1)>π2-τ0\theta(\sigma_{1})>\frac{\pi}{2}-\tau_{0} for all λ>μ1\lambda>\mu_{1}. Similarly, θ⁢(σ1)>3⁢π2-ξ0\theta(\sigma_{1})>\frac{3\pi}{2}-\xi_{0} for sufficiently large 𝜆 if θ⁢(σ0)∈[π2+ξ0,3⁢π2-ξ0]\theta(\sigma_{0})\in[\frac{\pi}{2}+\xi_{0},\frac{3\pi}{2}-\xi_{0}] for some ξ0∈(0,π2)\xi_{0}\in(0,\frac{\pi}{2}).ProofAs in Lemma 1, from (2.4), it follows thatθ⁢(σ1)=θ⁢(σ0)+∫σ0σ1θ′⁢(s)⁢ds≥θ⁢(σ0)+∫σ0σ1β⁢(s)⁢x2⁢(σ0)x2⁢(σ0)+y2⁢(s)⁢ds\theta(\sigma_{1})=\theta(\sigma_{0})+\int_{\sigma_{0}}^{\sigma_{1}}\theta^{\prime}(s)\,ds\geq\theta(\sigma_{0})+\int_{\sigma_{0}}^{\sigma_{1}}\beta(s)\frac{x^{2}(\sigma_{0})}{x^{2}(\sigma_{0})+y^{2}(s)}\,dsbecause α=0\alpha=0 on (σ0,σ1)(\sigma_{0},\sigma_{1}) and, hence, x⁢(s)≡x⁢(σ0)x(s)\equiv x(\sigma_{0}) therein. Thus, denotingς:=η⁢x2⁢(σ0)ε2⁢∫σ0σ1β⁢(s)⁢ds\varsigma:=\frac{\eta x^{2}(\sigma_{0})}{\varepsilon^{2}}\int_{\sigma_{0}}^{\sigma_{1}}\beta(s)\,dsand arguing as in Lemma 1, it is apparent thatθ(σ1)≥θ(σ0)+λς≥τ0+λς>π2-τ0 provided λ>π-4⁢τ02⁢ς=:μ1.\theta(\sigma_{1})\geq\theta(\sigma_{0})+\lambda\varsigma\geq\tau_{0}+\lambda\varsigma>\frac{\pi}{2}-\tau_{0}\quad\text{provided}\quad\lambda>\frac{\pi-4\tau_{0}}{2\varsigma}=:\mu_{1}.As highlighted in the proof of Lemma 1, the value of μ1\mu_{1} does not vary if either θ⁢(σ0)\theta(\sigma_{0}) or x2⁢(σ0)x^{2}(\sigma_{0}) increases. Similarly, θ⁢(σ1)\theta(\sigma_{1}) increases with 𝜆, and the second assertion of the lemma holds. This ends the proof. ∎Now, we are ready to prove Theorem 3.Proof of Theorem 3The proof is based on the version of the Poincaré–Birkhoff theorem collected in Theorem 2. First, we will prove that all small solutions in the disk DεD_{\varepsilon}, where 𝜀 is chosen sufficiently small so that (2.3) holds, have a rotation number greater than one. To prove this feature, we will distinguish between three different cases according to the precise location of their initial values, (x0,y0)(x_{0},y_{0}).Case 1: Assume that x0⁢y0>0x_{0}y_{0}>0. Then (x0,y0)(x_{0},y_{0}) lies either in the first or in the third quadrant. Both cases being similar, we will pay attention only to the case when x0>0x_{0}>0 and y0>0y_{0}>0. Then θ⁢(t01)∈(0,π2)\theta(t_{0}^{1})\in(0,\frac{\pi}{2}). Thus, by Lemma 1, there exists λ1>0\lambda_{1}>0 such that θ⁢(t11)>π-θ⁢(t01)\theta(t_{1}^{1})>\pi-\theta(t_{0}^{1}) for all λ>λ1\lambda>\lambda_{1}. Since α=β=0\alpha=\beta=0 in [t11,t21][t_{1}^{1},t_{2}^{1}], this implies thatθ⁢(t21)=θ⁢(t11)>π-θ⁢(t01).\theta(t_{2}^{1})=\theta(t_{1}^{1})>\pi-\theta(t_{0}^{1}).Thus, by Lemma 2, there exists λ2>0\lambda_{2}>0 such that θ⁢(t31)>π+θ⁢(t01)\theta(t_{3}^{1})>\pi+\theta(t_{0}^{1}) as soon as λ>max⁡{λ1,λ2}\lambda>\max\{\lambda_{1},\lambda_{2}\}. Also by Lemma 1, there exists λ3>0\lambda_{3}>0 such that θ⁢(t12)>2⁢π-θ⁢(t01)\theta(t_{1}^{2})>2\pi-\theta(t_{0}^{1}) for every λ>max⁡{λ1,λ2,λ3}\lambda>\max\{\lambda_{1},\lambda_{2},\lambda_{3}\}, and due to Lemma 2, there is λ4>0\lambda_{4}>0 such that θ⁢(t32)>2⁢π+θ⁢(t01)\theta(t_{3}^{2})>2\pi+\theta(t_{0}^{1}) for all λ>max⁡{λ1,λ2,λ3,λ4}\lambda>\max\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\}. Therefore, the solution with initial values (x0,y0)(x_{0},y_{0}) completes an entire turn in the interval [0,t32][0,t_{3}^{2}] for every λ>max⁡{λ1,λ2,λ3,λ4}\lambda>\max\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\}. In order to apply Theorem 2, it remains to show the existence of a uniform bound, Λ1>0\Lambda_{1}>0, such that the solutions with initial data in the sector of the circumference of radius r0r_{0} within the first quadrant,S1:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x>0,y>0},S_{1}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x>0,\,y>0\},have a rotation number greater than one for all λ>Λ1\lambda>\Lambda_{1} if 0<r0<ε0<r_{0}<\varepsilon. To prove it, we consider an angle ω~0∈(0,π2)\tilde{\omega}_{0}\in(0,\frac{\pi}{2}) and the sectors of S1S_{1} defined byS1,ω~0+:={z=(x,y)∈R2:∥z∥=r0,r0⁢cos⁡ω~0≥x>0,y>0},S1,ω~0-:=S1∖S1,ω~0+.S_{1,\tilde{\omega}_{0}}^{+}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0},\,r_{0}\cos\tilde{\omega}_{0}\geq x>0,\,y>0\},\quad S_{1,\tilde{\omega}_{0}}^{-}:=S_{1}\setminus S_{1,\tilde{\omega}_{0}}^{+}.Using recursively Lemmas 1 and 2 as above, it becomes apparent that there exists M1>0M_{1}>0 such that the solutions with initial data in S1,ω~0+S_{1,\tilde{\omega}_{0}}^{+} complete an entire turn in the interval [0,t32][0,t_{3}^{2}] for every λ>M1\lambda>M_{1}. For every t>0t>0, let us denote by Φt\Phi_{t} the Poincaré map at time 𝑡 of system (1.1), if defined. Then, by Lemma 2, there exists ω1>0\omega_{1}>0 such thatΦt31⁢(S1,ω~0-)⊂{(r,θ):0<r≤ε, 0<ω1≤θ<3⁢π2}.\Phi_{t_{3}^{1}}(S_{1,\tilde{\omega}_{0}}^{-})\subset\biggl{\{}(r,\theta):0<r\leq\varepsilon,\,0<\omega_{1}\leq\theta<\frac{3\pi}{2}\biggr{\}}.Therefore, as for S1,ω~0+S_{1,\tilde{\omega}_{0}}^{+}, there is M2M_{2} such that S1,ω~0-S_{1,\tilde{\omega}_{0}}^{-} completes one turn in [0,t33][0,t_{3}^{3}] for all λ>max⁡{M1,M2}\lambda>\max\{M_{1},M_{2}\}.Adapting the previous argument, it readily follows the existence of M~1,M~2>0\tilde{M}_{1},\tilde{M}_{2}>0 such that the solutions of (1.1) with initial data inS3:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x<0,y<0}S_{3}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x<0,\,y<0\}have a rotation number greater than one for all λ>max⁡{M~1,M~2}\lambda>\max\{\tilde{M}_{1},\tilde{M}_{2}\}. Consequently, for every z0∈S1∪S3z_{0}\in S_{1}\cup S_{3}, it is apparent that rot⁡(z0;[0,n⁢T])>1\operatorname{rot}(z_{0};[0,nT])>1 for all λ>Λ1:=max⁡{M1,M2,M~1,M~2}\lambda>\Lambda_{1}:=\max\{M_{1},M_{2},\tilde{M}_{1},\tilde{M}_{2}\}.Case 2: Assume that x0⁢y0<0x_{0}y_{0}<0. Most of the attention will be focused to the special case when x0<0x_{0}<0 and y0>0y_{0}>0, as the case x0>0x_{0}>0 and y0<0y_{0}<0 is analogous. Obviously, in this case, θ⁢(t01)∈(π2,π)\theta(t_{0}^{1})\in(\frac{\pi}{2},\pi). As in case 1, it should be proved the existence of a uniform bound, Λ2\Lambda_{2}, such that the solutions with initial data in the quadrant sectorS2:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x<0,y>0}S_{2}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x<0,\,y>0\}have rotation number greater than one for all λ>Λ2\lambda>\Lambda_{2} if 0<r0<ε0<r_{0}<\varepsilon. By Lemma 1, there exists ω2>π2\omega_{2}>\frac{\pi}{2} such thatΦt11⁢(S2)⊂{(r,θ):0<r≤ε,π2<ω2≤θ<π}.\Phi_{t_{1}^{1}}(S_{2})\subset\biggl{\{}(r,\theta):0<r\leq\varepsilon,\,\frac{\pi}{2}<\omega_{2}\leq\theta<\pi\biggr{\}}.Thus, as in case 1, we have already proven that, once the solution reaches the second quadrant, being separated away from π2\frac{\pi}{2}, it must have a rotation number greater than one for sufficiently large 𝜆 (which was a direct consequence from Lemmas 1 and 2), there exists Λ2>0\Lambda_{2}>0 such that the solution with θ⁢(t11)=ω2>π2\theta(t_{1}^{1})=\omega_{2}>\frac{\pi}{2} completes one turn for all λ>Λ2\lambda>\Lambda_{2}. Moreover, by the monotonicity properties of Lemmas 1 and 2, the solutions with initial data in S2S_{2} have rotation number greater than one for all λ>Λ2\lambda>\Lambda_{2}. Since the previous argument can be easily adapted to deal withS4:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x>0,y<0},S_{4}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x>0,\,y<0\},it becomes apparent that, for every z0∈S2∪S4z_{0}\in S_{2}\cup S_{4}, rot⁡(z0;[0,n⁢T])>1\operatorname{rot}(z_{0};[0,nT])>1 for all λ>Λ2\lambda>\Lambda_{2}.Case 3: Assume x0⁢y0=0x_{0}y_{0}=0, i.e., (x0,y0)(x_{0},y_{0}) lies on some coordinate axis. Without loss of generality, we can assume that x0>0x_{0}>0 and y0=0y_{0}=0, as the remaining cases can be treated similarly. Then, since y0=0y_{0}=0 and β=0\beta=0 on [t01,t11][t_{0}^{1},t_{1}^{1}], integrating (1.1) yields θ⁢(t01)=θ⁢(t21)=0\theta(t_{0}^{1})=\theta(t_{2}^{1})=0. Thus, by Lemma 2, for every ω∈(0,π2)\omega\in(0,\frac{\pi}{2}), there exists μ1:=μ1⁢(ω)\mu_{1}:=\mu_{1}(\omega) such that θ⁢(t31)>ω\theta(t_{3}^{1})>\omega for all λ>μ1\lambda>\mu_{1}. Thus, much like in case 1, owing to Lemmas 1 and 2, there exist μ2,μ3,μ4,μ5>0\mu_{2},\mu_{3},\mu_{4},\mu_{5}>0, depending on 𝜔, such thatθ⁢(t12)>π-ω\theta(t_{1}^{2})>\pi-\omegaif⁢λ>max⁡{μ1,μ2},\text{if}\ \lambda>\max\{\mu_{1},\mu_{2}\},θ⁢(t32)>π+ω\theta(t_{3}^{2})>\pi+\omegaif⁢λ>max⁡{μ1,μ2,μ3},\text{if}\ \lambda>\max\{\mu_{1},\mu_{2},\mu_{3}\},θ⁢(t13)>2⁢π-ω\theta(t_{1}^{3})>2\pi-\omegaif⁢λ>max⁡{μ1,μ2,μ3,μ4},\text{if}\ \lambda>\max\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\},θ⁢(t33)>2⁢π\theta(t_{3}^{3})>2\piifλ>max{μ1,μ2,μ3,μ4,μ5}=:Λ3,1.