Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities

Assume that conditions \((h_{i})\) (\(i=1,\ldots,4\)) and \((\tau)\) hold. Then system (1.1) has infinitely many 2π-periodic solutions \(\{(x_{k}(t), y_{k}(t))\}_{k=1}^{\infty}\) which satisfy Moreover, for each integer \(k\geq1\), both \(x_{k}(t)\) and \(y_{k}(t)\) have exactly 2n simple zeros in \([0, 2\pi)\).

Assume that a, b are two positive constants, \(e, p:\mathbf{R}\to\mathbf{R}\) are continuous and conditions \((h_{i})\) (\(i=1,2\)) and \((\tau)\) are satisfied. Then the same conclusions of Theorem 1.1 still hold for the system

From condition \((h_{4})\) we know that f can be written in the form where \(h: \mathbf{R}\to\mathbf{R}\) is continuous and satisfies Therefore, it suffices for us to prove the main theorem for the system where \(p_{i}\) (\(i=1,2\)) satisfy condition \((h_{3})\). In the case \(a=b\neq1\), by introducing a rescaling of the time \(s=at\), \(u(s)=x(\frac{s}{a})\), \(v(s)=y(\frac{s}{a})\), we find the equivalent system of (1.1)′ (having the classical form) Such a rescaling cannot be easily applied in the case \(a\neq b\) because we do not know when the solution will change its sign.

Assume that conditions \((h_{i})\) (\(i=1,2,3\)) hold. Then there exist constants \(\gamma>1\) and \(R_{0}>0\) such that In particular, under conditions \((h_{i})\) (\(i=1, 2\)), \(\rho(t)\) satisfies the inequality

Assume that conditions \((h_{i})\) (\(i=1,2,3\)) hold, and let Then, for any sufficiently small ε, there exists a positive constant ζ such that

Let \((\bar{x}(t), \bar{y}(t))=(\bar{x}(t, x_{0}, y_{0}), \bar{y}(t, x_{0}, y_{0}))\) be the solution of (2.1) with \((\bar{x}(0), \bar{y}(0))=(x_{0}, y_{0})\). It is noted that \((x(t), y(t))=(x(t, x_{0}, y_{0}), y(t, x_{0}, y_{0}))\) is a solution of system (1.1)′ with \((x(0), y(0))=(x_{0}, y_{0})\). Set Then we have Let \(d(t)=\sqrt{u^{2}(t)+v^{2}(t)}\). Then we get with \(\delta=\frac{1}{2}(\mu+L)\), \(\mu=\max\{a, b\}\). From condition \((h_{3})\) we have that, for any sufficiently small \(\eta>0\), there exists \(c_{\eta}>0\) such that and Therefore, we obtain Solving this inequality, we get where \(A_{\eta}=\frac{2c_{\eta} e^{2\pi\delta}}{\delta}\). It follows from Lemma 2.3 that, for \(t\in[0, 2\pi]\), where \(\beta=2\sqrt{2}\pi\gamma e^{2\pi\delta}\). Write \(\psi(t)=\psi(t, r_{0}, \theta_{0})=\varphi(t, r_{0}, \theta_{0})-\theta(t, r_{0},\theta_{0})\). Clearly, if \(|\psi(t)|<\pi\), then \(\psi(t)\) is just the angle between the vectors \((x(t), y(t))\) and \((\bar{x} (t), \bar{y}(t))\). Hence, we have It follows that According to Lemma 2.3, we have that if η is sufficiently small and \(r_{0}\) is large enough, then Since \(\psi(0)=0\) and \(\psi(t)\) varies continuously as t increases from 0 to 2π, we have Consequently, we have that there exists \(\zeta>0\) such that, for \(r_{0}\geq\zeta\),  □

Assume that conditions \((h_{i})\) (\(i=1,2\)) and \((\tau)\) hold. Then there exists a constant \(\omega>0\) such that, for \(t\in\mathbf{R}\) and k large enough, with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\) or \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{b_{k}}\).

