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Infinitely many periodic solutions of planar Hamiltonian systems via the Poincaré–Birkhoff theorem
Boundary Value Problems volume 2018, Article number: 102 (2018)
Abstract
In this paper, we study the multiplicity of periodic solutions of one kind of planar Hamiltonian systems with a nonlinear term satisfying semilinear conditions. Using a generalized Poincaré–Birkhoff fixed point theorem, we prove that the system has infinitely many periodic solutions, provided that the time map tends to zero.
1 Introduction
We are concerned with the multiplicity of periodic solutions of planar Hamiltonian systems of the type
where \(f, g: \mathbf{R}\to\mathbf{R}\) are continuous, and \(p_{i}: \mathbf{R^{3}}\to\mathbf{R}\) (\(i=1,2\)) are continuous and 2πperiodic with respect to the first variable t.
The periodic problem for planar Hamiltonian systems is a classical topic in nonlinear analysis and ordinary differential equations, which has been widely studied in literature by using various different methods such as phase plane analysis, topological degree, fixed point theorems, variational methods (see [1–5, 10, 11, 14, 19, 20] and the references therein). For instance, using the Poincaré–Bohl fixed point theorem, Fonda and Sfecci [11] studied the existence of periodic solutions of planar Hamiltonian systems
where \(J=\bigl( {\scriptsize\begin{matrix}{}0& 1\cr 1& 0 \end{matrix}} \bigr)\) is the standard symplectic matrix, and \(H: [0, T]\times\mathbf{R}^{2}\to \mathbf{R}\) is differentiable with respect to the second variable. When \(\nabla_{z}H(t, z)\) satisfies some semilinear conditions at infinity, it was proved in [11] that (1.2) has at least one Tperiodic solution. Using a generalized Poincaré–Bikhoff fixed point theorem, Boscaggin [4] studied the multiplicity of periodic solutions of (1.2), provided that \(\nabla_{z}H(t, z)\) satisfies some superlinear condition at infinity. It was pointed out that the main theorem (Theorem 2.3) in [4] applies to the forced Duffing equation
when g satisfies the superlinear condition
We note that Ding and Zanolin [7] proved the multiplicity of periodic solutions of Eq. (1.3) by replacing (1.4) with a weaker assumption on the time map of the autonomous equation \(x''+g(x)=0\), namely that the limit of the time map equals zero. Clearly, this condition is not included in [4].
In the present paper, we study the multiplicity of periodic solutions of (1.1) in terms of the time map. Assume that g satisfies the condition
and f satisfies the semilinear condition at infinity
Let \(G(x)=\int_{0}^{x}g(s)\,ds\). Define the function
From condition (\(h_{1}\)) we know that \(\tau(c)\) is continuous for \(c\) large enough; \(\tau(c)\) is usually called the time map related to the autonomous equation \(x''+g(x)=0\). The properties of \(\tau(c)\) were studied deeply in [6, 7, 18]. Assume that τ satisfies
From [18] we know that (\(h_{3}\)) holds if g satisfies the superlinear condition (1.4). Throughout the paper, we always assume that there exists a constant \(M>0\) such that
Moreover, there is a function \(\mathcal{U}: \mathbf{R}^{3}\to \mathbf{R}\) such that
In this case, system (1.1) is a Hamiltonian system. We can give simple examples of such functions. For example,
Clearly, if \(p_{1}(t, x, y)=p_{1}(t, y)\) and \(p_{2}(t, x, y)=p_{2}(t, x)\), then (1.1) is a Hamiltonian system.
Note that we can write system (1.1) in the form (1.2). Let
where \(F(y)=\int_{0}^{y}f(s)\,ds\). Then we have that \(\nabla_{z}H(t, z)=(g(x)p_{2}(t, x, y), f(y)+p_{1}(t, x, y))\). Since g only satisfies condition (\(h_{1}\)), we know that \(\nabla_{z}H(t, z)\) does not satisfy the semilinear condition as in [11] or the superlinear condition as in [4].
Using a generalized Poincaré–Birkhoff fixed point theorem and the phaseplane analysis method, we prove the following results.
