In this section, we consider a weakly coupled system
of the type

$\{\begin{array}{cc}\hfill J{\dot{z}}_{1}& ={A}_{1}\nabla {H}_{1}({z}_{1})+{\mathcal{\mathcal{R}}}_{1}(t,{z}_{1},\mathrm{\dots},{z}_{N}),\hfill \\ & \mathrm{\vdots}\hfill \\ \hfill J{\dot{z}}_{N}& ={A}_{N}\nabla {H}_{N}({z}_{N})+{\mathcal{\mathcal{R}}}_{N}(t,{z}_{1},\mathrm{\dots},{z}_{N}),\hfill \end{array}$(P)

where *J* is the $2\times 2$ standard symplectic matrix, namely

$J=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right),$

and ${A}_{1},\mathrm{\dots},{A}_{N}$ are positive real parameters.
For every $i=1,\mathrm{\dots},N$, we assume that ${H}_{i}:{\mathbb{R}}^{2}\to \mathbb{R}$ is continuously differentiable, and ${\mathcal{\mathcal{R}}}_{i}:\mathbb{R}\times {\mathbb{R}}^{2N}\to \mathbb{R}$ is continuous, *T*-periodic in *t* and continuously differentiable in $({z}_{1},\mathrm{\dots},{z}_{N})$.

We assume that system (P) can be reduced to a Hamiltonian system by a linear change of variables. More precisely, there exist *N* invertible $2\times 2$ matrices ${\mathbb{M}}_{1},\mathrm{\dots},{\mathbb{M}}_{N}$, having positive determinant, such that the linear operator $\mathcal{\mathcal{L}}:{\mathbb{R}}^{2N}\to {\mathbb{R}}^{2N}$, defined as

$\mathcal{\mathcal{L}}:({z}_{1},\mathrm{\dots},{z}_{N})\mapsto ({\mathbb{M}}_{1}{z}_{1},\mathrm{\dots},{\mathbb{M}}_{N}{z}_{N}),$(4.1)

transforms system (P) into a Hamiltonian system. With such an assumption, we will say that (P) is a *positive transformation of a Hamiltonian system*.

Let us introduce the following notation for a closed cone in ${\mathbb{R}}^{2}$ determined by two angles ${\vartheta}_{1}<{\vartheta}_{2}$:

$\mathrm{\Theta}({\vartheta}_{1},{\vartheta}_{2})=\{(\rho \mathrm{cos}\vartheta ,\rho \mathrm{sin}\vartheta ):\rho \ge 0,{\vartheta}_{1}\le \vartheta \le {\vartheta}_{2}\}.$

We are now ready to state the main theorem of this section.

*Let (P) be a positive transformation of a Hamiltonian system. For every $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, let the following assumptions hold:*

(A1)

*There exists *
${C}_{i}>0$
* such that
*

$\parallel \nabla {H}_{i}(w)\parallel \le {C}_{i}(\parallel w\parallel +1)\mathit{\hspace{1em}}\mathit{\text{for every}}w\in {\mathbb{R}}^{2}.$

(A2)

*There exist *
${r}_{i}>0$
* and *
${m}_{i}>0$
* such that*

$\u3008\nabla {H}_{i}(w),w\u3009\ge {m}_{i}{\parallel w\parallel}^{2}\mathit{\hspace{1em}}\mathit{\text{for every}}w\in \mathcal{\mathcal{B}}[0,{r}_{i}].$

(A3)

*For every *
$\sigma >0$
*, there exist *
${R}_{i}>0$
* and *
${\vartheta}_{1}^{i}<{\vartheta}_{2}^{i}$
*, with *
${\vartheta}_{2}^{i}-{\vartheta}_{1}^{i}\le 2\pi $
*, such that*

$sup\{\frac{\u3008\nabla {H}_{i}(w),w\u3009}{{\parallel w\parallel}^{2}}:w\in \mathrm{\Theta}({\vartheta}_{1}^{i},{\vartheta}_{2}^{i})\setminus \mathcal{\mathcal{B}}(0,{R}_{i})\}\le \sigma ({\vartheta}_{2}^{i}-{\vartheta}_{1}^{i}).$(4.2)