\text{if}\ \lambda>\max\{\mu_{1},\mu_{2},\mu_{3},\mu_{4},\mu_{5}\}=:\Lambda_{3,1}.Therefore, the solution completes one turn in the time interval [0,t33][0,t_{3}^{3}] for all λ>Λ3,1\lambda>\Lambda_{3,1}. Similarly, it can be easily shown that the solutions complete a turn in each of the remaining three cases when x0=0x_{0}=0 and y0>0y_{0}>0, x0<0x_{0}<0 and y0=0y_{0}=0, or x0=0x_{0}=0 and y0<0y_{0}<0, for λ>Λ3,2\lambda>\Lambda_{3,2}, λ>Λ3,3\lambda>\Lambda_{3,3} and λ>Λ3,4\lambda>\Lambda_{3,4}, respectively. Thus, takingΛ3:=max⁡{Λ3,1,Λ3,2,Λ3,3,Λ3,4},\Lambda_{3}:=\max\{\Lambda_{3,1},\Lambda_{3,2},\Lambda_{3,3},\Lambda_{3,4}\},it becomes apparent that rot⁡(z0;[0,n⁢T])>1\operatorname{rot}(z_{0};[0,nT])>1 provided λ>Λ3\lambda>\Lambda_{3} andz0∈S0:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x⁢y=0}.z_{0}\in S_{0}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ xy=0\}.Subsequently, we setλn:=max⁡{Λ1,Λ2,Λ3}.\lambda_{n}:=\max\{\Lambda_{1},\Lambda_{2},\Lambda_{3}\}.By Remark 2 and the analysis already done in the proof of the theorem, it is apparent that, for every λ>λn\lambda>\lambda_{n}, there exists 0<r0<δ⁢(n,λ,ε)0<r_{0}<\delta(n,\lambda,\varepsilon) such that, for every z0=(x0,y0)z_{0}=(x_{0},y_{0}) with ∥z0∥=r0\lVert z_{0}\rVert=r_{0},(x⁢(t),y⁢(t))∈Dε for all⁢t∈[0,n⁢T](x(t),y(t))\in D_{\varepsilon}\quad\text{for all}\ t\in[0,nT]and(2.5)rot⁡(z0;[0,n⁢T])>1.\operatorname{rot}(z_{0};[0,nT])>1.In order to apply Theorem 2, it remains to prove that, for sufficiently large λ>0\lambda>0, the large solutions do not rotate. As the proof of this feature follows the general scheme of the proof of [15, Theorem 2.1], we will simply sketch it here. Being analogous the remaining cases, the proof will be delivered in the special case when condition (g-)(g_{-}) holds in (1.4).We proceed by contradiction assuming that, regardless the size of the initial data (x0,y0)(x_{0},y_{0}), the solution (x⁢(t),y⁢(t))(x(t),y(t)) completes at least one turn for sufficiently large 𝜆. Thus, without loss of generality, changing the initial data if necessary, we can assume that (x⁢(t),y⁢(t))(x(t),y(t)) goes across the entire third quadrant. In such a case, there is an interval [s0,s1]⊂[0,n⁢T][s_{0},s_{1}]\subset[0,nT] such that y⁢(s0)=0=x⁢(s1)y(s_{0})=0=x(s_{1}) and x⁢(t)<0x(t)<0, y⁢(t)<0y(t)<0 for every t∈(s0,s1)t\in(s_{0},s_{1}). Thus, by (g-)(g_{-}), it becomes apparent that, setting B:=∫0Tβ⁢(s)⁢dsB:=\int_{0}^{T}\beta(s)\,ds,|y⁢(t)|=λ⁢|∫s0tβ⁢(s)⁢g⁢(x⁢(s))⁢ds|≤λ⁢M⁢∫0n⁢Tβ⁢(s)⁢ds=λ⁢M⁢n⁢B\lvert y(t)\rvert=\lambda\,\Biggl{\lvert}{}\int_{s_{0}}^{t}\beta(s)g(x(s))\,ds\Biggr{\rvert}\leq\lambda M\int_{0}^{nT}\beta(s)\,ds=\lambda MnBfor every t∈[s0,s1]t\in[s_{0},s_{1}]. Hence, defining N:=max⁡{|f⁢(y)|:|y|≤λ⁢M⁢n⁢B}N:=\max\{\lvert f(y)\rvert:\lvert y\rvert\leq\lambda MnB\}, it follows that, for every t∈[s0,s1]t\in[s_{0},s_{1}],|x⁢(t)|=|λ⁢∫ts1α⁢(s)⁢f⁢(y⁢(s))⁢ds|≤λ⁢N⁢∫0n⁢Tα⁢(s)⁢ds=λ⁢N⁢n⁢A,\lvert x(t)\rvert=\Biggl{\lvert}\lambda\int_{t}^{s_{1}}\alpha(s)f(y(s))\,ds\Biggr{\rvert}\leq\lambda N\int_{0}^{nT}\alpha(s)\,ds=\lambda NnA,where A:=∫0Tα⁢(s)⁢dsA:=\int_{0}^{T}\alpha(s)\,ds. Suppose that, for some t~∈[0,n⁢T]\tilde{t}\in[0,nT],x2⁢(t~)+y2⁢(t~)>λ2⁢n2⁢(M2⁢B2+N2⁢A2)≡R2x^{2}(\tilde{t})+y^{2}(\tilde{t})>\lambda^{2}n^{2}(M^{2}B^{2}+N^{2}A^{2})\equiv R^{2}with x⁢(t~)<0x(\tilde{t})<0 and y⁢(t~)<0y(\tilde{t})<0. Then the solution (x⁢(t),y⁢(t))(x(t),y(t)) cannot cross entirely the third quadrant. At this stage, the proof follows almost mutatis mutandis the steps of the proof of [15, Theorem 2.1], where the reader is sent for any further details. According to it, there exists a radius R0≥RR_{0}\geq R such that, for every solution with z0=x02+y02≥R0z_{0}=x_{0}^{2}+y_{0}^{2}\geq R_{0},(2.6)rot⁡(z0;[0,n⁢T])<1.\operatorname{rot}(z_{0};[0,nT])<1.By (2.5) and (2.6), the twist condition holds, and hence, by Theorem 2, system (1.1) admits at least two nontrivial n⁢TnT-periodic solutions belonging to different periodicity classes with rotation number ω≤n\omega\leq n for sufficiently large 𝜆. This concludes the proof. ∎In order to apply Theorem 2, the distribution of the weight functions settled by (2.2) is optimal. Indeed, if αi=0\alpha_{i}=0 or βi=0\beta_{i}=0 for some i∈{1,2,3}i\in\{1,2,3\}, then each of the points (-r0,0)(-r_{0},0) and (r0,0)(r_{0},0), for sufficiently small r0>0r_{0}>0, have rotation number less than one in the interval [0,T][0,T].Remark 3As already observed in [15, Remark 3], without any significant change in the proof, a slightly more general version of Theorem 2 can be proven by assuming f,gf,g only continuous (and not locally Lipschitz) and replacing the condition on the derivatives in (1.3) with the following one:0<lim inf|y|→0⁡f⁢(y)y≤lim sup|y|→0⁡f⁢(y)y<∞,0<lim inf|x|→0⁡g⁢(x)x≤lim sup|x|→0⁡g⁢(x)x<∞.0<\liminf_{\lvert y\rvert\to 0}\frac{f(y)}{y}\leq\limsup_{\lvert y\rvert\to 0}\frac{f(y)}{y}<\infty,\quad 0<\liminf_{\lvert x\rvert\to 0}\frac{g(x)}{x}\leq\limsup_{\lvert x\rvert\to 0}\frac{g(x)}{x}<\infty.To this aim, instead of Theorem 2, one can apply the generalized version of the Poincaré–Birkhoff theorem due to Fonda and Ureña [11] for Hamiltonian systems where the uniqueness of the solutions of the initial value problems is not required (see also [8, Theorem 10.6.1] for the precise statement).3Counting 𝑇-Periodic Solutions and Subharmonics of (1.1)This section applies Theorem 3 to model (1.1) when condition (2.2) holds. Recall that either k=ℓk=\ell, or |k-ℓ|=1\lvert k-\ell\rvert=1 and m=min⁡{k,ℓ}m=\min\{k,\ell\}. Based on Theorem 3, the next result holds.Theorem 4Assume that n⁢m≥3nm\geq 3 for some integer n≥1n\geq 1. Then there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, (1.1) possesses at least σ⁢(n)\sigma(n) periodic solutions with period n⁢TnT, whereσ⁢(n):={2⁢h⁢mif⁢n=3⁢h,2⁢(h⁢m+[m3])if⁢n=3⁢h+1,2⁢(h⁢m+[2⁢m3])if⁢n=3⁢h+2.\sigma(n):=\begin{cases}2hm&\text{if}\ n=3h,\\ 2\bigl{(}hm+\bigl{[}\frac{m}{3}\bigr{]}\bigr{)}&\text{if}\ n=3h+1,\\ 2\bigl{(}hm+\bigl{[}\frac{2m}{3}\bigr{]}\bigr{)}&\text{if}\ n=3h+2.