Theorem 1.1
Assume that conditions (\(h_{i}\)) (\(i=1,\ldots,4\)) hold. Then system (1.1) has infinitely many 2πperiodic solutions \(\{(x_{j}(t),y_{j}(t))\}_{j=1}^{\infty}\) that satisfy
Theorem 1.2
Assume that conditions (\(h_{i}\)) (\(i=1,\ldots,4\)) hold. Then for any given integer \(m\geq2\), system (1.1) has infinitely many \(2m\pi\)periodic solutions \(\{(x_{j}(t), y_{j}(t))\}_{j=1}^{\infty}\) that are not \(2k\pi\)periodic for \(1\leq k\leq m1\) and satisfy
Corollary 1.3
Assume that conditions (\(h_{2}\)), (\(h_{4}\)), and (1.4) hold. Then the conclusions of Theorems 1.1 and 1.2 still hold.
Remark 1.4
Ding and Zanolin [7] proved the multiplicity of periodic solutions of Eq. (1.3) when conditions (\(h_{1}\)) and (\(h_{3}\)) hold and \(p(t,x)\) is bounded. Note that Eq. (1.3) is equivalent to the planar Hamiltonian system \(x'=y\), \(y'=g(x)+p(t,x)\), which is a particular form of (1.1). Therefore our conclusions generalize the main results in [7].
Remark 1.5
We will prove the above results under the additional assumption that the solutions to Cauchy problems of (1.1) are unique. It is shown in Sect. 4 that this requirement is not restrictive and that our results are valid when the uniqueness of the solutions to Cauchy problems is not satisfied.
Throughout this paper, by R and N we denote the sets of real and natural numbers, respectively.
2 Several lemmas
In this section, we perform some phaseplane analysis for system (1.1). Let \((x(t), y(t))=(x(t,x_{0},y_{0}), y(t, x_{0},y_{0}))\) be the solution of (1.1) satisfying the initial value
We denote
Lemma 2.1
Assume that conditions (\(h_{i}\)) (\(i=1,2,4\)) hold. Then every solution \((x(t), y(t))\) of (1.1) exists on the whole taxis.
Proof
In view of (\(h_{1}\)), there exists a constant \(c_{0}>0\) such that
Let us define two Lyapunovlike functions \(V_{\pm}:\mathbf{R^{2}}\to \mathbf{R}\):
where \(h:\mathbf{R}\to\mathbf{R}\) is a continuously differentiable function such that \(h(x)=M \operatorname{sgn}(x)\) for \(x\geq c_{0}\) with M from (\(h_{4}\)). We will prove that \(V_{\pm}\) are coercive, that is, \(V_{\pm}(x, y)\to+\infty\) as \(x+y\to+\infty\). From (\(h_{1}\)) we know that \(\lim_{x\to+\infty}G(x)=+\infty\). From (\(h_{2}\)) we get that
Since \(h(x)\) is bounded, we further see that the inequalities
hold uniformly with respect to \(x\in\mathbf{R}\). From (2.2) we have that the limits
hold uniformly with respect to \(x\in\mathbf{R}\). Therefore \(V_{\pm}(x, y)\to+\infty\) as \(x+y\to+\infty\).