*Then, for every fixed positive integers ${\nu}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{\nu}_{N}$, there exist $A\mathrm{>}\mathrm{0}$ and $\epsilon \mathrm{>}\mathrm{0}$ such that if ${A}_{i}\mathrm{\ge}A$ and*

$\parallel {\mathcal{\mathcal{R}}}_{i}(t,{w}_{1},\mathrm{\dots},{w}_{N})\parallel \le \epsilon \mathit{\hspace{1em}}\mathit{\text{for every}}t\in [0,T]\mathit{\text{and}}{w}_{1},\mathrm{\dots},{w}_{N}\in {\mathbb{R}}^{2}$(4.3)

*for every $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, then system (P) has at least $N\mathrm{+}\mathrm{1}$ distinct **T*-periodic solutions

${z}^{k}(t)=({z}_{1}^{k}(t),\mathrm{\dots},{z}_{N}^{k}(t))$

*such that for every $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N\mathrm{+}\mathrm{1}$, each planar component ${z}_{i}^{k}\mathit{}\mathrm{(}t\mathrm{)}$, with $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, makes exactly ${\nu}_{i}$ clockwise rotations around the origin in the time interval $\mathrm{[}\mathrm{0}\mathrm{,}T\mathrm{[}$.*

Some comments on the hypotheses of Theorem 4.1 are in order.
Assumption (A1) is needed to ensure the global existence of the solutions to the Cauchy problems associated with (P). Concerning (A2), it will guarantee that the small amplitude planar components of the solutions do rotate around the origin, clockwise, with a least positive angular speed.
Our hypothesis (A3), on the contrary, will ensure a small rotation number for large amplitude components. It could be compared with assumption $({H}_{\mathrm{\infty}}^{\prime})$ in [14, Theorem 4.1].

We now start the proof of Theorem 4.1. For a solution $z(t)$ of system (P), whose *i*-th component is such that ${z}_{i}(t)=({x}_{i}(t),{y}_{i}(t))\in {\mathbb{R}}^{2}\setminus \{0\}$ for every $t\in [0,T]$, we denote by $Rot({z}_{i}(t);[0,T])$ the standard clockwise winding number of the path $t\mapsto {z}_{i}(t)$ around the origin, namely

$Rot({z}_{i}(t);[0,T])=\frac{1}{2\pi}{\int}_{0}^{T}\frac{\u3008J{\dot{z}}_{i}(t),{z}_{i}(t)\u3009}{{\parallel {z}_{i}(t)\parallel}^{2}}dt.$

Our first lemma concerns solutions $z(t)$ whose *i*-th component ${z}_{i}(t)$ is small. We assume without loss of generality that ${H}_{i}(0)=0$, and consider the level set

${\mathrm{\Gamma}}_{i}^{h}=\{w\in {\mathbb{R}}^{2}:{H}_{i}(w)=h\}.$

By (A2), if $h>0$ is sufficiently small, then ${\mathrm{\Gamma}}_{i}^{h}$ is a strictly star-shaped Jordan curve around the origin. We will denote by ${D}_{i}^{h}$ the bounded, closed and connected region of
${\mathbb{R}}^{2}$ with $\partial {D}_{i}^{h}={\mathrm{\Gamma}}_{i}^{h}$.

*For any $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$ and every positive integer ${\nu}_{i}$, if (A1) and (A2) hold, there exist three positive constants ${\overline{A}}_{i}$, ${\overline{\epsilon}}_{i}$ and ${\overline{h}}_{i}$ such that, if ${A}_{i}\mathrm{\ge}{\overline{A}}_{i}$, $h\mathrm{\in}\mathrm{]}\mathrm{0}\mathrm{,}{\overline{h}}_{i}\mathrm{]}$ and*