\end{cases}Moreover, settingγ⁢(n):=min⁡{γ≥0:gcd⁡(n,σ⁢(n)2-γ)=1},\gamma(n):=\min\biggl{\{}\gamma\geq 0:\gcd\biggl{(}n,\frac{\sigma(n)}{2}-\gamma\biggr{)}=1\biggr{\}},it turns out that, for every λ>λn\lambda>\lambda_{n}, (1.1) has at least σ⁢(n)-2⁢γ⁢(n)\sigma(n)-2\gamma(n) periodic solutions with minimal period n⁢TnT.ProofSuppose k=ℓk=\ell. Then m=k=ℓm=k=\ell. Hence, according to (1.7), the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] equals(3.1)2⁢n⁢k=2⁢n⁢m.2nk=2nm.Thus, if n=3⁢hn=3h for some integer h≥1h\geq 1, the sum of 𝛼-intervals and 𝛽-intervals in [0,n⁢T][0,nT] is 6⁢h⁢k6hk. Hence, by Theorem 3, there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, the solutions of (1.1) with sufficiently small z0=(x0,y0)z_{0}=(x_{0},y_{0}) complete h⁢khk turns, whereas the solutions with sufficiently large z0z_{0} cannot complete any. Therefore, by Theorem 2, (1.1) has at least two n⁢TnT-periodic coexistence states with rotation number j∈{1,2,…,h⁢k}j\in\{1,2,\ldots,hk\}. Consequently, (1.1) possesses at least 2⁢h⁢k=σ⁢(n)2hk=\sigma(n) coexistence states with period n⁢TnT.Now, assume that n=3⁢h+1n=3h+1 for some integer h≥0h\geq 0. Then there are a total of2⁢m⁢n=2⁢k⁢(3⁢h+1)=6⁢h⁢k+2⁢k=6⁢(h⁢k+k3)2mn=2k(3h+1)=6hk+2k=6\biggl{(}hk+\frac{k}{3}\biggr{)}𝛼 and 𝛽-intervals in [0,n⁢T][0,nT]. Thus, by Theorem 3, there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, the solutions of (1.1) with sufficiently small z0z_{0} complete h⁢k+[k3]hk+\bigl{[}\frac{k}{3}\bigr{]} turns, while the solutions with large initial data cannot rotate. Therefore, thanks to Theorem 2, (1.1) possesses at least2⁢(h⁢k+[k3])=σ⁢(n)2\biggl{(}hk+\biggl{[}\frac{k}{3}\biggr{]}\biggr{)}=\sigma(n)periodic coexistence states of period n⁢TnT.Similarly, according to Theorems 2 and 3, when n=3⁢h+2n=3h+2 for some integer h≥0h\geq 0, there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, (1.1) possesses at least2⁢(h⁢k+[2⁢k3])=σ⁢(n)2\biggl{(}hk+\biggl{[}\frac{2k}{3}\biggr{]}\biggr{)}=\sigma(n)coexistence states with period n⁢TnT.The last assertion of the theorem will be derived from the fact that, owing to Remark 1, any n⁢TnT-periodic coexistence state of (1.1) such that, for some 0<r0<R00<r_{0}<R_{0}, it satisfies(3.2){rot⁡(z0;[0,n⁢T])>ωif⁢∥z0∥=r0,rot⁡(z0;[0,n⁢T])<ωif⁢∥z0∥=R0,\begin{cases}\operatorname{rot}(z_{0};[0,nT])>\omega&\text{if}\ \lVert z_{0}\rVert=r_{0},\\ \operatorname{rot}(z_{0};[0,nT])<\omega&\text{if}\ \lVert z_{0}\rVert=R_{0},\end{cases}has minimal period n⁢TnT if gcd⁡(n,ω)=1\gcd(n,\omega)=1. In all the cases covered by Theorem 4, we have actually proven the existence of 0<r0<R00<r_{0}<R_{0} such that{rot⁡(z0;[0,n⁢T])>σ⁢(n)2if⁢∥z0∥=r0,rot⁡(z0;[0,n⁢T])<1if⁢∥z0∥=R0,\begin{cases}\operatorname{rot}(z_{0};[0,nT])>\frac{\sigma(n)}{2}&\text{if}\ \lVert z_{0}\rVert=r_{0},\\ \operatorname{rot}(z_{0};[0,nT])<1&\text{if}\ \lVert z_{0}\rVert=R_{0},\end{cases}by the definition of σ⁢(n)\sigma(n). Thus, (3.2) holds for the choice ω=σ⁢(n)2\omega=\frac{\sigma(n)}{2}. In the case gcd⁡(n,σ⁢(n)2)=1\gcd(n,\frac{\sigma(n)}{2})=1, by Remark 1, problem (1.1) possesses at least σ⁢(n)\sigma(n) coexistence states with minimal period n⁢TnT. This ends the proof in this case because we can take γ=0\gamma=0 in (3.1), and hence, γ⁢(n)=0\gamma(n)=0.Subsequently, we assume that gcd⁡(n,σ⁢(n)2)≠1\gcd(n,\frac{\sigma(n)}{2})\neq 1 and consider the unique integer j≥1j\geq 1 such that(3.3)gcd⁡(n,σ⁢(n)2-j)=1 and gcd⁡(n,σ⁢(n)2-i)≠1 for all⁢ 0≤i<j.\gcd\biggl{(}n,\frac{\sigma(n)}{2}-j\biggr{)}=1\quad\text{and}\quad\gcd\biggl{(}n,\frac{\sigma(n)}{2}-i\biggr{)}\neq 1\quad\text{for all}\ 0\leq i<j.In such a case, we can make the choice ω=σ⁢(n)2-j\omega=\frac{\sigma(n)}{2}-j. By (3.3), gcd⁡(n,ω)=1\gcd(n,\omega)=1. Moreover, as soon as ∥z0∥=r0\lVert z_{0}\rVert=r_{0}, we have thatrot⁡(z0;[0,n⁢T])>σ⁢(n)2>σ⁢(n)2-j=ω.\operatorname{rot}(z_{0};[0,nT])>\frac{\sigma(n)}{2}>\frac{\sigma(n)}{2}-j=\omega.And due to (3.3), it is apparent that, whenever ∥z0∥=R0\lVert z_{0}\rVert=R_{0},rot⁡(z0;[0,n⁢T])<1≤σ⁢(n)2-j=ω.\operatorname{rot}(z_{0};[0,nT])<1\leq\frac{\sigma(n)}{2}-j=\omega.Indeed, if σ⁢(n)2-j<1\frac{\sigma(n)}{2}-j<1, then there exists 0≤i<j0\leq i<j such that σ⁢(n)2-i=1\frac{\sigma(n)}{2}-i=1, and hence,gcd⁡(n,σ⁢(n)2-i)=gcd⁡(n,1)=1,\gcd\biggl{(}n,\frac{\sigma(n)}{2}-i\biggr{)}=\gcd(n,1)=1,contradicting the minimality of 𝑗. Therefore, by Remark 1, it becomes apparent that (1.1) has at least2⁢ω=2⁢(σ⁢(n)2-j)=σ⁢(n)-2⁢j=σ⁢(n)-2⁢γ⁢(n)2\omega=2\biggl{(}\frac{\sigma(n)}{2}-j\biggr{)}=\sigma(n)-2j=\sigma(n)-2\gamma(n)coexistence states with minimal period n⁢TnT. The proof is complete when k=ℓk=\ell.Now, assume that k=ℓ+1k=\ell+1. Then m=ℓm=\ell. Thus, according to (1.8), the total number of the 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] is n⁢m+1+n⁢m=2⁢n⁢m+1nm+1+nm=2nm+1. As the integers 2⁢n⁢m+12nm+1 and 2⁢n⁢m2nm, going back to (3.1), have the same divisibility properties by 6, the result when k=ℓ+1k=\ell+1 follows the same patterns as for k=ℓk=\ell. Similarly, the same result holds when ℓ=k+1\ell=k+1. This concludes the proof. ∎Remark 4As far as it concerns the cases not treated in this paper when n⁢(k+ℓ)≤5n(k+\ell)\leq 5, so far, it is known that if n⁢(k+ℓ)≤3n(k+\ell)\leq 3, then (1.1) does not admit any n⁢TnT-periodic solutions because the condition |Z|=0\lvert Z\rvert=0 ensures that no solution of (1.1) different from (0,0)(0,0), say (x⁢(t),y⁢(t))(x(t),y(t)), can complete one turn around the origin. Thus, it cannot satisfy (x⁢(0),y⁢(0))=(x⁢(n⁢T),y⁢(n⁢T))(x(0),y(0))=(x(nT),y(nT)) for some n≥1n\geq 1. The cases when n⁢(k+ℓ)=4,5n(k+\ell)=4,5 remain outside the general scope of this paper and will be analyzed elsewhere.4An Application to a Class of Predator-Prey ModelsThe non-autonomous planar Hamiltonian system (1.1) covers a large number of mathematical models of physical and biological nature. In particular, for the special choice f⁢(y)=ey-1f(y)=e^{y}-1 and g⁢(x)=ex-1g(x)=e^{x}-1, system (1.1) can be written, through the change of variables x=log⁡ux=\log u and y=log⁡vy=\log v, as(4.1){u′=λ⁢α⁢(t)⁢u⁢(1-v),v′=λ⁢β⁢(t)⁢v⁢(-1+u),\left\{\begin{aligned} \displaystyle{}u^{\prime}&\displaystyle=\lambda\alpha(t)u(1-v),\\ \displaystyle v^{\prime}&\displaystyle=\lambda\beta(t)v(-1+u),\end{aligned}\right.which is a non-autonomous 𝑇-periodic predator-prey model of Volterra type. As shown in [1, Section 5] and in [15, Introduction], system (4.1) can be obtained from the Volterra system with periodic coefficients{p′=λ⁢p⁢(a⁢(t)⁢p-b⁢(t)⁢q),q′=λ⁢q⁢(-c⁢(t)+d⁢(t)⁢p),\left\{\begin{aligned} \displaystyle{}p^{\prime}&\displaystyle=\lambda p\bigl{(}a(t)p-b(t)q\bigr{)},\\ \displaystyle q^{\prime}&\displaystyle=\lambda q\bigl{(}-c(t)+d(t)p\bigr{)},\end{aligned}\right.after a suitable change of variables. It is clear that the (nontrivial) n⁢TnT-periodic solutions of (1.1) are the n⁢TnT-periodic coexistence states of (4.1). By a coexistence state, it is meant a component-wise positive solution pair. This model was introduced in a degenerate setting in [16, 13] and later analyzed in [14] in the very special case when supp⁡α⊂[0,T2]\operatorname{supp}\alpha\subset\bigl{[}0,\frac{T}{2}\bigr{]} and supp⁡β⊂[T2,T]\operatorname{supp}\beta\subset\bigl{[}\frac{T}{2},T\bigr{]}. Since the functions f⁢(y)=ey-1f(y)=e^{y}-1, g⁢(x)=ex-1g(x)=e^{x}-1 satisfy (1.3) and (1.4), according to Theorems 2, 3 and 4, system (4.1) has at least σ⁢(n)\sigma(n) coexistence states with period n⁢TnT provided n⁢(k+ℓ)≥6n(k+\ell)\geq 6, among them, σ⁢(n)-2⁢γ⁢(n)\sigma(n)-2\gamma(n) with minimal period n⁢TnT. By Remark 4, system (4.1) cannot admit any n⁢TnT-periodic coexistence state if n⁢(k+ℓ)≤3n(k+\ell)\leq 3. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advanced Nonlinear Studies de Gruyter

The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems

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de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1536-1365
eISSN
2169-0375
DOI
10.1515/ans-2021-2137
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Abstract

