proposition collects several homogeneity properties of the continuous dynamics:

\[thm:prop2\] Let $(m(t),w(t),f(t))$ be a trajectory solution of the dynamics –, with the initial markings $(m_p(0))_{p \in {\mathcal{P}}}$, and the holding times $(\tau_p)_{p \in {\mathcal{P}}}$ and let $\alpha \in {\mathbb{R}_{> 0}}$, then:

(i) $(\alpha m(t), \alpha w(t), \alpha f(t))$ is a trajectory solution of the dynamics, associated with the initial markings $(\alpha m_p(0))_{p \in {\mathcal{P}}}$.

(ii) $(m(t/\alpha),w(t/\alpha),(1/\alpha)f(t/\alpha))$ is a trajectory solution of the dynamics, associated with the holding times $(\alpha \tau_p)_{p \in {\mathcal{P}}}$ and the same initial conditions.

(iii) let $x$ be a vector of the kernel of $C$, and $D = \text{diag}(\tau)$ be the ${\mathcal{P}}\times {\mathcal{P}}$ diagonal matrix such that $D_{pp} = \tau_p$, then $(m(t) + \alpha D C^+ x, w(t), f(t) + \alpha x)$ is a trajectory solution of the dynamics, associated with the initial markings $(m(0) + \alpha D C^+ x)$.

The first two statements derive easily from the homogeneity properties of Equations –. For the third statement, one can note that adding $\alpha x_q$ to the $f_q(t)$ and adding $\alpha \sum_{q\in p{^{\textrm{in}}}} a_{qp}^+ x_q$ to the $m_p(t)/\tau_p$ in – does not change the right hand sides of  and , or the expression within the minimum in . 