$\parallel {\mathcal{\mathcal{R}}}_{i}(t,{w}_{1},\mathrm{\dots},{w}_{N})\parallel \le {\overline{\epsilon}}_{i}\mathit{\hspace{1em}}\mathit{\text{for every}}t\in [0,T]\mathit{\text{and}}{w}_{1},\mathrm{\dots},{w}_{N}\in {\mathbb{R}}^{2},$(4.4)

*then any solution $z\mathit{}\mathrm{(}t\mathrm{)}$ of (P) with ${z}_{i}\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{\in}{\mathrm{\Gamma}}_{i}^{h}$ satisfies*

$Rot({z}_{i}(t);[0,T])>{\nu}_{i}.$

#### Proof.

Let $i\in \{1,\mathrm{\dots},N\}$ and ${\nu}_{i}$ be fixed.
We can choose $h>0$ and $\widehat{r}\in ]0,r{}_{i}[$, where ${r}_{i}$ is as in assumption (A2), in such a way that

$\mathcal{\mathcal{B}}(0,\widehat{r})\subset {D}_{i}^{h}\subset {D}_{i}^{2h}\subset {D}_{i}^{3h}\subset \mathcal{\mathcal{B}}(0,{r}_{i}).$(4.5)

We now claim that if (4.4) holds with a suitable choice of ${\overline{\epsilon}}_{i}$, then for every solution $z(t)$ of (P) with ${z}_{i}(0)\in {\mathrm{\Gamma}}_{i}^{2h}$ one has

$h<{H}_{i}({z}_{i}(t))<3h\mathit{\hspace{1em}}\text{for every}t\in [0,T].$

Indeed, set

$C=\mathrm{max}\left\{\parallel \nabla {H}_{i}(w)\parallel :w\in \mathcal{\mathcal{B}}[0,{r}_{i}]\right\},{\overline{\epsilon}}_{i}=\frac{h}{2CT},$

and assume by contradiction that ${z}_{i}(0)\in {\mathrm{\Gamma}}_{i}^{2h}$ and there exists ${t}_{1}\in [0,T]$ such that
$h<{H}_{i}({z}_{i}(t))<3h$ for every $t\in [0,{t}_{1}[$, and either ${H}_{i}({z}_{i}({t}_{1}))=h$ or ${H}_{i}({z}_{i}({t}_{1}))=3h$. In view of (4.5),

$\left|{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{H}_{i}({z}_{i}(t))\right|=\left|\u3008J\nabla {H}_{i}({z}_{i}(t)),{A}_{i}\nabla {H}_{i}({z}_{i}(t))+{\mathcal{\mathcal{R}}}_{i}(t,{z}_{1},\mathrm{\dots},{z}_{N})\u3009\right|$$=\left|\u3008J\nabla {H}_{i}({z}_{i}(t)),{\mathcal{\mathcal{R}}}_{i}(t,{z}_{1},\mathrm{\dots},{z}_{N})\u3009\right|\le C{\overline{\epsilon}}_{i}={\displaystyle \frac{h}{2T}}$

for every $t\in [0,{t}_{1}]$, so that

$|{H}_{i}({z}_{i}({t}_{1}))-{H}_{i}({z}_{i}(0))|\le \frac{h}{2T}{t}_{1}<h,$

a contradiction.

Consequently, if ${z}_{i}(0)\in {\mathrm{\Gamma}}_{i}^{2h}$, we have that

$\widehat{r}<\parallel {z}_{i}(t)\parallel \le {r}_{i}\mathit{\hspace{1em}}\text{for every}t\in [0,T],$

so that the rotation number of ${z}_{i}(t)$ around the origin is well defined.
Writing ${z}_{i}(t)$ in polar coordinates, namely

${z}_{i}(t)=({\rho}_{i}(t)\mathrm{cos}{\vartheta}_{i}(t),{\rho}_{i}(t)\mathrm{sin}{\vartheta}_{i}(t)),$

using (A2) and (4.4), we thus have

$-{\vartheta}_{i}^{\prime}(t)=\frac{\u3008J{\dot{z}}_{i}(t),{z}_{i}(t)\u3009}{{\parallel {z}_{i}(t)\parallel}^{2}}=\frac{\u3008{A}_{i}\nabla {H}_{i}({z}_{i}(t))+{\mathcal{\mathcal{R}}}_{i}(t,{z}_{1},\mathrm{\dots},{z}_{N}),{z}_{i}(t)\u3009}{{\parallel {z}_{i}(t)\parallel}^{2}}\ge {A}_{i}{m}_{i}-\frac{{\overline{\epsilon}}_{i}}{\widehat{r}}.$

