$$\forall\varepsilon>0~,\quad\exists c~:~H(t,x)\leq{\frac{\varepsilon}{2}}\left\|x\right\|^{2}+c~.$$
2
2 + c . **(26)**
Assume also that H is C
2
, and H′′(t, x) **is positive definite everywhere. Then**
there is a sequence xk, k ∈ IN, of kT **-periodic solutions of the system**

$${\dot{x}}=J H^{\prime}(t,x)$$
x˙ = JH′(t, x**) (27)**
such that, for every k ∈ IN, there is some po ∈ IN **with:**
p ≥ po ⇒ xpk 6= xk . **(28)**

$$(27)$$
$$(28)$$
$$\mathbb{D}$$
$$(30)$$
$$(31)$$

Example 1 (External forcing). **Consider the system:**
x˙ = JH′(x) + f(t**) (29)**
where the Hamiltonian H is (0, b∞**)-subquadratic, and the forcing term is a**
distribution on the circle:

$$f={\frac{d}{d t}}F+f_{o}$$
dtF + fo **with** F ∈ L
$$\mathrm{with}\;\;F\in L^{2}\left(\mathbb{R}/T Z;\mathbb{R}^{2n}\right)\;,$$
where $f_{o}:=T^{-1}\int_{o}^{T}f(t)dt.$ For instance,
$$f(t)=\sum_{k\in\mathbb{N}}\delta_{k}\xi\ ,$$

where δk is the Dirac mass at t = k and ξ ∈ IR2nis a constant, fits the prescription. This means that the system ˙x = JH′(x**) is being excited by a series**
of identical shocks at interval T .

Definition 1. Let A∞(t) and B∞(t) be symmetric operators in IR2n**, depending**
continuously on t ∈ [0, T ], such that A∞(t) ≤ B∞(t) **for all** t.

A Borelian function H : [0, T ] × IR2n → IR is called (A∞, B∞)**-subquadratic**
at infinity if there exists a function N(t, x) **such that:**

$$H(t,x)=\frac{1}{2}\left(A_{\infty}(t)x,x\right)+N(t,x)\tag{32}$$  $$\forall t\,\quad N(t,x)\qquad\mbox{is convex with respect to}\ \ x\tag{33}$$  $$N(t,x)\geq n\left(\|x\|\right)\qquad\mbox{with}\ \ n(s)s^{-1}\to+\infty\ \ \mbox{as}\ \ s\to+\infty\tag{34}$$  $$\exists c\in{\rm I\!R}\ :\quad H(t,x)\leq\frac{1}{2}\left(B_{\infty}(t)x,x\right)+c\quad\forall x.\tag{35}$$  _If $A_{\infty}(t)=a_{\infty}I$ and $B_{\infty}(t)=b_{\infty}I$, with $a_{\infty}\leq b_{\infty}\in{\rm I\!R}$, we shall say that 
$$(32)$$
$$(33)$$
$$(35)$$

H is (a∞, b∞)**-subquadratic at infinity. As an example, the function** kxk α**, with**

1 ≤ α < 2, is (0, ε)-subquadratic at infinity for every ε > 0**. Similarly, the** Hamiltonian
$$H(t,x)={\frac{1}{2}}k\left\|k\right\|^{2}+\left\|x\right\|^{\alpha}$$
α**(36)**
is (k, k + ε)-subquadratic for every ε > 0. Note that, if k < 0, it is not convex.