1IntroductionThis paper studies the non-autonomous planar Hamiltonian system(1.1){x′=-λ⁢α⁢(t)⁢f⁢(y),y′=λ⁢β⁢(t)⁢g⁢(x),\left\{\begin{aligned} \displaystyle{}x^{\prime}&\displaystyle=-\lambda\alpha(t)f(y),\\ \displaystyle y^{\prime}&\displaystyle=\lambda\beta(t)g(x),\end{aligned}\right.where λ>0\lambda>0 is regarded as a real parameter, and given a real number T>0T>0, 𝛼 and 𝛽 are nonnegative 𝑇-periodic continuous functions such thatA:=∫0Tα>0,B:=∫0Tβ>0,A:=\int_{0}^{T}\alpha>0,\quad B:=\int_{0}^{T}\beta>0,for which the set(1.2)Z:=supp⁡α∩supp⁡βZ:=\operatorname{supp}\alpha\cap\operatorname{supp}\betahas Lebesgue measure zero, |Z|=0\lvert Z\rvert=0. This is why model (1.1) is said to be degenerate.In (1.1), f,g∈C⁢(R)f,g\in\mathcal{C}({{\mathbb{R}}}) are locally Lipschitz continuous functions such that f,g∈C1⁢(-ρ,ρ)f,g\in\mathcal{C}^{1}(-\rho,\rho) for some ρ>0\rho>0 and(1.3){f⁢(0)=0,f⁢(y)⁢y>0for all⁢y≠0,g⁢(0)=0,g⁢(x)⁢x>0for all⁢x≠0,f′⁢(0)>0,g′⁢(0)>0.\left\{\begin{aligned} \displaystyle{}f(0)&\displaystyle=0,&\displaystyle f(y)y&\displaystyle>0&&\displaystyle\text{for all}\ y\neq 0,\\ \displaystyle g(0)&\displaystyle=0,&\displaystyle g(x)x&\displaystyle>0&&\displaystyle\text{for all}\ x\neq 0,\\ \displaystyle f^{\prime}(0)&\displaystyle>0,&\displaystyle g^{\prime}(0)&\displaystyle>0.\end{aligned}\right.Moreover, it is assumed that either 𝑓 or 𝑔 satisfies one of the following four conditions:(1.4)(f-)(f_{-})f⁢is bounded in⁢R-,f\ \text{is bounded in}\ \mathbb{R}^{-},(f+)(f_{+})f⁢is bounded in⁢R+,f\ \text{is bounded in}\ \mathbb{R}^{+},(g-)(g_{-})g⁢is bounded in⁢R-,g\ \text{is bounded in}\ \mathbb{R}^{-},(g+)(g_{+})g⁢is bounded in⁢R+.g\ \text{is bounded in}\ \mathbb{R}^{+}.This paper analyzes the existence of n⁢TnT-periodic solutions of model (1.1) for any integer n≥1n\geq 1. Those with n≥2n\geq 2 (and 𝑛 minimal) are often referred to as subharmonics of order 𝑛. Besides |Z|=0\lvert Z\rvert=0, through this paper, we assume that, given two positive integers k,ℓ≥1k,\ell\geq 1 such that |k-ℓ|≤1\lvert k-\ell\rvert\leq 1, there exist k+ℓk+\ell continuous functions in the interval [0,T][0,T], αi⪈0\alpha_{i}\gneq 0, 1≤i≤k1\leq i\leq k, and βj⪈0\beta_{j}\gneq 0, 1≤j≤ℓ1\leq j\leq\ell, such that α=α1+α2+⋯+αk\alpha=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{k}, β=β1+β2+⋯+βℓ\beta=\beta_{1}+\beta_{2}+\cdots+\beta_{\ell}, with(1.5)supp⁡αi⊆[t0i,t1i] and supp⁡βj⊆[t2j,t3j]\operatorname{supp}\alpha_{i}\subseteq[t_{0}^{i},t_{1}^{i}]\quad\text{and}\quad\operatorname{supp}\beta_{j}\subseteq[t_{2}^{j},t_{3}^{j}]for some partition of [0,T][0,T],0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0k<t1k≤t2k<t3k≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{k}<t_{1}^{k}\leq t_{2}^{k}<t_{3}^{k}\leq Tif⁢k=ℓ,or\text{if}\ k=\ell,\,\text{or}0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0k<t1k≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{k}<t_{1}^{k}\leq Tif⁢k=ℓ+1.\text{if}\ k=\ell+1.Similarly, we also consider the case when, instead of (1.5),(1.6)supp⁡βj⊆[t0j,t1j] and supp⁡αi⊆[t2i,t3i]\operatorname{supp}\beta_{j}\subseteq[t_{0}^{j},t_{1}^{j}]\quad\text{and}\quad\operatorname{supp}\alpha_{i}\subseteq[t_{2}^{i},t_{3}^{i}]for some partition of [0,T][0,T],0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0ℓ<t1ℓ≤t2ℓ<t3ℓ≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{\ell}<t_{1}^{\ell}\leq t_{2}^{\ell}<t_{3}^{\ell}\leq Tif⁢ℓ=k,or\text{if}\ \ell=k,\,\text{or}0≤t01<t11≤t21<t31≤t02<t12≤t22<t32≤⋯≤t0ℓ<t1ℓ≤T0\leq t_{0}^{1}<t_{1}^{1}\leq t_{2}^{1}<t_{3}^{1}\leq t_{0}^{2}<t_{1}^{2}\leq t_{2}^{2}<t_{3}^{2}\leq\cdots\leq t_{0}^{\ell}<t_{1}^{\ell}\leq Tif⁢ℓ=k+1.\text{if}\ \ell=k+1.We will refer to an 𝛼-interval (resp. 𝛽-interval) as the maximal interval 𝐼, where |supp⁡β|I|=0\lvert\operatorname{supp}\beta|_{I}\rvert=0 and |supp⁡α|I|>0\lvert\operatorname{supp}\alpha|_{I}\rvert>0 (resp. |supp⁡α|I|=0\lvert\operatorname{supp}\alpha|_{I}\rvert=0 and |supp⁡β|I|>0\lvert\operatorname{supp}\beta|_{I}\rvert>0). So the total number of 𝛼-intervals and 𝛽-intervals in [0,T][0,T] is k+ℓk+\ell. However, ascertaining the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] when n≥2n\geq 2 is slightly more subtle, as it depends on whether k=ℓk=\ell or |k-ℓ|=1\lvert k-\ell\rvert=1. If k=ℓk=\ell, it is apparent that the number of 𝛼-intervals is n⁢knk, whereas the number of 𝛽-intervals is n⁢ℓn\ell. Thus, the total number of 𝛼 and 𝛽-intervals in this case equals(1.7)n⁢(k+ℓ)=2⁢n⁢k.n(k+\ell)=2nk.Now, assume that |k-ℓ|=1\lvert k-\ell\rvert=1. Then, in case k=ℓ+1k=\ell+1, there are n⁢ℓ+1n\ell+1 𝛼-intervals and n⁢ℓn\ell 𝛽-intervals in [0,n⁢T][0,nT]. Indeed, as for every i∈{0,1,…,n-2}i\in\{0,1,\ldots,n-2\} the last 𝛼-interval of [i⁢T,(i+1)⁢T][iT,(i+1)T] and the first one of [(i+1)⁢T,(i+2)⁢T][(i+1)T,(i+2)T] produce a unique 𝛼-interval in [0,n⁢T][0,nT], the total number of 𝛼-intervals in [0,n⁢T][0,nT] is given by n⁢(ℓ+1)-(n-1)=n⁢ℓ+1n(\ell+1)-(n-1)=n\ell+1. Obviously, the number of 𝛽-intervals in [0,n⁢T][0,nT] is n⁢ℓn\ell. Thus, the total number of 𝛼 and 𝛽-intervals equals n⁢ℓ+1+n⁢ℓ=2⁢n⁢ℓ+1n\ell+1+n\ell=2n\ell+1. Similarly, in case ℓ=k+1\ell=k+1, the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] is n⁢k+1+n⁢k=2⁢n⁢k+1nk+1+nk=2nk+1. Therefore, setting m:=min⁡{k,ℓ}m:=\min\{k,\ell\}, the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] in case |k-ℓ|=1\lvert k-\ell\rvert=1 is(1.8)2⁢n⁢m+1.2nm+1.Figure 1 shows a series of examples satisfying the previous requirements. Note that the support of αi\alpha_{i} and βj\beta_{j} on each of the intervals [tri,tr+1i][t_{r}^{i},t_{r+1}^{i}], 1≤i≤k1\leq i\leq k, and [trj,tr+1j][t_{r}^{j},t_{r+1}^{j}], 1≤j≤ℓ1\leq j\leq\ell, might not be connected.Figure 1Some admissible distributions of 𝛼 and 𝛽.On each of the intervals [trs,tr+1s][t_{r}^{s},t_{r+1}^{s}], s∈{i,j}s\in\{i,j\}, r∈{0,2}r\in\{0,2\}, the structure of the support of αi\alpha_{i} or βj\beta_{j} might be rather involved topologically, as illustrated by Figure 2, where we have plotted a sketch of the graph of a function αi\alpha_{i} or βj\beta_{j}, vanishing on the tertiary Cantor set of the interval [trs,tr+1s][t_{r}^{s},t_{r+1}^{s}] and being positive on the interior of its complement.Figure 2The internal complexity of the weights on each of their support intervals.Note that the cases when k+ℓ≠2k+\ell\neq 2, or k+ℓ=2k+\ell=2 under condition (1.6), were not dealt with in any previous references. In particular, they stay outside the general scope of [16, 13, 14]. The main result of this paper can be stated as follows. (Subsequently, for every r∈Rr\in{\mathbb{R}}, we are denoting by [r][r] the integer part of 𝑟.)Theorem 1Assume n⁢m≥3nm\geq 3 for some integer n=3⁢h+in=3h+i, with i∈{0,1,2}i\in\{0,1,2\}, where m:=min⁡{k,ℓ}m:=\min\{k,\ell\}. Then there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, (1.1) possesses at leastσ⁢(n):=2⁢(h⁢m+[i⁢m3])\sigma(n):=2\biggl{(}hm+\biggl{[}\frac{im}{3}\biggr{]}\biggr{)}periodic solutions with period n⁢TnT. Moreover, settingγ⁢(n):=min⁡{γ≥0:gcd⁡(n,σ⁢(n)2-γ)=1},\gamma(n):=\min\biggl{\{}\gamma\geq 0:\gcd\biggl{(}n,\frac{\sigma(n)}{2}-\gamma\biggr{)}=1\biggr{\}},it turns out that, for every λ>λn\lambda>\lambda_{n}, (1.1) has at least σ⁢(n)-2⁢γ⁢(n)\sigma(n)-2\gamma(n) periodic solutions with minimal period n⁢TnT.The main technical device to prove Theorem 1 is the Poincaré–Birkhoff twist theorem collected in Theorem 2. Theorem 1 deals with a degenerate case in the context of Hamiltonian systems not previously studied in the literature, because neither the monotonicity of α⁢(t)⁢f⁢(y)\alpha(t)f(y) or β⁢(t)⁢g⁢(x)\beta(t)g(x)for all 𝑡, nor the non-degeneration of α⁢(t)\alpha(t) and β⁢(t)\beta(t) are required (see [12, 5, 18, 10], and [15, § 1]).In Section 2, we state the version of the Poincaré–Birkhoff theorem invoked in the proof of Theorem 1 and make sure that it can be applied to deal with the degenerate model (1.1). Then, in Section 3, the proof of Theorem 1.1 is delivered. We refer to [9, 3] for a general discussion about the applications of the Poincaré–Birkhoff theorem to non-autonomous equations.2The Poincaré–Birkhoff Theorem in a Degenerate SettingIn this section, we adapt the Poincaré–Birkhoff theorem to deal with problem (1.1) in the degenerate case when |Z|=0\lvert Z\rvert=0 (see (1.2), if necessary). The Poincaré–Birkhoff theorem has been applied, very successfully, to study some non-degenerate Volterra predator-prey models of type (1.1) (see, e.g., [12, 4, 5, 18, 1, 10, 7] and the recent paper by the authors [15]).According to [15, Theorem 2.2], as soon as |Z|>0\lvert Z\rvert>0, system (1.1) possesses at least two n⁢TnT-periodic solutions for every integer n≥1n\geq 1 and sufficiently large λ>0\lambda>0. The main result of this section, Theorem 3, provides with some general sufficient conditions on the coefficients α⁢(t)\alpha(t) and β⁢(t)\beta(t) for the validity of the same result in the case when |Z|=0\lvert Z\rvert=0.Subsequently, for any given nontrivial solution (x⁢(t),y⁢(t))(x(t),y(t)) with initial data z0:=(x⁢(0),y⁢(0))≠(0,0)z_{0}:=(x(0),y(0))\neq(0,0), we denote by θ⁢(t)\theta(t) the angular polar coordinate so that, on any interval [0,n⁢T][0,nT], the rotation number of the solution can be defined throughrot⁡(z0;[0,n⁢T]):=θ⁢(n⁢T)-θ⁢(0)2⁢π.