Next, we show that every solution \((x(t), y(t))\) of (1.1) is defined on the interval \([0, +\infty)\). Set \(V_{+}(t)=V_{+}(x(t), y(t))\). We first prove that there exist constants \(c_{1}>0\) and \(c_{2}>0\) such that
For simplicity, we denote \(p_{1}(t)=p_{1}(t, x(t), y(t))\), \(p_{2}(t)=p_{2}(t, x(t), y(t))\). By (1.1) we have
If \(x(t)\geq c_{0}\), then we infer from (\(h_{4}\)), the definition of \(h(x)\), and (2.1) that
If \(x(t)\leq c_{0}\), then it follows from (\(h_{4}\)) and the continuity of \(g(x)\) and \(h(x)\) that there exists \(\alpha_{0}>0\) such that
In view of (\(h_{2}\)), we conclude that there exist \(l_{0}>0\) and \(\alpha_{1}>0\) such that
Since \(h'(x)=0\) for \(x\geq c_{0}\), we know that \(h'(x)\) is bounded, and then there exists \(\beta_{0}>0\) such that \(h'(x)\leq\beta_{0}\), \(x\in\mathbf{R}\). It follows from (2.4) that
Meanwhile, in view of (\(h_{4}\)), we have
From (\(h_{4}\)) and (2.4) we get that
Since \(h(x)\) is bounded for \(x\in\mathbf{R}\), there exists \(\beta_{0}'>0\) such that \(h(x)\leq\beta_{0}'\), \(x\in\mathbf{R}\). Consequently, we have
Therefore, we obtain
with \(\beta_{1}=\beta_{0} l_{0}+\frac{1}{2}\) and \(\beta_{1}'=\frac{1}{2}(\alpha_{1}\beta_{0}+M\beta_{0}+Ml_{0})^{2}+(\alpha _{0}+M\alpha_{1}+M\beta_{0}')\). From (2.2) we know that there exist \(l_{1}>0\) and \(\beta_{2}>0\) such that
Combining (2.5) and (2.6), we get
Since \(\lim_{x\to+\infty}G(x)=+\infty\), there exists \(G_{0}>0\) such that \(G(x)+G_{0}\geq0\) for \(x\in\mathbf{R}\). We conclude that
Set \(c_{1}=\beta_{1}l_{1}\) and \(c_{2}=\beta_{1}l_{1}G_{0}+\beta_{1}\beta_{2}+\beta_{1}'\). We get that \(V_{+}'(t)\leq c_{1}V_{+}(t)+c_{2}\). Then, for any finite \(\omega>0\), we have
Since \(V_{+}\) is coercive, there is no blowup for \((x(t), y(t))\) on any finite interval \([0, \omega)\). Therefore, \((x(t), y(t))\) exists on the interval \([0, +\infty)\).
To prove that \((x(t), y(t))\) exists on the interval \((\infty, 0]\), we consider another Lyapunovlike function \(V_{}(x, y)\). Set \(V_{}(t)=V_{}(x(t), y(t))\). Using the same methods as before, we can show that there exist two positive constants \(d_{1}\), \(d_{2}\) such that
Then, for any \(\omega>0\), we have
Since \(V_{}\) is coercive, there is also no blowup for \((x(t), y(t))\) on any finite interval \((\omega, 0]\). Therefore, \((x(t), y(t))\) exists on the interval \((\infty, 0]\). The proof is complete. □
Since the uniqueness of the solutions to Cauchy problems of (1.1) is assumed, we can define the Poincaré map \(P: \mathbf{R}^{2}\to\mathbf{R}^{2}\) as follows:
It is well known that the Poincaré map P is an areapreserving homeomorphism. The fixed points of P correspond to the 2πperiodic solutions of (1.1).
On the basis of the global existence of solutions of (1.1), we can get the elasticity property of solutions of (1.1) by using a classical result (Theorem 6.5 in [15]).
Lemma 2.2
Assume that conditions (\(h_{i}\)) (\(i=1,2,4\)) hold. Then, for any \(T>0\) and \(R_{1}>0\), there exists \(R_{2}>R_{1}\) such that:

(1)
If \(x_{0}^{2}+y_{0}^{2}\leq R_{1}^{2}\), then \(x(t)^{2}+y(t)^{2}\leq R_{2}^{2}\), \(t\in[0, T]\).

(2)
If \(x_{0}^{2}+y_{0}^{2}\geq R_{2}^{2}\), then \(x(t)^{2}+y(t)^{2}\geq R_{1}^{2}\), \(t\in[0, T]\).
From Lemma 2.2 we know that if \(x_{0}^{2}+y_{0}^{2}\) is large enough, then \(x^{2}(t)+y^{2}(t)\neq0\), \(t\in[0, T]\). Thus we can take the polar coordinate transformation
Under this transformation, system (1.1) becomes
where \(p_{1}(t,r,\theta)=p_{1}(t, r\cos\theta,r\sin\theta)\) and \(p_{2}(t,r,\theta)=p_{2}(t,r\cos\theta,r\sin\theta)\). Let us denote by \((r(t), \theta(t))=(r(t, r_{0}, \theta_{0}), \theta(t, r_{0}, \theta_{0}))\) the solution of (2.8) satisfying the initial value
with \(x_{0}=r_{0}\cos\theta_{0}\), \(y_{0}=r_{0}\sin\theta_{0}\). We can rewrite the Poincaré map P in the polar coordinate form \(P: (r_{0}, \theta_{0})\to(r_{1}, \theta_{1})\),
where l is an arbitrary integer.