Choosing finally

${\overline{A}}_{i}=\frac{2\pi \widehat{r}{\nu}_{i}+{\overline{\epsilon}}_{i}T}{{m}_{i}\widehat{r}T},$

we easily conclude the proof.
∎

Now we need a control on the rotation number of the large planar components of the solutions.

*For any $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, let ${\overline{A}}_{i}$ and ${\overline{\epsilon}}_{i}$ be as in Lemma 4.2, and assume that ${A}_{i}\mathrm{\ge}{\overline{A}}_{i}$ and (4.4) holds. Then, there exists ${\overline{R}}_{i}\mathrm{>}\mathrm{0}$ such that any solution $z\mathit{}\mathrm{(}t\mathrm{)}$ of (P) with $\mathrm{\parallel}{z}_{i}\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{\parallel}\mathrm{\ge}{\overline{R}}_{i}$ satisfies*

$Rot({z}_{i}(t);[0,T])<1.$

#### Proof.

Fix $\sigma =1/(2{A}_{i}T)$ and let ${R}_{i}>0$ and ${\vartheta}_{1}^{i}<{\vartheta}_{2}^{i}$, with ${\vartheta}_{2}^{i}-{\vartheta}_{1}^{i}\le 2\pi $, be as in (A3). Choose ${\widehat{R}}_{i}\ge {R}_{i}$ such that

${\widehat{R}}_{i}>\frac{2{\overline{\epsilon}}_{i}T}{{\vartheta}_{2}^{i}-{\vartheta}_{1}^{i}}.$

In view of assumption (A1), there exists ${\overline{R}}_{i}\ge {\widehat{R}}_{i}$ such that if $\parallel {z}_{i}(0)\parallel \ge {\overline{R}}_{i}$, then $\parallel {z}_{i}(t)\parallel \ge {\widehat{R}}_{i}$ for every $t\in [0,T]$. In particular, the rotation number of ${z}_{i}(t)$ is well defined. Let us assume, by contradiction, that $\parallel {z}_{i}(0)\parallel \ge {\overline{R}}_{i}$ and $Rot({z}_{i}(t);[0,T])\ge 1$. Then, writing

${z}_{i}(t)=({\rho}_{i}(t)\mathrm{cos}{\vartheta}_{i}(t),{\rho}_{i}(t)\mathrm{sin}{\vartheta}_{i}(t)),$

as long as ${\vartheta}_{i}(t)\in \mathrm{\Theta}({\vartheta}_{1}^{i},{\vartheta}_{2}^{i})$, since ${\rho}_{i}(t)\ge {\widehat{R}}_{i}\ge {R}_{i}$, we can use (4.2) and (4.4) to obtain

$-{\vartheta}_{i}^{\prime}(t)={\displaystyle \frac{\u3008{A}_{i}\nabla {H}_{i}({z}_{i}(t))+{\mathcal{\mathcal{R}}}_{i}(t,{z}_{1},\mathrm{\dots},{z}_{N}),{z}_{i}(t)\u3009}{{\parallel {z}_{i}(t)\parallel}^{2}}}$$\le {A}_{i}{\displaystyle \frac{1}{2{A}_{i}T}}({\vartheta}_{2}^{i}-{\vartheta}_{1}^{i})+{\displaystyle \frac{{\overline{\epsilon}}_{i}}{{\widehat{R}}_{i}}}<{\displaystyle \frac{{\vartheta}_{2}^{i}-{\vartheta}_{1}^{i}}{T}}.$

Consequently, the time needed to clockwise cross the sector $\mathrm{\Theta}({\vartheta}_{1}^{i},{\vartheta}_{2}^{i})$ is greater than *T*, a contradiction.
∎

