Finite escape time

A trajectory ${\vec {x}}(t)$ of a dynamical system is said to have finite escape time if $\lim _{t\to t_{e}^{-}}\|{\vec {x}}(t)\|=\infty$ for some $t_{0}<t_{e}<\infty$. That is to say, the trajectory blows up to infinity (and ceases to exist) at a finite time in the future.

Finite escape time in mass action systems

Finite escape time can be exhibited by mass action systems. For example, consider the dynamical system

${\frac {dc(t)}{dt}}=1+c(t)^{2},\;\;\;c(0)=c_{0}>0$

corresponding to the reactions $0\rightarrow {\mathcal {A}}$ and $2{\mathcal {A}}\rightarrow 3{\mathcal {A}}$, with both rate constants equal to one and a positive initial concentration . This system has solution $c(t)=\tan(t+\arctan(c_{0}))$ which satisfies $\lim _{t\to t_{e}^{-}}c(t)=\infty$ for $t_{e}={\frac {\pi }{2}}-\arctan(c_{0})$. In other words, the solution blows up to infinity within a finite time.

It is worth noting that finite escape time is not simply the capacity of a trajectory to blow up. Consider the dynamical system

${\frac {dc(t)}{dt}}=c(t),\;\;\;c(0)=c_{0}>0$

corresponding to the reaction $0\;\rightarrow \;{\mathcal {A}}$ with associated rate constant $k=1$ and a strictly positive initial concentration. The solution $c(t)=c_{0}e^{t}$ blows up but there is no point $t_{e}>0$ such that $\lim _{t\to t_{e}^{-}}c(t)=\infty$. Consequently, no trajectory of this mechanism exhibits finite escape time.