Lemma 2.3
Assume that conditions (\(h_{i}\)) (\(i=1,2,4\)) hold. Then, for any \(T>0\), there exists a constant \(R>0\) such that if \(r(t)\geq R\), \(t\in[0, T]\), then
Proof
It follows from (\(h_{1}\)) that there exists \(a_{1}>0\) such that
which, together with (\(h_{4}\)), implies that
From (\(h_{2}\)) and (\(h_{4}\)) we know that there exist two constants \(\gamma>0\) and \(a_{2}>0\) such that
Therefore, if \(x(t)\geq a_{1}\) and \(y(t)\geq a_{2}\), then we have
If \(x(t)\leq a_{1}\) and \(y(t)\geq a_{2}\), then we have
where \(\alpha=\arctan\frac{a_{2}}{a_{1}}\). On the other hand, there exists \(A_{1}>0\) such that
It follows that if \(r(t)\) is large enough and \(x(t)\leq a_{1}\), then
Consequently, if \(r(t)\) is large enough and \(x(t)\leq a_{1}\), \(y(t)\geq a_{2}\), then we get
Finally, we know that there exists \(A_{2}>0\) such that
If \(y(t)\leq a_{2}\) and \(r(t)\) is large enough, then we have that \(x(t)\) is also large enough. Therefore we get from (\(h_{1}\)), (\(h_{4}\)), and (2.11) that, for \(r(t)\) large enough,
Furthermore
Combining (2.9), (2.10), and (2.12), we get the conclusion of Lemma 2.3. □
Lemma 2.4
Assume that conditions (\(h_{i}\)) (\(i=1,\ldots,4\)) hold. Let m be a positive integer. Then, for any given integer \(n\geq1\), there exists a constant \(\varrho_{n}>0\) such that, for \(r_{0}\geq\varrho_{n}\),
Proof
From conditions (\(h_{1}\)) and (\(h_{2}\)) we know that there exists \(d>0\) such that
From Lemma 2.3 we know that there is a constant \(c_{m}\geq d\) such that
and
Then the solution \((r(t), \theta(t))\) moves clockwise in the region
We now decompose the set D into eight regions as follows:
Next, we will estimate the time needed for the solution \((x(t), y(t))\) to pass through the regions \(D_{i}\) (\(i=1,\ldots, 8\)), respectively. Without loss of generality, we assume that \((x_{0}, y_{0})\in D_{1}\). Then, \((x(t), y(t))\) rotates following the cycle
Given k (\(k=1,\ldots, 8\)), let \([t_{1}, t_{2}]\subset[0, 2n\pi]\) be such that
and
We first treat the case \((x(t),y(t))\in D_{1}\), \(t\in[t_{1}, t_{2}]\). It follows from (\(h_{2}\)) that there exist constants \(\beta_{0}\geq \alpha_{0}>0\) and \(M_{0}>\) such that
Therefore, if \((x(t), y(t))\in D_{1}\), then we have
with \(M_{1}=M_{0}+M\). From Lemma 2.2 we know that, for any constant l (\(>\sqrt{c_{m}^{2}+\frac{M_{1}^{2}}{\alpha_{0}^{2}}}\)), there is a constant \(l_{0}>l\) such that, for \(r_{0}\geq l_{0}\),
As a result, we get that, for \(r_{0}\geq l_{0}\) and \((x(t),y(t))\in D_{1}\), \(t\in[t_{1}, t_{2}]\),
Consequently,
which implies that, for any sufficiently small \(\varepsilon>0\),
provided that l is sufficiently large. According to Lemma 2.2, we further know that (2.15) holds when \(r_{0}\) is sufficiently large.
Similarly, we have that if \((x(t), y(t))\in D_{5}\), \(t\in[t_{1}, t_{2}]\), then
provided that \(r_{0}\) is sufficiently large.