#### Proof of Theorem 4.1.

For any $i\in \{1,\mathrm{\dots},N\}$, let ${\overline{A}}_{i}>0$ and ${\overline{\epsilon}}_{i}>0$ be as in Lemma 4.2, and set

$A=\mathrm{max}\left\{{\overline{A}}_{i}:i=1,\mathrm{\dots},N\right\},\epsilon =\mathrm{min}\left\{{\overline{\epsilon}}_{i}:i=1,\mathrm{\dots},N\right\}.$

Take ${A}_{i}\ge A$ and assume that (4.3) holds. Then, take ${\overline{R}}_{i}$ as in Lemma 4.3 for
every $i=1,\mathrm{\dots},N$, and consider the annulus ${\mathcal{\mathcal{A}}}_{i}=\mathcal{\mathcal{B}}(0,{\overline{R}}_{i})\setminus {D}_{i}^{{\overline{h}}_{i}}$. Recall that, taking ${\overline{h}}_{i}>0$ sufficiently small, the inner boundary of ${\mathcal{\mathcal{A}}}_{i}$ is star-shaped. Then, by Lemmas 4.2 and 4.3, for every solution $z(t)$ of (P), if ${z}_{i}(0)$ belongs to the inner boundary of ${\mathcal{\mathcal{A}}}_{i}$, then ${z}_{i}(t)$ makes more than ${\nu}_{i}$ clockwise rotations around the origin in the time *T*, while if $\parallel {z}_{i}(0)\parallel ={\overline{R}}_{i}$, it makes less than one clockwise turn in the same time.

We now use the fact that (P) is a *positive transformation of a Hamiltonian system*, and consider the linear transformation $\mathcal{\mathcal{L}}$ defined in (4.1). Being all the matrices ${\mathbb{M}}_{i}$ invertible with positive determinant, the set

$\mathcal{\mathcal{A}}=\mathcal{\mathcal{L}}({\mathcal{\mathcal{A}}}_{1}\times \mathrm{\dots}\times {\mathcal{\mathcal{A}}}_{N})$

is thus of the type ${\stackrel{~}{\mathcal{\mathcal{A}}}}_{1}\times \mathrm{\dots}\times {\stackrel{~}{\mathcal{\mathcal{A}}}}_{N}$, where each ${\stackrel{~}{\mathcal{\mathcal{A}}}}_{i}$ is a planar annulus with star-shaped boundaries with respect to the origin. Since the change of variables preserves the above described rotational properties of the solutions, we can apply [31, Theorem 8.2] to the Hamiltonian system obtained from (P) through the change of variables given by $\mathcal{\mathcal{L}}$. We thus obtain at least $N+1$ distinct *T*-periodic solutions

${\stackrel{~}{z}}^{k}(t)=({\stackrel{~}{z}}_{1}^{k}(t),\mathrm{\dots},{\stackrel{~}{z}}_{N}^{k}(t))$

such that for every $k=1,\mathrm{\dots},N+1$, each component ${\stackrel{~}{z}}_{i}^{k}(t)$, with $i=1,\mathrm{\dots},N$, makes exactly ${\nu}_{i}$ clockwise rotations around the origin in the time interval $[0,T[$. Setting

${z}^{k}(t)=({\mathbb{M}}_{1}^{-1}{\stackrel{~}{z}}_{1}^{k}(t),\mathrm{\dots},{\mathbb{M}}_{N}^{-1}{\stackrel{~}{z}}_{N}^{k}(t)),$

we obtain the solutions of (P) we are looking for, and the proof is thus completed.
∎

As a particular case, we can deal with a system of scalar second order equations like