\operatorname{rot}(z_{0};[0,nT]):=\frac{\theta(nT)-\theta(0)}{2\pi}.To obtain the main result of this section, we need the following version of the Poincaré–Birkhoff twist theorem. It is, essentially, an application of Ding’s version of the twist theorem for planar annuli (see [6]), as presented in [17, Theorem A] (see also [2] for another application).Theorem 2Assume that, for some 0<r0<R00<r_{0}<R_{0} and an integer ω≥1\omega\geq 1, the next twist condition holds:(2.1)rot⁡(z0;[0,n⁢T])>ω⁢if⁢∥z0∥=r0 and rot⁡(z0;[0,n⁢T])<ω⁢if⁢∥z0∥=R0.\operatorname{rot}(z_{0};[0,nT])>\omega\ \text{if\/}\ \lVert z_{0}\rVert=r_{0}\quad\text{and}\quad\operatorname{rot}(z_{0};[0,nT])<\omega\ \text{if\/}\ \lVert z_{0}\rVert=R_{0}.Then system (1.1) has at least 2 nontrivial n⁢TnT-periodic solutions belonging to different periodicity classes with rotation number 𝜔.As observed in [5, § 3], given any n⁢TnT-periodic solution (x,y)(x,y) with n≥2n\geq 2, for every j∈{1,2,…,n-1}j\in\{1,2,\ldots,n-1\}, also xj⁢(t):=x⁢(t+j⁢T)x_{j}(t):=x(t+jT), yj⁢(t):=y⁢(t+j⁢T)y_{j}(t):=y(t+jT) is an n⁢TnT-periodic solution. In Theorem 2, all these solutions are considered to be equivalent, and it is said that they belong to the same periodicity class.Remark 1The information on the rotation number provided by Theorem 2 is very relevant. First, because the solutions with different rotation numbers are essentially different since, as paths in R2∖{(0,0)}{\mathbb{R}}^{2}\setminus\{(0,0)\}, they have a different fundamental group. Moreover, because Theorem 2, as stated, does not guarantee the minimality of the period n⁢TnT, except in the special case when gcd⁡(n,ω)=1\gcd(n,\omega)=1. Indeed, if (x⁢(t),y⁢(t))(x(t),y(t)) is ℓ⁢T\ell T-periodic for some integer ℓ<n\ell<n, then the rotation number in the interval [0,ℓ⁢T][0,\ell T] must be an integer, say ω1≥1\omega_{1}\geq 1. Thus, by the additivity property of the rotation numbers, it becomes apparent that rot⁡(z0;[0,n⁢ℓ⁢T])=ℓ⁢ω=ω1⁢n\operatorname{rot}(z_{0};[0,n\ell T])=\ell\omega=\omega_{1}n, where z0=(x⁢(0),y⁢(0))z_{0}=(x(0),y(0)), which contradicts the fact that gcd⁡(n,ω)=1\gcd(n,\omega)=1. Consequently, Theorem 2 is providing us with solutions of minimal period n⁢TnT if ω=1\omega=1.Remark 2By the continuous dependence of the solutions of (1.1) with respect to the initial conditions, for every ε>0\varepsilon>0, λ>0\lambda>0 and any integer n≥1n\geq 1, there exists δ≡δ⁢(n,λ,ε)>0\delta\equiv\delta(n,\lambda,\varepsilon)>0 such that the unique solution of (1.1), (x⁢(t),y⁢(t))(x(t),y(t)), satisfies (x⁢(t),y⁢(t))∈Dε(x(t),y(t))\in D_{\varepsilon} for all t∈[0,n⁢T]t\in[0,nT] if (x⁢(0),y⁢(0))∈Dδ(x(0),y(0))\in D_{\delta} (see [15, Proposition 2.1]). For every R>0R>0, we are denoting by DRD_{R} the disk of radius R>0R>0 centered at the origin.According to (1.7) and (1.8) and recalling that m=min⁡{k,ℓ}m=\min\{k,\ell\}, throughout the rest of this paper, we will assume that 2⁢n⁢m≥62nm\geq 6 if k=ℓk=\ell, and 2⁢n⁢m+1≥72nm+1\geq 7 if |k-ℓ|=1\lvert k-\ell\rvert=1. Thus, unifying both conditions, throughout the rest of this paper, we will actually assume that(2.2)n⁢m≥3.nm\geq 3.This condition entails, essentially, at least five alternations between the components of the supports of α⁢(t)\alpha(t) and β⁢(t)\beta(t), as illustrated in Figure 3.Figure 3An example of five alternations between the supports of 𝛼 and 𝛽.The main result of this paper reads as follows.Theorem 3Assume n⁢m≥3nm\geq 3. Then there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, the twist condition (2.1) in model (1.1) holds for ω≥1\omega\geq 1.This theorem analyzes a degenerate case for Hamiltonian systems of the form of (1.1), through the Poincaré–Birkhoff twist theorem, which had not been previously studied in this context, for as neither the monotonicity of α⁢(t)⁢f⁢(y)\alpha(t)f(y) or β⁢(t)⁢g⁢(x)\beta(t)g(x)for all 𝑡, nor the non-degeneration of α⁢(t)\alpha(t) and β⁢(t)\beta(t) are required (see [12, 5, 18, 10] and [15, § 1]).The technical details of the proof will be given in the special case when 𝛼 and 𝛽 satisfy (1.5), as the case when (1.6) holds follows similarly. By (1.3),min⁡{f′⁢(0),g′⁢(0)}>η\min\{f^{\prime}(0),g^{\prime}(0)\}>\etafor some constant η>0\eta>0. Thus, for sufficiently small |ζ|≤ε\lvert\zeta\rvert\leq\varepsilon,(2.3)f⁢(ζ)⁢ζ≥η⁢ζ2,g⁢(ζ)⁢ζ≥η⁢ζ2.f(\zeta)\zeta\geq\eta\zeta^{2},\quad g(\zeta)\zeta\geq\eta\zeta^{2}.Hence, since up to an additive constant,θ⁢(t)=arctan⁡y⁢(t)x⁢(t),\theta(t)=\arctan\frac{y(t)}{x(t)},differentiating with respect to 𝑡 and using the fact that (x⁢(t),y⁢(t))(x(t),y(t)) solves (1.1) yieldsθ′⁢(t)=y′⁢(t)⁢x⁢(t)-x′⁢(t)⁢y⁢(t)x2⁢(t)+y2⁢(t)=λ⁢β⁢(t)⁢g⁢(x⁢(t))⁢x⁢(t)+λ⁢α⁢(t)⁢f⁢(y⁢(t))⁢y⁢(t)x2⁢(t)+y2⁢(t)\theta^{\prime}(t)=\frac{y^{\prime}(t)x(t)-x^{\prime}(t)y(t)}{x^{2}(t)+y^{2}(t)}=\frac{\lambda\beta(t)g(x(t))x(t)+\lambda\alpha(t)f(y(t))y(t)}{x^{2}(t)+y^{2}(t)}for every t∈[0,n⁢T]t\in[0,nT]. So, owing to (2.3), we have that(2.4)θ′⁢(t)≥λ⁢η⁢[β⁢(t)⁢x2⁢(t)x2⁢(t)+y2⁢(t)+α⁢(t)⁢y2⁢(t)x2⁢(t)+y2⁢(t)]=λ⁢η⁢[β⁢(t)⁢cos2⁡θ⁢(t)+α⁢(t)⁢sin2⁡θ⁢(t)]≥0\theta^{\prime}(t)\geq\lambda\eta\biggl{[}\beta(t)\frac{x^{2}(t)}{x^{2}(t)+y^{2}(t)}+\alpha(t)\frac{y^{2}(t)}{x^{2}(t)+y^{2}(t)}\biggr{]}=\lambda\eta[\beta(t)\cos^{2}\theta(t)+\alpha(t)\sin^{2}\theta(t)]\geq 0for every t∈[0,n⁢T]t\in[0,nT]. In particular, θ⁢(t)\theta(t) is non-decreasing. Moreover, by Remark 2, for every integer n≥1n\geq 1, there exists δ>0\delta>0 such that (x⁢(t),y⁢(t))∈Dε(x(t),y(t))\in D_{\varepsilon} for all t∈[0,n⁢T]t\in[0,nT] if (x0,y0):=(x⁢(0),y⁢(0))∈Dδ(x_{0},y_{0}):=(x(0),y(0))\in D_{\delta}. This condition will be kept throughout the next lemmas and the proof of Theorem 3 in order to guarantee that the solution cannot escape from DεD_{\varepsilon}. Naturally, the bigger 𝜆 is, the smaller is 𝛿.Based on (2.4) and Remark 2, the next result holds. Essentially, it establishes that each of the components of the support of αi\alpha_{i} pushes the solutions of (1.1) from the first quadrant towards the second one, as well as from the third towards the fourth.Lemma 1Assume that there exists (ρ0,ρ1)⊊supp⁡α(\rho_{0},\rho_{1})\subsetneq\operatorname{supp}\alpha such that θ⁢(ρ0)∈[ω0,π-ω0]\theta(\rho_{0})\in[\omega_{0},\pi-\omega_{0}] for some ω0∈(0,π2)\omega_{0}\in(0,\frac{\pi}{2}). Then there exists λ1>0\lambda_{1}>0 such that θ⁢(ρ1)>π-ω0\theta(\rho_{1})>\pi-\omega_{0} for all λ>λ1\lambda>\lambda_{1}. Similarly, if θ⁢(ρ0)∈[π+ζ0,2⁢π-ζ0]\theta(\rho_{0})\in[\pi+\zeta_{0},2\pi-\zeta_{0}] for some ζ0∈(0,π2)\zeta_{0}\in(0,\frac{\pi}{2}), then θ⁢(ρ1)>2⁢π-ζ0\theta(\rho_{1})>2\pi-\zeta_{0} for sufficiently large 𝜆.ProofSince we are dealing with small solutions, it is apparent from (2.4) thatθ⁢(ρ1)=θ⁢(ρ0)+∫ρ0ρ1θ′⁢(s)⁢ds≥θ⁢(ρ0)+λ⁢η⁢∫ρ0ρ1α⁢(s)⁢y2⁢(ρ0)x2⁢(s)+y2⁢(ρ0)⁢ds\theta(\rho_{1})=\theta(\rho_{0})+\int_{\rho_{0}}^{\rho_{1}}\theta^{\prime}(s)\,ds\geq\theta(\rho_{0})+\lambda\eta\int_{\rho_{0}}^{\rho_{1}}\alpha(s)\frac{y^{2}(\rho_{0})}{x^{2}(s)+y^{2}(\rho_{0})}\,dsbecause β=0\beta=0 on (ρ0,ρ1)(\rho_{0},\rho_{1}) and, hence, y⁢(s)≡y⁢(ρ0)y(s)\equiv y(\rho_{0}) therein. Thus, settingν:=η⁢y2⁢(ρ0)ε2⁢∫ρ0ρ1α⁢(s)⁢ds\nu:=\frac{\eta y^{2}(\rho_{0})}{\varepsilon^{2}}\int_{\rho_{0}}^{\rho_{1}}\alpha(s)\,dsand taking into account that (x⁢(s),y⁢(s))=(x⁢(s),y⁢(ρ0))∈Dε(x(s),y(s))=(x(s),y(\rho_{0}))\in D_{\varepsilon} for all s∈(ρ0,ρ1)s\in(\rho_{0},\rho_{1}), it follows thatθ(ρ1)≥θ(ρ0)+λν≥ω0+λν>π-ω0 provided λ>π-2⁢ω0ν=:λ1.