We next treat the case \((x(t), y(t))\in D_{2}\), \(t\in[t_{1}, t_{2}]\). Let us define \(W_{+}:\mathbf{R^{2}}\to\mathbf{R}\) as follows:
Set
If \(x(t)\geq c_{m}\) and \(y(t)\geq d\), then we get from (\(h_{4}\)) and (2.14) that
which implies that \(W_{+}(t)\) is decreasing when \((x(t), y(t))\) lies in the field \(D_{2}\). Hence, we get that, for \(t\in[t_{1}, t_{2}]\),
Consequently,
Since \(y(t_{2})=d\), there is a constant \(B>0\) such that \(F(y(t_{2}))\leq B\). It follows from (2.16) that
where \(M_{1}=B+Md\). According to (2.14), we have that, for \(y\geq0\),
Hence we get that, if \(t\in[t_{1}, t_{2}]\), then we infer from (2.17) and (2.18) that
Let us take \(\eta>\beta_{0}\) such that, for \(y\geq d\),
Then we obtain
Using the mean value theorem, we get
where \(\xi\in[x(t), x(t_{2})]\). Since \(x(t)\geq c_{m}\), \(t\in[t_{1}, t_{2}]\), we can take \(c_{m}\) large enough such that \(g(\xi)\geq2M\). Therefore, we obtain that, for \(t\in[t_{1}, t_{2}]\),
which, together with (2.19), implies that
Since \(x'(t)=f(y(t))+p_{1}(t,x(t),y(t))\), we infer from (2.14) and (2.20) that
Let us take a fixed positive constant L. Then we have that, for \(x(t)\in[c_{m}, x(t_{2})L]\),
which implies that there exists a positive constant \(\eta_{0}<\frac{\alpha_{0}}{\sqrt{\eta}}\) such that, for \(x(t)\in [c_{m}, x(t_{2})L]\),
Consequently, for \(x(t)\in[c_{m}, x(t_{2})L]\), we get
Let \(t_{*}\in[t_{1}, t_{2}]\) be such that \(x(t_{*})=x(t_{2})L\). Then we have that, for any sufficiently small \(\varepsilon>0\),
provided that \(x(t_{2})\) is large enough. Consequently, we have that \(t_{*}t_{1}<\frac{\varepsilon}{2}\), provided that \(r_{0}\) is sufficiently large. We next estimate \(t_{2}t_{*}\). If \(t\in[t_{*}, t_{2}]\), then we have
where \(\nu(x(t_{2}))=\min\{g(x):x(t_{2})L\leq x\leq x(t_{2})\}\). Obviously, \(\nu(x(t_{2}))\to+\infty\) as \(x(t_{2})\to+\infty\). On the other hand, it follows from (2.14) that
Therefore, we get that, for \(x(t_{2})\) large enough,
which, together with \(\nu(x(t_{2}))\to+\infty\) as \(x(t_{2})\to+\infty\), implies that, for any sufficiently small \(\varepsilon>0\),
provided that \(x(t_{2})\) is large enough or \(r_{0}\) is large enough. From (2.21) and (2.22) we know that, for any sufficiently small \(\varepsilon>0\),
provided that \(r_{0}\) is large enough.
Similarly, we have that, if \((x(t), y(t))\in D_{i}\), \(i=4, 6, 8\), \(t\in[t_{1}, t_{2}]\), then
provided that \(r_{0}\) is large enough.
We now consider the case \((x(t), y(t))\in D_{3}\), \(t\in[t_{1}, t_{2}]\). In this case, we have
Integrating both sides of \(y'=g(x(t))+p_{2}(t, x(t), y(t))\) over \([t_{1}, t_{2}]\) and using \(y(t_{1})=d\) and \(y(t_{2})=d\), we get
where \(\mu_{*}=\min\{g(x(t)): t_{1}\leq t\leq t_{2}\}\). From (2.23), (\(h_{1}\)), and Lemma 2.2 we get that \(\mu_{*}\to+\infty\) as \(r_{0}\to\infty\). Therefore, we have that, for any sufficiently small \(\varepsilon>0\),
provided that \(r_{0}\) is large enough.