$\{\begin{array}{cc}\hfill {\ddot{x}}_{1}+{A}_{1}^{2}{f}_{1}({x}_{1})& =\frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{1}}(t,{x}_{1},\mathrm{\dots},{x}_{N}),\hfill \\ & \mathrm{\vdots}\hfill \\ \hfill {\ddot{x}}_{N}+{A}_{N}^{2}{f}_{N}({x}_{N})& =\frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{N}}(t,{x}_{1},\mathrm{\dots},{x}_{N}),\hfill \end{array}$(4.6)

where the continuous function $\mathcal{\mathcal{W}}:\mathbb{R}\times {\mathbb{R}}^{N}\to \mathbb{R}$ is *T*-periodic in *t*, and continuously differentiable with respect to $({x}_{1},\mathrm{\dots},{x}_{N})$. Indeed, we can write the equivalent system

$\{\begin{array}{cc}\hfill -{\dot{y}}_{i}& ={A}_{i}{f}_{i}({x}_{i})-\frac{1}{{A}_{i}}\frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{i}}(t,{x}_{1},\mathrm{\dots},{x}_{n}),\hfill \\ \hfill {\dot{x}}_{i}& ={A}_{i}{y}_{i},\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}}i=1,\mathrm{\dots},N,$

which is in the form (P), with ${z}_{i}=({x}_{i},{y}_{i})$, taking

${H}_{i}({x}_{i},{y}_{i})=\frac{1}{2}{y}_{i}^{2}+{F}_{i}({x}_{i}),$

where ${F}_{i}$ is a primitive of ${f}_{i}$ and

${\mathcal{\mathcal{R}}}_{i}(t,{x}_{1},{y}_{1},\mathrm{\dots},{x}_{N},{y}_{N})=-\frac{1}{{A}_{i}}\left(\begin{array}{c}\hfill \frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{i}}(t,{x}_{1},\mathrm{\dots},{x}_{n})\hfill \\ \hfill 0\hfill \end{array}\right).$

Notice that (4.6) is a *positive transformation of a Hamiltonian system*, with the linear function $\mathcal{\mathcal{L}}$ in (4.1) given by

${\mathbb{M}}_{i}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {A}_{i}\hfill \end{array}\right),i=1,\mathrm{\dots},N.$

As a consequence, we have the following statement, where, for simplicity, we only consider the case ${\nu}_{1}=\mathrm{\dots}={\nu}_{N}=1$.

*Assume that the continuous functions ${f}_{i}\mathrm{:}\mathrm{R}\mathrm{\to}\mathrm{R}$
satisfy*

$\underset{s\to 0}{lim\; inf}\frac{{f}_{i}(s)}{s}>0,\underset{s\to +\mathrm{\infty}}{lim}\frac{{f}_{i}(s)}{s}=0.$

*Moreover, for every $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, let ${K}_{i}\mathrm{>}\mathrm{0}$ be such that*

$\left|\frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{i}}(t,{x}_{1},\mathrm{\dots},{x}_{n})\right|\le K{}_{i}\mathit{\hspace{1em}}\mathit{\text{for every}}t\in [0,T]\mathit{\text{and}}{x}_{1},\mathrm{\dots},{x}_{N}\in \mathbb{R}.$(4.7)

*Then, there exists $\overline{A}\mathrm{>}\mathrm{0}$ such that if ${A}_{i}\mathrm{\ge}\overline{A}$ for every $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, system (4.6) has at least $N\mathrm{+}\mathrm{1}$ distinct periodic solutions*

${x}^{k}(t)=({x}_{1}^{k}(t),\mathrm{\dots},{x}_{N}^{k}(t))$

*with minimal period **T*.
Moreover, for every $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N\mathrm{+}\mathrm{1}$, each component ${x}_{i}^{k}\mathit{}\mathrm{(}t\mathrm{)}$, with $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}N$, has exactly two simple zeros in the interval $\mathrm{[}\mathrm{0}\mathrm{,}T\mathrm{[}$.