From the definition of \(T(c)\) and condition \((\tau)\) we know that, for each \(k\in\mathbf{N}\), In what follows, without loss of generality, we assume that the sequence \(\{T(b_{k})\}\) is bounded. Otherwise, we can replace the sequence \(\{T(b_{k})\}\) with a bounded one because \(T(c)\) is continuous for c large enough. We shall only deal with the first case, and the second one can be proved similarly. Let us set Obviously, \(d_{k}\to+\infty\), \(w_{k}\to-\infty\) as \(k\to\infty\). Next, we prove that there exist two positive constants \(\nu_{i}\) (\(i=1, 2\)) such that Assume by contradiction that Then there exists a subsequence of \(\{d_{k}\}\) (we still denote it by \(\{d_{k}\}\)) such that Set We have that \(\varepsilon_{k}\to0\) as \(k\to\infty\). From condition \((h_{2})\) we know that, for \(0< x\leq d_{k}\), For simplicity, we assume \(g(0)=0\). Then we get that, for \(0< x\leq d_{k}\), Consequently, we have that, for \(0\leq x\leq d_{k}\), It follows that, for \(0\leq x\leq d_{k}\), From the definition of T we have that Since we have that This is a contradiction because \(T(a_{k})\) is a bounded sequence. Therefore, there exists a constant \(\nu_{1}>0\) such that Similarly, there exists \(\nu_{2}>0\) such that From condition \((h_{1})\) and (2.5), (2.6) we know that there exists sufficiently small \(\varepsilon_{0}>0\) such that, for k large enough and \(x\in[(1-\varepsilon_{0})d_{k}, d_{k}]\), Therefore, if \(\rho(t)\cos\varphi(t)\in[(1-\varepsilon_{0})d_{k}, d_{k}]\), then we have where \(\omega_{1}=\min\{a, b, \frac{1}{4}\nu_{1}\}\). Next, we deal with the case \(\rho(t)\cos\varphi(t)\in[0, (1-\varepsilon_{0})d_{k}]\). Set \(x_{k}=(1-\varepsilon_{0})d_{k}\). Assume that the line \(x=x_{k}\) intersects with the curve \(\Gamma_{a_{k}}\) at two points \((x_{k}, y_{k}^{+})\) and \((x_{k}, y_{k}^{-})\) with \(y_{k}^{-}<0<y_{k}^{+}\). Then we have Therefore, we get From (2.7) we have and Set From condition \((h_{1})\) we know that there exists \(A>0\) such that \(g(x)\geq0\) for \(x\geq A\). If \(\rho(t)\cos\varphi(t)\in[A, (1-\varepsilon_{0})d_{k}]\) and \(\rho(t)\sin\varphi(t)\geq0\), then we have If \(\rho(t)\cos\varphi(t)\in[0, A]\) and \(\rho(t)\sin\varphi (t)\geq 0\), then we have that, for \(\rho_{0}\) large enough, Similarly, we have that, if \(\rho(t)\cos\varphi(t)\in[A, (1-\varepsilon_{0})d_{k}]\) and \(\rho(t)\sin\varphi(t)\leq0\), then we have If \(\rho(t)\cos\varphi(t)\in[0, A]\) and \(\rho(t)\sin\varphi (t)\leq 0\), then we have that, for \(\rho_{0}\) large enough, In conclusion, we have proved that there exists \(\omega_{2}>0\) such that with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\), \(\rho(t)\cos\varphi(t)\geq0\) and k large enough. Similarly, we can prove that there exists \(\omega_{2}'>0\) such that with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\), \(\rho(t)\cos\varphi(t)\leq0\) and k large enough. Let us set \(\omega=\min\{\omega_{2}, \omega_{2}'\}\). Then we have that with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\) and k large enough. □

Assume that conditions \((h_{i})\) (\(i=1,2\)) and \((\tau)\) hold, and let \(\Phi(\rho_{0}, \varphi_{0})=\varphi(2\pi, \rho_{0}, \varphi_{0})-\varphi_{0}\). Then there exist two positive constants δ and \(\varrho_{0}\) such that

From Lemma 2.5 we have that there exists \(\varrho_{0}>0\) such that, for \(a_{k}\geq\varrho_{0}\) or \(b_{k}\geq\varrho_{0}\), Write \(\Phi(\rho_{0}, \varphi_{0})=-2l\pi-\phi\), where \(l\geq0\) is an integer, \(0\leq\phi<2\pi\). Let us denote by \(t_{\phi}\) the time for \(\varphi(t)\) to decrease from \(-2l\pi\) to \(-2l\pi-\phi\). If \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\), then we have Since \(t_{\phi}\leq T(a_{k})=m\tau(a_{k})\), we have It follows that \(l\geq n\). If \(l\geq n+1\), then we have If \(l=n\), then we have Therefore, we get Furthermore, Set \(\delta=\min\{2\pi, nm\sigma\omega\}\). Then we have Similarly, we can prove  □