Similarly, we have that, if \((x(t), y(t))\in D_{7}\), \(t\in[t_{1}, t_{2}]\), then
provided that \(r_{0}\) is large enough.
From the previous conclusion we get that, for any sufficiently small \(\varepsilon>0\), there is \(\varrho_{1}>0\) such that if \(r_{0}\geq \varrho_{1}\), then \((x(t), y(t))\in D\), and if
then
Consequently, the motion \((x(t),y(t))\) rotates clockwise a turn in a period less than 8ε. Therefore, for any integer \(n\geq 1\), there is \(\varrho_{n}>0\) such that, for \(r_{0}\geq\varrho_{n}\), the motion \((x(t), y(t))\) can rotate more than n turns during the period \(2m\pi\).
The proof of Lemma 2.4 is thus complete. □
3 Proof of main theorems
First, we recall a generalized version of the Poincaré–Birkhoff fixed point theorem by Rebelo [19].
A generalized form of the Poincaré–Birkhoff fixed point theorem. Let \(\mathcal{A}\) be an annular region bounded by two strictly starshaped curves around the origin, \(\Gamma_{1}\) and \(\Gamma_{2}\), \(\Gamma_{1}\subset\operatorname{int}(\Gamma_{2})\), where \(\operatorname{int}(\Gamma_{2})\) denotes the interior domain bounded by \(\Gamma_{2}\). Suppose that \(F:\overline{\operatorname{int}(\Gamma_{2})}\to R^{2}\) is an areapreserving homeomorphism and \(F\mathcal{A}\) admits a lifting, with the standard covering projection \(\it\Pi:(r, \theta)\to z=(r\cos\theta, r\sin\theta)\), of the form
where w and h are continuous functions of period 2π in the second variable. Correspondingly, for \(\tilde{\Gamma}_{1}={\it\Pi}^{1}(\Gamma_{1})\) and \(\tilde{\Gamma}_{2}={\it\Pi}^{1}(\Gamma_{2})\), assume the twist condition
Then, F has two fixed points \(z_{1}\), \(z_{2}\) in the interior of \(\mathcal{A}\) such that
Remark 3.1
The assumption on the starshaped boundaries of the annulus is a delicate hypothesis. Martins and Ureña [17] showed that the starshapedness assumption on the interior boundary is not eliminable. Le Calvez and Wang [16] then proved that the starshapedness of the exterior boundary is also necessary, although this assumption was not made in Ding’s theorem [8].
Proof of Theorem 1.1
Let us take \(m=1\) in Lemma 2.4. From Lemmas 2.2 and 2.3 we have that there is \(c_{1}'>0\) such that, for \(r_{0}>c_{1}'\),
Let \(a_{1}>c_{1}'\) be a fixed constant. Then there exists an integer \(n_{1}>0\) such that, for \(r_{0}=a_{1}\),
According to Lemma 2.4, there exists a constant \(b_{1}>a_{1}\) such that, for \(r_{0}=b_{1}\),
Let us define the annulus
Consider the Poincaré map \(P: R^{2}\to R^{2}\). Write the map P in the form
with
Set
Then we have
From (3.1) and (3.2) we have that
Since system (1.1) is conservative, P is an areapreserving mapping. It follows from (3.3) and (3.4) that the areapreserving map P is twisting on the annulus \(D_{1}\). According to the generalized Poincaré–Birkhoff fixed point theorem, P has at least two fixed points in \(D_{1}\). Similarly, we can find a sequence
such that the areapreserving map P is twisting on the annuluses
