introducing a second dynamical variable [@pomeau84]: $$\begin{aligned}
y^{t+1}_k & = & x^{t}_k \nonumber \\
 x^{t+1}_k &= & y^{t}_k\,  \phi\left( \sum_{i\in V_k} x^{t}_{i} \right).
 \label{q2r}\end{aligned}$$ The rule (\[q2r\]) is complemented with the initial condition $x_k^{t=0} $ and $ y_k^{t=0} $.

As shown by Pomeau [@pomeau84], the energy $$\begin{aligned}
E[\left\{x^t,y^t\right\}]  = -\frac{1}{2} \sum_{\left< i,k\right>}  x^{t}_k y^{t}_i,
\label{energy}\end{aligned}$$ is preserved, $E[\left\{x^t,y^t\right\}] = E[\left\{x^{t=0},y^{t=0}\right\}] $ under the dynamics defined by the Q2R rule (\[q2r\]). Moreover, the energy is bounded by $ -2N \leq E\leq 2 N$.

Despite the existence of an energy-like quantity, it is not possible to speak about a Hamiltonian discrete dynamic because, the variables $x^t$ and $y^t$ and the energy (\[energy\]) are discrete quantities [@pomeau84]. Moreover, supported by the existence of a large number of periodic orbits (see next), it is believed that Q2R possesses a large number of other invariants, however, up to date, other conserved quantities are not known yet.

“Long-time” dynamics of the Q2R cellular automata. {#longtime}
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Numerical simulations of the Q2R model in two space dimensions for large system sizes, [*eg.*]{} $N=256\times 256$, and random initial conditions, shows that the dynamics displays a fluctuating spatio-temporal pattern showing regions with states +1 and sectors with states $-1$, 