\theta(\rho_{1})\geq\theta(\rho_{0})+\lambda\nu\geq\omega_{0}+\lambda\nu>\pi-\omega_{0}\quad\text{provided}\quad\lambda>\frac{\pi-2\omega_{0}}{\nu}=:\lambda_{1}.Note that the bound λ1\lambda_{1} remains invariant if either θ⁢(ρ0)\theta(\rho_{0}) or y2⁢(ρ0)y^{2}(\rho_{0}) increases. Moreover, θ⁢(ρ1)\theta(\rho_{1}) increases with 𝜆 for any given (fixed) θ⁢(ρ0)\theta(\rho_{0}) and y2⁢(ρ0)y^{2}(\rho_{0}). Similarly, we have θ⁢(ρ1)>2⁢π-ζ0\theta(\rho_{1})>2\pi-\zeta_{0} for sufficiently large 𝜆 if θ⁢(ρ0)∈[π+ζ0,2⁢π-ζ0]\theta(\rho_{0})\in[\pi+\zeta_{0},2\pi-\zeta_{0}] for some ζ0∈(0,π2)\zeta_{0}\in(0,\frac{\pi}{2}). This ends the proof. ∎Analogously, the next result establishes that each of the components of the support of βi\beta_{i} pushes the solutions of (1.1) from the fourth quadrant towards the first, while it moves them from the second towards the third one.Lemma 2Assume that there exists (σ0,σ1)⊊supp⁡β(\sigma_{0},\sigma_{1})\subsetneq\operatorname{supp}\beta such that θ⁢(σ0)∈[-π2+τ0,π2-τ0]\theta(\sigma_{0})\in[-\frac{\pi}{2}+\tau_{0},\frac{\pi}{2}-\tau_{0}] for some τ0∈(0,π2)\tau_{0}\in(0,\frac{\pi}{2}). Then there exists μ1\mu_{1} such that θ⁢(σ1)>π2-τ0\theta(\sigma_{1})>\frac{\pi}{2}-\tau_{0} for all λ>μ1\lambda>\mu_{1}. Similarly, θ⁢(σ1)>3⁢π2-ξ0\theta(\sigma_{1})>\frac{3\pi}{2}-\xi_{0} for sufficiently large 𝜆 if θ⁢(σ0)∈[π2+ξ0,3⁢π2-ξ0]\theta(\sigma_{0})\in[\frac{\pi}{2}+\xi_{0},\frac{3\pi}{2}-\xi_{0}] for some ξ0∈(0,π2)\xi_{0}\in(0,\frac{\pi}{2}).ProofAs in Lemma 1, from (2.4), it follows thatθ⁢(σ1)=θ⁢(σ0)+∫σ0σ1θ′⁢(s)⁢ds≥θ⁢(σ0)+∫σ0σ1β⁢(s)⁢x2⁢(σ0)x2⁢(σ0)+y2⁢(s)⁢ds\theta(\sigma_{1})=\theta(\sigma_{0})+\int_{\sigma_{0}}^{\sigma_{1}}\theta^{\prime}(s)\,ds\geq\theta(\sigma_{0})+\int_{\sigma_{0}}^{\sigma_{1}}\beta(s)\frac{x^{2}(\sigma_{0})}{x^{2}(\sigma_{0})+y^{2}(s)}\,dsbecause α=0\alpha=0 on (σ0,σ1)(\sigma_{0},\sigma_{1}) and, hence, x⁢(s)≡x⁢(σ0)x(s)\equiv x(\sigma_{0}) therein. Thus, denotingς:=η⁢x2⁢(σ0)ε2⁢∫σ0σ1β⁢(s)⁢ds\varsigma:=\frac{\eta x^{2}(\sigma_{0})}{\varepsilon^{2}}\int_{\sigma_{0}}^{\sigma_{1}}\beta(s)\,dsand arguing as in Lemma 1, it is apparent thatθ(σ1)≥θ(σ0)+λς≥τ0+λς>π2-τ0 provided λ>π-4⁢τ02⁢ς=:μ1.\theta(\sigma_{1})\geq\theta(\sigma_{0})+\lambda\varsigma\geq\tau_{0}+\lambda\varsigma>\frac{\pi}{2}-\tau_{0}\quad\text{provided}\quad\lambda>\frac{\pi-4\tau_{0}}{2\varsigma}=:\mu_{1}.As highlighted in the proof of Lemma 1, the value of μ1\mu_{1} does not vary if either θ⁢(σ0)\theta(\sigma_{0}) or x2⁢(σ0)x^{2}(\sigma_{0}) increases. Similarly, θ⁢(σ1)\theta(\sigma_{1}) increases with 𝜆, and the second assertion of the lemma holds. This ends the proof. ∎Now, we are ready to prove Theorem 3.Proof of Theorem 3The proof is based on the version of the Poincaré–Birkhoff theorem collected in Theorem 2. First, we will prove that all small solutions in the disk DεD_{\varepsilon}, where 𝜀 is chosen sufficiently small so that (2.3) holds, have a rotation number greater than one. To prove this feature, we will distinguish between three different cases according to the precise location of their initial values, (x0,y0)(x_{0},y_{0}).Case 1: Assume that x0⁢y0>0x_{0}y_{0}>0. Then (x0,y0)(x_{0},y_{0}) lies either in the first or in the third quadrant. Both cases being similar, we will pay attention only to the case when x0>0x_{0}>0 and y0>0y_{0}>0. Then θ⁢(t01)∈(0,π2)\theta(t_{0}^{1})\in(0,\frac{\pi}{2}). Thus, by Lemma 1, there exists λ1>0\lambda_{1}>0 such that θ⁢(t11)>π-θ⁢(t01)\theta(t_{1}^{1})>\pi-\theta(t_{0}^{1}) for all λ>λ1\lambda>\lambda_{1}. Since α=β=0\alpha=\beta=0 in [t11,t21][t_{1}^{1},t_{2}^{1}], this implies thatθ⁢(t21)=θ⁢(t11)>π-θ⁢(t01).\theta(t_{2}^{1})=\theta(t_{1}^{1})>\pi-\theta(t_{0}^{1}).Thus, by Lemma 2, there exists λ2>0\lambda_{2}>0 such that θ⁢(t31)>π+θ⁢(t01)\theta(t_{3}^{1})>\pi+\theta(t_{0}^{1}) as soon as λ>max⁡{λ1,λ2}\lambda>\max\{\lambda_{1},\lambda_{2}\}. Also by Lemma 1, there exists λ3>0\lambda_{3}>0 such that θ⁢(t12)>2⁢π-θ⁢(t01)\theta(t_{1}^{2})>2\pi-\theta(t_{0}^{1}) for every λ>max⁡{λ1,λ2,λ3}\lambda>\max\{\lambda_{1},\lambda_{2},\lambda_{3}\}, and due to Lemma 2, there is λ4>0\lambda_{4}>0 such that θ⁢(t32)>2⁢π+θ⁢(t01)\theta(t_{3}^{2})>2\pi+\theta(t_{0}^{1}) for all λ>max⁡{λ1,λ2,λ3,λ4}\lambda>\max\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\}. Therefore, the solution with initial values (x0,y0)(x_{0},y_{0}) completes an entire turn in the interval [0,t32][0,t_{3}^{2}] for every λ>max⁡{λ1,λ2,λ3,λ4}\lambda>\max\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\}. In order to apply Theorem 2, it remains to show the existence of a uniform bound, Λ1>0\Lambda_{1}>0, such that the solutions with initial data in the sector of the circumference of radius r0r_{0} within the first quadrant,S1:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x>0,y>0},S_{1}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x>0,\,y>0\},have a rotation number greater than one for all λ>Λ1\lambda>\Lambda_{1} if 0<r0<ε0<r_{0}<\varepsilon. To prove it, we consider an angle ω~0∈(0,π2)\tilde{\omega}_{0}\in(0,\frac{\pi}{2}) and the sectors of S1S_{1} defined byS1,ω~0+:={z=(x,y)∈R2:∥z∥=r0,r0⁢cos⁡ω~0≥x>0,y>0},S1,ω~0-:=S1∖S1,ω~0+.S_{1,\tilde{\omega}_{0}}^{+}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0},\,r_{0}\cos\tilde{\omega}_{0}\geq x>0,\,y>0\},\quad S_{1,\tilde{\omega}_{0}}^{-}:=S_{1}\setminus S_{1,\tilde{\omega}_{0}}^{+}.Using recursively Lemmas 1 and 2 as above, it becomes apparent that there exists M1>0M_{1}>0 such that the solutions with initial data in S1,ω~0+S_{1,\tilde{\omega}_{0}}^{+} complete an entire turn in the interval [0,t32][0,t_{3}^{2}] for every λ>M1\lambda>M_{1}. For every t>0t>0, let us denote by Φt\Phi_{t} the Poincaré map at time 𝑡 of system (1.1), if defined. Then, by Lemma 2, there exists ω1>0\omega_{1}>0 such thatΦt31⁢(S1,ω~0-)⊂{(r,θ):0<r≤ε, 0<ω1≤θ<3⁢π2}.\Phi_{t_{3}^{1}}(S_{1,\tilde{\omega}_{0}}^{-})\subset\biggl{\{}(r,\theta):0<r\leq\varepsilon,\,0<\omega_{1}\leq\theta<\frac{3\pi}{2}\biggr{\}}.Therefore, as for S1,ω~0+S_{1,\tilde{\omega}_{0}}^{+}, there is M2M_{2} such that S1,ω~0-S_{1,\tilde{\omega}_{0}}^{-} completes one turn in [0,t33][0,t_{3}^{3}] for all λ>max⁡{M1,M2}\lambda>\max\{M_{1},M_{2}\}.Adapting the previous argument, it readily follows the existence of M~1,M~2>0\tilde{M}_{1},\tilde{M}_{2}>0 such that the solutions of (1.1) with initial data inS3:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x<0,y<0}S_{3}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x<0,\,y<0\}have a rotation number greater than one for all λ>max⁡{M~1,M~2}\lambda>\max\{\tilde{M}_{1},\tilde{M}_{2}\}. Consequently, for every z0∈S1∪S3z_{0}\in S_{1}\cup S_{3}, it is apparent that rot⁡(z0;[0,n⁢T])>1\operatorname{rot}(z_{0};[0,nT])>1 for all λ>Λ1:=max⁡{M1,M2,M~1,M~2}\lambda>\Lambda_{1}:=\max\{M_{1},M_{2},\tilde{M}_{1},\tilde{M}_{2}\}.Case 2: Assume that x0⁢y0<0x_{0}y_{0}<0. Most of the attention will be focused to the special case when x0<0x_{0}<0 and y0>0y_{0}>0, as the case x0>0x_{0}>0 and y0<0y_{0}<0 is analogous. Obviously, in this case, θ⁢(t01)∈(π2,π)\theta(t_{0}^{1})\in(\frac{\pi}{2},\pi). As in case 1, it should be proved the existence of a uniform bound, Λ2\Lambda_{2}, such that the solutions with initial data in the quadrant sectorS2:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x<0,y>0}S_{2}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x<0,\,y>0\}have rotation number greater than one for all λ>Λ2\lambda>\Lambda_{2} if 0<r0<ε0<r_{0}<\varepsilon. By Lemma 1, there exists ω2>π2\omega_{2}>\frac{\pi}{2} such thatΦt11⁢(S2)⊂{(r,θ):0<r≤ε,π2<ω2≤θ<π}.