Using the generalized Poincaré–Birkhoff fixed point theorem, we know that P has at least two fixed points in each annulus \(D_{j}\), \(j=2,3,\dots \). Thus we have obtained the existence of a sequence of 2πperiodic solutions \(\{(x_{j}(t),y_{j}(t))\}\) of system (1.1) satisfying
The proof of Theorem 1.1 is complete. □
Proof of Theorem 1.2
Let \(m\geq2\) in Lemma 2.4. From Lemmas 2.2 and 2.3 we have that there is \(c_{m}'>0\) such that, for \(r_{0}\geq c_{m}'\),
Let \(a_{1}'>c_{m}'\) be a sufficiently large constant. Then there is a prime \(q_{1}\geq 2\) such that, for \(r_{0}=a_{1}'\),
According to Lemma 2.4, we know that there is \(b_{1}'>a_{1}'\) such that, for \(r_{0}=b_{1}'\),
Set
Then the areapreserving iteration map \(P^{m}\) is twisting on the annulus \(D_{1}'\). Using the generalized Poincaré–Birkhoff fixed point theorem, \(P^{m}\) has at least two fixed points \((x_{1i}, y_{1i})\) (\(i=1,2\)) in \(D_{1}'\). Hence
are two \(2m\pi\)periodic solutions of system (1.1). Since \(q_{1}\) is a prime, we can further prove that \((x_{1i}(t), y_{1i}(t))\) are not \(2k\pi\)periodic for \(1\leq k\leq m1\) by the same method as in [7]. Similarly, we can find two sequences
and
with \(q_{j}\) (\(j=1,2, \ldots\)) being prime numbers such that, for \(r_{0}=a_{j}'\),
and, for \(r_{0}=b_{j}'\),
Therefore, \(P^{m}\) is twisting on the annuluses
It follows that \(P^{m}\) has at least two fixed points in each annulus \(D_{j}'\), \(j=2 ,3, \dots \), which correspond to two \(2m\pi\)periodic solutions of system (1.1). In the same way, these \(2m\pi\)periodic solutions are not \(2k\pi\)periodic for \(1\leq k\leq m1\). Consequently, system (1.1) has infinitely many \(2m\pi\)periodic solutions \(\{(x_{j}(t), y_{j}(t))\}_{j=1}^{\infty }\) that are not \(2k\pi\)periodic for \(1\leq k\leq m1\) and satisfy
The proof of Theorem 1.2 is thus complete. □
4 Remarks
The assumption on the uniqueness of the solutions to Cauchy problems of (1.1) made in the proofs of the previous sections can be removed. In fact, Lemmas 2.3 and 2.4 guarantee the applicability of the nonuniqueness version of the Poincaré–Birkhoff theorem, which was proved by Fonda and Ureña [13]. We now state this theorem for a general Hamiltonian system in \(\mathbf {R}^{2N}\). Let us consider the Hamiltonian system
where the continuous function \(H:\mathbf{R}\times\mathbf{R}^{N}\times \mathbf{R}^{N}\to\mathbf{R}\), \(H=H(t, x, y)\) is Tperiodic in its first variable t and continuously differentiable with respect to \((x, y)\), \(x=(x_{1}, \ldots, x_{N})\), \(y=(y_{1}, \ldots, y_{N})\).
We next introduce the definition of the rotation number of a continuous path in \(\mathbf{R}^{2}\). Let \(\sigma:[t_{1}, t_{2}]\to\mathbf{R}^{2}\) be a continuous path such that \(\sigma(t)\neq(0,0)\) for every \(t\in[t_{1}, t_{2}]\). The rotation number of σ around the origin is defined as
where \(\theta:[t_{1}, t_{2}]\to\mathbf{R}\) is a continuous determination of the argument along σ, that is, \(\sigma(t)=\sigma(t)(\cos \theta(t), \sin\theta(t))\).