#### Proof.

First, we notice that (A1) is fulfilled, in view of the growth assumption on the nonlinearities.
Let us now check (A2). We know that there exist ${\alpha}_{i}>0$ and ${\beta}_{i}>0$ such that

$0<|s|<{\beta}_{i}\Rightarrow \frac{{f}_{i}(s)}{s}\ge {\alpha}_{i}.$

Then, if $\parallel ({x}_{i},{y}_{i})\parallel \le {\beta}_{i}$,

$\frac{\u3008\nabla {H}_{i}({x}_{i},{y}_{i}),({x}_{i},{y}_{i})\u3009}{{\parallel ({x}_{i},{y}_{i})\parallel}^{2}}}={\displaystyle \frac{{x}_{i}{f}_{i}({x}_{i})+{y}_{i}^{2}}{{x}_{i}^{2}+{y}_{i}^{2}}}\ge \mathrm{min}\{{\alpha}_{i},1\}>0,$

as desired.

We now verify (A3). Fix $\sigma \in ]0,\pi [$, and take ${\vartheta}_{1}^{i}=0$, ${\vartheta}_{2}^{i}=\sigma /2$. Writing

${z}_{i}=({x}_{i},{y}_{i})=({\rho}_{i}\mathrm{cos}{\vartheta}_{i},{\rho}_{i}\mathrm{sin}{\vartheta}_{i}),$

we have that if ${z}_{i}\in \mathrm{\Theta}(0,\sigma /2)$, then

$\frac{\u3008\nabla {H}_{i}({z}_{i}),{z}_{i}\u3009}{{\parallel {z}_{i}\parallel}^{2}}}={\displaystyle \frac{({\rho}_{i}\mathrm{cos}{\vartheta}_{i}){f}_{i}({\rho}_{i}\mathrm{cos}{\vartheta}_{i})+{({\rho}_{i}\mathrm{sin}{\vartheta}_{i})}^{2}}{{\rho}_{i}^{2}}$$\le {\mathrm{sin}}^{2}{\vartheta}_{i}+\left|{\displaystyle \frac{{f}_{i}({\rho}_{i}\mathrm{cos}{\vartheta}_{i})}{{\rho}_{i}\mathrm{cos}{\vartheta}_{i}}}\right|\le {\displaystyle \frac{{\sigma}^{2}}{4}}+\left|{\displaystyle \frac{{f}_{i}({\rho}_{i}\mathrm{cos}{\vartheta}_{i})}{{\rho}_{i}\mathrm{cos}{\vartheta}_{i}}}\right|.$

Taking ${R}_{i}>0$ large enough, if ${z}_{i}\in \mathrm{\Theta}(0,\sigma /2)\setminus \mathcal{\mathcal{B}}(0,{R}_{i})$, then

$\frac{\u3008\nabla {H}_{i}({z}_{i}),{z}_{i}\u3009}{{\parallel {z}_{i}\parallel}^{2}}\le \frac{{\sigma}^{2}}{4}+\frac{{\sigma}^{2}}{4}=\sigma ({\vartheta}_{2}^{i}-{\vartheta}_{1}^{i}).$

The proof is thus completed, noticing that it suffices to choose ${A}_{i}$ large enough in order to make ${\mathcal{\mathcal{R}}}_{i}(t,{z}_{1},\mathrm{\dots},{z}_{N})$ as small as desired.
∎

As an example, Corollary 4.5 directly applies to the following system of *N* coupled pendulums,

$\{\begin{array}{cc}\hfill {\ddot{x}}_{1}+{A}_{1}^{2}\mathrm{sin}{x}_{1}& =\frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{1}}(t,{x}_{1},\mathrm{\dots},{x}_{N}),\hfill \\ & \mathrm{\vdots}\hfill \\ \hfill {\ddot{x}}_{N}+{A}_{N}^{2}\mathrm{sin}{x}_{N}& =\frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{N}}(t,{x}_{1},\mathrm{\dots},{x}_{N}),\hfill \end{array}$

where $\frac{\partial \mathcal{\mathcal{W}}}{\partial {x}_{i}}(t,{x}_{1},\mathrm{\dots},{x}_{N})$ is continuous and bounded for $i=1,\mathrm{\dots},N$, and the constants ${A}_{1},\mathrm{\dots},{A}_{N}$ are large enough. We are thus able to recover the results obtained in [32], by the use of the Poincaré–Birkhoff theorem, for a single equation modeling a forced pendulum having a very small length.

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