\Phi_{t_{1}^{1}}(S_{2})\subset\biggl{\{}(r,\theta):0<r\leq\varepsilon,\,\frac{\pi}{2}<\omega_{2}\leq\theta<\pi\biggr{\}}.Thus, as in case 1, we have already proven that, once the solution reaches the second quadrant, being separated away from π2\frac{\pi}{2}, it must have a rotation number greater than one for sufficiently large 𝜆 (which was a direct consequence from Lemmas 1 and 2), there exists Λ2>0\Lambda_{2}>0 such that the solution with θ⁢(t11)=ω2>π2\theta(t_{1}^{1})=\omega_{2}>\frac{\pi}{2} completes one turn for all λ>Λ2\lambda>\Lambda_{2}. Moreover, by the monotonicity properties of Lemmas 1 and 2, the solutions with initial data in S2S_{2} have rotation number greater than one for all λ>Λ2\lambda>\Lambda_{2}. Since the previous argument can be easily adapted to deal withS4:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x>0,y<0},S_{4}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ x>0,\,y<0\},it becomes apparent that, for every z0∈S2∪S4z_{0}\in S_{2}\cup S_{4}, rot⁡(z0;[0,n⁢T])>1\operatorname{rot}(z_{0};[0,nT])>1 for all λ>Λ2\lambda>\Lambda_{2}.Case 3: Assume x0⁢y0=0x_{0}y_{0}=0, i.e., (x0,y0)(x_{0},y_{0}) lies on some coordinate axis. Without loss of generality, we can assume that x0>0x_{0}>0 and y0=0y_{0}=0, as the remaining cases can be treated similarly. Then, since y0=0y_{0}=0 and β=0\beta=0 on [t01,t11][t_{0}^{1},t_{1}^{1}], integrating (1.1) yields θ⁢(t01)=θ⁢(t21)=0\theta(t_{0}^{1})=\theta(t_{2}^{1})=0. Thus, by Lemma 2, for every ω∈(0,π2)\omega\in(0,\frac{\pi}{2}), there exists μ1:=μ1⁢(ω)\mu_{1}:=\mu_{1}(\omega) such that θ⁢(t31)>ω\theta(t_{3}^{1})>\omega for all λ>μ1\lambda>\mu_{1}. Thus, much like in case 1, owing to Lemmas 1 and 2, there exist μ2,μ3,μ4,μ5>0\mu_{2},\mu_{3},\mu_{4},\mu_{5}>0, depending on 𝜔, such thatθ⁢(t12)>π-ω\theta(t_{1}^{2})>\pi-\omegaif⁢λ>max⁡{μ1,μ2},\text{if}\ \lambda>\max\{\mu_{1},\mu_{2}\},θ⁢(t32)>π+ω\theta(t_{3}^{2})>\pi+\omegaif⁢λ>max⁡{μ1,μ2,μ3},\text{if}\ \lambda>\max\{\mu_{1},\mu_{2},\mu_{3}\},θ⁢(t13)>2⁢π-ω\theta(t_{1}^{3})>2\pi-\omegaif⁢λ>max⁡{μ1,μ2,μ3,μ4},\text{if}\ \lambda>\max\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\},θ⁢(t33)>2⁢π\theta(t_{3}^{3})>2\piifλ>max{μ1,μ2,μ3,μ4,μ5}=:Λ3,1.\text{if}\ \lambda>\max\{\mu_{1},\mu_{2},\mu_{3},\mu_{4},\mu_{5}\}=:\Lambda_{3,1}.Therefore, the solution completes one turn in the time interval [0,t33][0,t_{3}^{3}] for all λ>Λ3,1\lambda>\Lambda_{3,1}. Similarly, it can be easily shown that the solutions complete a turn in each of the remaining three cases when x0=0x_{0}=0 and y0>0y_{0}>0, x0<0x_{0}<0 and y0=0y_{0}=0, or x0=0x_{0}=0 and y0<0y_{0}<0, for λ>Λ3,2\lambda>\Lambda_{3,2}, λ>Λ3,3\lambda>\Lambda_{3,3} and λ>Λ3,4\lambda>\Lambda_{3,4}, respectively. Thus, takingΛ3:=max⁡{Λ3,1,Λ3,2,Λ3,3,Λ3,4},\Lambda_{3}:=\max\{\Lambda_{3,1},\Lambda_{3,2},\Lambda_{3,3},\Lambda_{3,4}\},it becomes apparent that rot⁡(z0;[0,n⁢T])>1\operatorname{rot}(z_{0};[0,nT])>1 provided λ>Λ3\lambda>\Lambda_{3} andz0∈S0:={z=(x,y)∈R2:∥z∥=r0⁢and⁢x⁢y=0}.z_{0}\in S_{0}:=\{z=(x,y)\in{\mathbb{R}}^{2}:\lVert z\rVert=r_{0}\ \text{and}\ xy=0\}.Subsequently, we setλn:=max⁡{Λ1,Λ2,Λ3}.\lambda_{n}:=\max\{\Lambda_{1},\Lambda_{2},\Lambda_{3}\}.By Remark 2 and the analysis already done in the proof of the theorem, it is apparent that, for every λ>λn\lambda>\lambda_{n}, there exists 0<r0<δ⁢(n,λ,ε)0<r_{0}<\delta(n,\lambda,\varepsilon) such that, for every z0=(x0,y0)z_{0}=(x_{0},y_{0}) with ∥z0∥=r0\lVert z_{0}\rVert=r_{0},(x⁢(t),y⁢(t))∈Dε for all⁢t∈[0,n⁢T](x(t),y(t))\in D_{\varepsilon}\quad\text{for all}\ t\in[0,nT]and(2.5)rot⁡(z0;[0,n⁢T])>1.\operatorname{rot}(z_{0};[0,nT])>1.In order to apply Theorem 2, it remains to prove that, for sufficiently large λ>0\lambda>0, the large solutions do not rotate. As the proof of this feature follows the general scheme of the proof of [15, Theorem 2.1], we will simply sketch it here. Being analogous the remaining cases, the proof will be delivered in the special case when condition (g-)(g_{-}) holds in (1.4).We proceed by contradiction assuming that, regardless the size of the initial data (x0,y0)(x_{0},y_{0}), the solution (x⁢(t),y⁢(t))(x(t),y(t)) completes at least one turn for sufficiently large 𝜆. Thus, without loss of generality, changing the initial data if necessary, we can assume that (x⁢(t),y⁢(t))(x(t),y(t)) goes across the entire third quadrant. In such a case, there is an interval [s0,s1]⊂[0,n⁢T][s_{0},s_{1}]\subset[0,nT] such that y⁢(s0)=0=x⁢(s1)y(s_{0})=0=x(s_{1}) and x⁢(t)<0x(t)<0, y⁢(t)<0y(t)<0 for every t∈(s0,s1)t\in(s_{0},s_{1}). Thus, by (g-)(g_{-}), it becomes apparent that, setting B:=∫0Tβ⁢(s)⁢dsB:=\int_{0}^{T}\beta(s)\,ds,|y⁢(t)|=λ⁢|∫s0tβ⁢(s)⁢g⁢(x⁢(s))⁢ds|≤λ⁢M⁢∫0n⁢Tβ⁢(s)⁢ds=λ⁢M⁢n⁢B\lvert y(t)\rvert=\lambda\,\Biggl{\lvert}{}\int_{s_{0}}^{t}\beta(s)g(x(s))\,ds\Biggr{\rvert}\leq\lambda M\int_{0}^{nT}\beta(s)\,ds=\lambda MnBfor every t∈[s0,s1]t\in[s_{0},s_{1}]. Hence, defining N:=max⁡{|f⁢(y)|:|y|≤λ⁢M⁢n⁢B}N:=\max\{\lvert f(y)\rvert:\lvert y\rvert\leq\lambda MnB\}, it follows that, for every t∈[s0,s1]t\in[s_{0},s_{1}],|x⁢(t)|=|λ⁢∫ts1α⁢(s)⁢f⁢(y⁢(s))⁢ds|≤λ⁢N⁢∫0n⁢Tα⁢(s)⁢ds=λ⁢N⁢n⁢A,\lvert x(t)\rvert=\Biggl{\lvert}\lambda\int_{t}^{s_{1}}\alpha(s)f(y(s))\,ds\Biggr{\rvert}\leq\lambda N\int_{0}^{nT}\alpha(s)\,ds=\lambda NnA,where A:=∫0Tα⁢(s)⁢dsA:=\int_{0}^{T}\alpha(s)\,ds. Suppose that, for some t~∈[0,n⁢T]\tilde{t}\in[0,nT],x2⁢(t~)+y2⁢(t~)>λ2⁢n2⁢(M2⁢B2+N2⁢A2)≡R2x^{2}(\tilde{t})+y^{2}(\tilde{t})>\lambda^{2}n^{2}(M^{2}B^{2}+N^{2}A^{2})\equiv R^{2}with x⁢(t~)<0x(\tilde{t})<0 and y⁢(t~)<0y(\tilde{t})<0. Then the solution (x⁢(t),y⁢(t))(x(t),y(t)) cannot cross entirely the third quadrant. At this stage, the proof follows almost mutatis mutandis the steps of the proof of [15, Theorem 2.1], where the reader is sent for any further details. According to it, there exists a radius R0≥RR_{0}\geq R such that, for every solution with z0=x02+y02≥R0z_{0}=x_{0}^{2}+y_{0}^{2}\geq R_{0},(2.6)rot⁡(z0;[0,n⁢T])<1.\operatorname{rot}(z_{0};[0,nT])<1.By (2.5) and (2.6), the twist condition holds, and hence, by Theorem 2, system (1.1) admits at least two nontrivial n⁢TnT-periodic solutions belonging to different periodicity classes with rotation number ω≤n\omega\leq n for sufficiently large 𝜆. This concludes the proof. ∎In order to apply Theorem 2, the distribution of the weight functions settled by (2.2) is optimal. Indeed, if αi=0\alpha_{i}=0 or βi=0\beta_{i}=0 for some i∈{1,2,3}i\in\{1,2,3\}, then each of the points (-r0,0)(-r_{0},0) and (r0,0)(r_{0},0), for sufficiently small r0>0r_{0}>0, have rotation number less than one in the interval [0,T][0,T].Remark 3As already observed in [15, Remark 3], without any significant change in the proof, a slightly more general version of Theorem 2 can be proven by assuming f,gf,g only continuous (and not locally Lipschitz) and replacing the condition on the derivatives in (1.3) with the following one:0<lim inf|y|→0⁡f⁢(y)y≤lim sup|y|→0⁡f⁢(y)y<∞,0<lim inf|x|→0⁡g⁢(x)x≤lim sup|x|→0⁡g⁢(x)x<∞.0<\liminf_{\lvert y\rvert\to 0}\frac{f(y)}{y}\leq\limsup_{\lvert y\rvert\to 0}\frac{f(y)}{y}<\infty,\quad 0<\liminf_{\lvert x\rvert\to 0}\frac{g(x)}{x}\leq\limsup_{\lvert x\rvert\to 0}\frac{g(x)}{x}<\infty.To this aim, instead of Theorem 2, one can apply the generalized version of the Poincaré–Birkhoff theorem due to Fonda and Ureña [11] for Hamiltonian systems where the uniqueness of the solutions of the initial value problems is not required (see also [8, Theorem 10.6.1] for the precise statement).3Counting 𝑇-Periodic Solutions and Subharmonics of (1.1)This section applies Theorem 3 to model (1.1) when condition (2.2) holds. Recall that either k=ℓk=\ell, or |k-ℓ|=1\lvert k-\ell\rvert=1 and m=min⁡{k,ℓ}m=\min\{k,\ell\}. Based on Theorem 3, the next result holds.Theorem 4Assume that n⁢m≥3nm\geq 3 for some integer n≥1n\geq 1. Then there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, (1.1) possesses at least σ⁢(n)\sigma(n) periodic solutions with period n⁢TnT, whereσ⁢(n):={2⁢h⁢mif⁢n=3⁢h,2⁢(h⁢m+[m3])if⁢n=3⁢h+1,2⁢(h⁢m+[2⁢m3])if⁢n=3⁢h+2.\sigma(n):=\begin{cases}2hm&\text{if}\ n=3h,\\ 2\bigl{(}hm+\bigl{[}\frac{m}{3}\bigr{]}\bigr{)}&\text{if}\ n=3h+1,\\ 2\bigl{(}hm+\bigl{[}\frac{2m}{3}\bigr{]}\bigr{)}&\text{if}\ n=3h+2.