From Lemmas 2.4 and 2.6 we know that there exists an integer \(k_{0}>0\) such that, for any \(k\geq k_{0}\), Without loss of generality, we assume that \(a_{k}< b_{k}< a_{k+1}\) for \(k\geq k_{0}\). Let \(D_{k}\) be an annular region with boundary \(\Gamma_{a_{k}}\) and \(\Gamma_{b_{k}}\). Consider the Poincaré map \(P: D_{k}\to\mathbf{R}^{2}\). Write the Poincaré map in the form where \(\Theta^{*}(r_{0}, \theta_{0})=\theta(2\pi, r_{0}, \theta_{0})-\theta_{0}+2n\pi\). From (3.1) and (3.2) we know that, for \(k\geq k_{0}\), Therefore, the area-preserving homeomorphism P is twisting on the annulus \(D_{k}\). On the other hand, by Lemma 2.1, we have that both \(\Gamma_{a_{k}}\) and \(\Gamma_{b_{k}}\) are star-shaped with respect to the origin O for k large enough. Consequently, all assumptions of the generalized form of the Poincaré-Birkhoff fixed point theorem are satisfied. Therefore, the Poincaré map P has at least two fixed points in annulus \(D_{k}\). Furthermore, system (1.1)′ has infinitely many 2π-periodic solutions \(\{(x_{k}(t), y_{k}(t))\}_{k=1}^{\infty}\) which satisfy Clearly, each 2π-periodic solution \((x_{k}(t), y_{k}(t))\) rotates clockwise strictly n turns around the origin in the interval \([0, 2\pi]\). It follows that both \(x_{k}(t)\) and \(y_{k}(t)\) have exactly 2n simple zeros in \([0, 2\pi)\). Hence, the conclusion of Theorem 1.1 holds. □

Assume that condition \((h_{1})\) holds, g is odd and \(G_{+}< G^{+}\). Then where \(\tau_{+}=\liminf_{c\to+\infty}\tau(c)\), \(\tau^{+}=\limsup_{c\to+\infty}\tau(c)\).

Assume that conditions \((h_{i})\) (\(i=1, \ldots, 4\)) hold. Let \(g(x)\) be an odd function and \(G_{+}< G^{+}\). If then system (1.1) has infinitely many 2π-periodic solutions \(\{(x_{k}(t), y_{k}(t))\}_{k=1}^{\infty}\) which satisfy Moreover, for each integer \(k\geq1\), both \(x_{k}(t)\) and \(y_{k}(t)\) have exactly 2n simple zeros in \([0, 2\pi)\).

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 \(\nu_{1}, \ldots, \nu_{N}\in \mathbf{Z}\) such that, for each \(i=1, \ldots, N\), either or Then Hamiltonian system (4.1) has at least \(N+1\) distinct T-periodic solutions \(z^{0}(t), \ldots, z^{N}(t)\), with \(z^{0}(0), \ldots, z^{N}(0)\in\mathcal{A}\), such that

It is noted that there is no requirement of uniqueness of Cauchy problems associated to system (4.1) in this higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows. In [17, 18], Theorem 4.1 is applied to prove the multiplicity of periodic solutions of weakly coupled Hamiltonian systems. Theorem 1.1 in the present paper can also be extended to a weakly coupled system of the type where \(x=(x_{1}, \ldots, x_{N})\), \(y=(y_{1}, \ldots, y_{N})\), \(f_{i}, g_{i}:\mathbf{R}\to\mathbf{R}\) (\(i=1, \ldots, N\)) are continuous, \(p_{ji}: \mathbf{R}^{2N+1}\to\mathbf{R}\) (\(j=1, 2\), \(i=1, \ldots, N\)) are continuous and 2π-periodic with the variable t. Assume that there is a function \(U: \mathbf{R}^{2N+1}\to \mathbf{R}\) such that In this case system (4.2) is a Hamiltonian system. Simple examples of such functions can be given. For example, Assume that the following conditions hold: Set \(G_{i}(u)=\int_{0}^{u}g_{i}(s)\,ds\), \(i=1, \ldots, N\). Let us define the time maps for \(c>0\) large enough, where \(w_{i}(c)\) and \(d_{i}(c)\) satisfy \(w_{i}(c)<0<d_{i}(c)\) and \(G_{i}(w_{i}(c))=G_{i}(d_{i}(c))=c\). Assume that the time map \(\tau_{i}(c)\) satisfies the condition:

Assume that conditions \((h_{i}')\) (\(i=1, \ldots, 4\)) and \((\tau')\) hold. Then Hamiltonian system (4.2) has infinitely many 2π-periodic solutions \(\{(x^{k}(t), y^{k}(t))\}_{k=1}^{\infty}\) which satisfy where \(|x^{k}(t)|=\sum_{i=1}^{i=N}|x_{i}^{k}(t)|\), \(|y^{k}(t)|=\sum_{i=1}^{i=N}|y_{i}^{k}(t)|\). Moreover, for each index i, both \(x^{k}_{i}(t)\) and \(y^{k}_{i}(t)\) have exactly \(2n_{i}\) simple zeros in the interval \([0, 2\pi)\).