Assume that, for each \(i=1,\ldots,N\), there are two strictly starshaped Jordan closed curves around the origin \(\Gamma^{i}_{1}, \Gamma ^{i}_{2}\subset\mathbf{R}^{2}\) such that
where \(\mathcal{D}(\Gamma)\) is the open region bounded by the Jordan closed curve Γ. Consider the generalized annular region
Theorem 4.1
([13])
Under the framework above, denoting \(z_{i}(t)=(x_{i}(t), y_{i}(t))\), assume that every solution \(z(t)=(z_{1}(t), \ldots, z_{N}(t))\) of (4.1) departing from \(z(0)\in \partial\mathcal{A}\) is defined on \([0, T]\) and satisfies
Assume that there are integer numbers \(v_{1}, \ldots, v_{N}\in\mathbf {Z}\) such that, for each \(i=1, \ldots, N\), either
or
Then Hamiltonian system (4.1) has at least \(N+1\) distinct Tperiodic solutions \(z^{0}(t), \ldots, z^{N}(t)\), with \(z^{0}(0), \ldots, z^{N}(0)\in \mathcal{A}\), such that
Remark 4.2
Note that there is no requirement of uniqueness of the solutions to Cauchy problems of (4.1) in this higherdimensional Poincaré–Birkhoff theorem for Hamiltonian flows. Theorem 4.1 can be applied to deal with the multiplicity of periodic solutions of higherdimensional Hamiltonian systems [9, 12]. Fonda and Sfecci [12] studied the multiplicity of periodic solutions of weakly coupled Hamiltonian systems
where all the functions \(h_{i}: \mathbf{R}\times\mathbf{R}\to\mathbf {R}\) are continuous, Tperiodic in the first variable t, and such that
The functions \(p_{i}: \mathbf{R}\times\mathbf{R}^{N}\) are continuous, Tperiodic in the first variable t, and bounded. Moreover, there is a function \(\mathcal{U}:\mathbf{R}\times\mathbf{R}^{N}\to\mathbf {R}\) such that
In this case, (4.2) is a superlinear Hamiltonian system. Under these conditions, it was proved in [12] that (4.2) has infinitely many periodic solutions by using Theorem 4.1. Theorems 1.1 and 1.2 in the present paper can also be extended to a weakly coupled Hamiltonian system of the type
where \(x=(x_{1}, \ldots, x_{N})\), \(y=(y_{1}, \ldots, y_{N})\), \(f_{i}, g_{i}:\mathbf{R}\to\mathbf{R}\) are continuous, \(p_{ji}:\mathbf {R}\times\mathbf{R}^{N}\times\mathbf{R}^{N}\to\mathbf{R}\) (\(j=1,2\), \(i=1,\ldots, N\)) are continuous and 2πperiodic in the first variable t. Assume that there is a function \(\mathcal{W}: \mathbf {R}\times\mathbf{R}^{N}\times\mathbf{R}^{N}\to\mathbf{R}\) such that
It follows that system (4.3) is a Hamiltonian system. Assume that the following conditions hold:
 (\(h_{1}'\)):

\(\lim_{s\to\infty}\operatorname{sgn}(s)g_{i}(s)=+\infty\) (\(i=1,\ldots, N\)).
 (\(h_{2}'\)):

There are positive constants \(\alpha_{i}\), \(\beta_{i}\) such that
$$0< \alpha_{i}=\liminf_{s\to+\infty}\frac{f_{i}(s)}{s}\leq \limsup_{s\to+\infty}\frac{f_{i}(s)}{s}=\beta_{i}< +\infty. $$  (\(h_{3}'\)):

There are positive constants \(M_{i}\) such that
$$\bigl\vert p_{1i}(t, x, y) \bigr\vert \leq M_{i}, \qquad \bigl\vert p_{2i}(t, x, y) \bigr\vert \leq M_{i}\quad \mbox{for all } t, x, y\in\mathbf{R} \mbox{ and } i=1, \ldots, N. $$  (\(h_{4}'\)):

The time maps \(\tau_{i}(c)\) satisfy \(\lim_{c\to \infty}\tau_{i}(c)=0\), where \(\tau_{i}(c)\) are defined like \(\tau(c)\) in Sect. 1.
Using Theorem 4.1, we can prove that (4.3) has infinitely many 2πperiodic solutions and, for any integer \(m\geq2\), (4.3) has infinitely many \(2m\pi\)periodic solutions that are not \(2k\pi \)periodic for \(1\leq k\leq m1\), provided that conditions (\(h_{i}'\)) (\(i=1, \ldots, 4\)) hold. For brevity, we omit the details.
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The authors are grateful to the referees for many valuable suggestions to make the paper more readable.
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Research supported by the National Nature Science Foundation of China, No. 11501381.
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ZW proved the global existence of the solution of any Cauchy problem. TM proved the other conclusions and helped to draft the manuscript. Both authors read and approved the final manuscript.
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Wang, Z., Ma, T. Infinitely many periodic solutions of planar Hamiltonian systems via the Poincaré–Birkhoff theorem. Bound Value Probl 2018, 102 (2018). https://doi.org/10.1186/s136610181022y
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DOI: https://doi.org/10.1186/s136610181022y