\end{cases}Moreover, settingγ⁢(n):=min⁡{γ≥0:gcd⁡(n,σ⁢(n)2-γ)=1},\gamma(n):=\min\biggl{\{}\gamma\geq 0:\gcd\biggl{(}n,\frac{\sigma(n)}{2}-\gamma\biggr{)}=1\biggr{\}},it turns out that, for every λ>λn\lambda>\lambda_{n}, (1.1) has at least σ⁢(n)-2⁢γ⁢(n)\sigma(n)-2\gamma(n) periodic solutions with minimal period n⁢TnT.ProofSuppose k=ℓk=\ell. Then m=k=ℓm=k=\ell. Hence, according to (1.7), the total number of 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] equals(3.1)2⁢n⁢k=2⁢n⁢m.2nk=2nm.Thus, if n=3⁢hn=3h for some integer h≥1h\geq 1, the sum of 𝛼-intervals and 𝛽-intervals in [0,n⁢T][0,nT] is 6⁢h⁢k6hk. Hence, by Theorem 3, there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, the solutions of (1.1) with sufficiently small z0=(x0,y0)z_{0}=(x_{0},y_{0}) complete h⁢khk turns, whereas the solutions with sufficiently large z0z_{0} cannot complete any. Therefore, by Theorem 2, (1.1) has at least two n⁢TnT-periodic coexistence states with rotation number j∈{1,2,…,h⁢k}j\in\{1,2,\ldots,hk\}. Consequently, (1.1) possesses at least 2⁢h⁢k=σ⁢(n)2hk=\sigma(n) coexistence states with period n⁢TnT.Now, assume that n=3⁢h+1n=3h+1 for some integer h≥0h\geq 0. Then there are a total of2⁢m⁢n=2⁢k⁢(3⁢h+1)=6⁢h⁢k+2⁢k=6⁢(h⁢k+k3)2mn=2k(3h+1)=6hk+2k=6\biggl{(}hk+\frac{k}{3}\biggr{)}𝛼 and 𝛽-intervals in [0,n⁢T][0,nT]. Thus, by Theorem 3, there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, the solutions of (1.1) with sufficiently small z0z_{0} complete h⁢k+[k3]hk+\bigl{[}\frac{k}{3}\bigr{]} turns, while the solutions with large initial data cannot rotate. Therefore, thanks to Theorem 2, (1.1) possesses at least2⁢(h⁢k+[k3])=σ⁢(n)2\biggl{(}hk+\biggl{[}\frac{k}{3}\biggr{]}\biggr{)}=\sigma(n)periodic coexistence states of period n⁢TnT.Similarly, according to Theorems 2 and 3, when n=3⁢h+2n=3h+2 for some integer h≥0h\geq 0, there exists λn>0\lambda_{n}>0 such that, for every λ>λn\lambda>\lambda_{n}, (1.1) possesses at least2⁢(h⁢k+[2⁢k3])=σ⁢(n)2\biggl{(}hk+\biggl{[}\frac{2k}{3}\biggr{]}\biggr{)}=\sigma(n)coexistence states with period n⁢TnT.The last assertion of the theorem will be derived from the fact that, owing to Remark 1, any n⁢TnT-periodic coexistence state of (1.1) such that, for some 0<r0<R00<r_{0}<R_{0}, it satisfies(3.2){rot⁡(z0;[0,n⁢T])>ωif⁢∥z0∥=r0,rot⁡(z0;[0,n⁢T])<ωif⁢∥z0∥=R0,\begin{cases}\operatorname{rot}(z_{0};[0,nT])>\omega&\text{if}\ \lVert z_{0}\rVert=r_{0},\\ \operatorname{rot}(z_{0};[0,nT])<\omega&\text{if}\ \lVert z_{0}\rVert=R_{0},\end{cases}has minimal period n⁢TnT if gcd⁡(n,ω)=1\gcd(n,\omega)=1. In all the cases covered by Theorem 4, we have actually proven the existence of 0<r0<R00<r_{0}<R_{0} such that{rot⁡(z0;[0,n⁢T])>σ⁢(n)2if⁢∥z0∥=r0,rot⁡(z0;[0,n⁢T])<1if⁢∥z0∥=R0,\begin{cases}\operatorname{rot}(z_{0};[0,nT])>\frac{\sigma(n)}{2}&\text{if}\ \lVert z_{0}\rVert=r_{0},\\ \operatorname{rot}(z_{0};[0,nT])<1&\text{if}\ \lVert z_{0}\rVert=R_{0},\end{cases}by the definition of σ⁢(n)\sigma(n). Thus, (3.2) holds for the choice ω=σ⁢(n)2\omega=\frac{\sigma(n)}{2}. In the case gcd⁡(n,σ⁢(n)2)=1\gcd(n,\frac{\sigma(n)}{2})=1, by Remark 1, problem (1.1) possesses at least σ⁢(n)\sigma(n) coexistence states with minimal period n⁢TnT. This ends the proof in this case because we can take γ=0\gamma=0 in (3.1), and hence, γ⁢(n)=0\gamma(n)=0.Subsequently, we assume that gcd⁡(n,σ⁢(n)2)≠1\gcd(n,\frac{\sigma(n)}{2})\neq 1 and consider the unique integer j≥1j\geq 1 such that(3.3)gcd⁡(n,σ⁢(n)2-j)=1 and gcd⁡(n,σ⁢(n)2-i)≠1 for all⁢ 0≤i<j.\gcd\biggl{(}n,\frac{\sigma(n)}{2}-j\biggr{)}=1\quad\text{and}\quad\gcd\biggl{(}n,\frac{\sigma(n)}{2}-i\biggr{)}\neq 1\quad\text{for all}\ 0\leq i<j.In such a case, we can make the choice ω=σ⁢(n)2-j\omega=\frac{\sigma(n)}{2}-j. By (3.3), gcd⁡(n,ω)=1\gcd(n,\omega)=1. Moreover, as soon as ∥z0∥=r0\lVert z_{0}\rVert=r_{0}, we have thatrot⁡(z0;[0,n⁢T])>σ⁢(n)2>σ⁢(n)2-j=ω.\operatorname{rot}(z_{0};[0,nT])>\frac{\sigma(n)}{2}>\frac{\sigma(n)}{2}-j=\omega.And due to (3.3), it is apparent that, whenever ∥z0∥=R0\lVert z_{0}\rVert=R_{0},rot⁡(z0;[0,n⁢T])<1≤σ⁢(n)2-j=ω.\operatorname{rot}(z_{0};[0,nT])<1\leq\frac{\sigma(n)}{2}-j=\omega.Indeed, if σ⁢(n)2-j<1\frac{\sigma(n)}{2}-j<1, then there exists 0≤i<j0\leq i<j such that σ⁢(n)2-i=1\frac{\sigma(n)}{2}-i=1, and hence,gcd⁡(n,σ⁢(n)2-i)=gcd⁡(n,1)=1,\gcd\biggl{(}n,\frac{\sigma(n)}{2}-i\biggr{)}=\gcd(n,1)=1,contradicting the minimality of 𝑗. Therefore, by Remark 1, it becomes apparent that (1.1) has at least2⁢ω=2⁢(σ⁢(n)2-j)=σ⁢(n)-2⁢j=σ⁢(n)-2⁢γ⁢(n)2\omega=2\biggl{(}\frac{\sigma(n)}{2}-j\biggr{)}=\sigma(n)-2j=\sigma(n)-2\gamma(n)coexistence states with minimal period n⁢TnT. The proof is complete when k=ℓk=\ell.Now, assume that k=ℓ+1k=\ell+1. Then m=ℓm=\ell. Thus, according to (1.8), the total number of the 𝛼 and 𝛽-intervals in [0,n⁢T][0,nT] is n⁢m+1+n⁢m=2⁢n⁢m+1nm+1+nm=2nm+1. As the integers 2⁢n⁢m+12nm+1 and 2⁢n⁢m2nm, going back to (3.1), have the same divisibility properties by 6, the result when k=ℓ+1k=\ell+1 follows the same patterns as for k=ℓk=\ell. Similarly, the same result holds when ℓ=k+1\ell=k+1. This concludes the proof. ∎Remark 4As far as it concerns the cases not treated in this paper when n⁢(k+ℓ)≤5n(k+\ell)\leq 5, so far, it is known that if n⁢(k+ℓ)≤3n(k+\ell)\leq 3, then (1.1) does not admit any n⁢TnT-periodic solutions because the condition |Z|=0\lvert Z\rvert=0 ensures that no solution of (1.1) different from (0,0)(0,0), say (x⁢(t),y⁢(t))(x(t),y(t)), can complete one turn around the origin. Thus, it cannot satisfy (x⁢(0),y⁢(0))=(x⁢(n⁢T),y⁢(n⁢T))(x(0),y(0))=(x(nT),y(nT)) for some n≥1n\geq 1. The cases when n⁢(k+ℓ)=4,5n(k+\ell)=4,5 remain outside the general scope of this paper and will be analyzed elsewhere.4An Application to a Class of Predator-Prey ModelsThe non-autonomous planar Hamiltonian system (1.1) covers a large number of mathematical models of physical and biological nature. In particular, for the special choice f⁢(y)=ey-1f(y)=e^{y}-1 and g⁢(x)=ex-1g(x)=e^{x}-1, system (1.1) can be written, through the change of variables x=log⁡ux=\log u and y=log⁡vy=\log v, as(4.1){u′=λ⁢α⁢(t)⁢u⁢(1-v),v′=λ⁢β⁢(t)⁢v⁢(-1+u),\left\{\begin{aligned} \displaystyle{}u^{\prime}&\displaystyle=\lambda\alpha(t)u(1-v),\\ \displaystyle v^{\prime}&\displaystyle=\lambda\beta(t)v(-1+u),\end{aligned}\right.which is a non-autonomous 𝑇-periodic predator-prey model of Volterra type. As shown in [1, Section 5] and in [15, Introduction], system (4.1) can be obtained from the Volterra system with periodic coefficients{p′=λ⁢p⁢(a⁢(t)⁢p-b⁢(t)⁢q),q′=λ⁢q⁢(-c⁢(t)+d⁢(t)⁢p),\left\{\begin{aligned} \displaystyle{}p^{\prime}&\displaystyle=\lambda p\bigl{(}a(t)p-b(t)q\bigr{)},\\ \displaystyle q^{\prime}&\displaystyle=\lambda q\bigl{(}-c(t)+d(t)p\bigr{)},\end{aligned}\right.after a suitable change of variables. It is clear that the (nontrivial) n⁢TnT-periodic solutions of (1.1) are the n⁢TnT-periodic coexistence states of (4.1). By a coexistence state, it is meant a component-wise positive solution pair. This model was introduced in a degenerate setting in [16, 13] and later analyzed in [14] in the very special case when supp⁡α⊂[0,T2]\operatorname{supp}\alpha\subset\bigl{[}0,\frac{T}{2}\bigr{]} and supp⁡β⊂[T2,T]\operatorname{supp}\beta\subset\bigl{[}\frac{T}{2},T\bigr{]}. Since the functions f⁢(y)=ey-1f(y)=e^{y}-1, g⁢(x)=ex-1g(x)=e^{x}-1 satisfy (1.3) and (1.4), according to Theorems 2, 3 and 4, system (4.1) has at least σ⁢(n)\sigma(n) coexistence states with period n⁢TnT provided n⁢(k+ℓ)≥6n(k+\ell)\geq 6, among them, σ⁢(n)-2⁢γ⁢(n)\sigma(n)-2\gamma(n) with minimal period n⁢TnT. By Remark 4, system (4.1) cannot admit any n⁢TnT-periodic coexistence state if n⁢(k+ℓ)≤3n(k+\ell)\leq 3.

Journal

Advanced Nonlinear Studiesde Gruyter

Published: Aug 1, 2021

Keywords: Periodic Predator-Prey Model of Volterra Type; Subharmonic Coexistence States; Poincaré–Birkhoff Twist Theorem; Degenerate Versus Non-Degenerate Models; Point-Wise Behavior of the Low-Order Subharmonics as the Model Degenerates; 34C25; 37B55; 37E40

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