# 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 [1]. 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.

## References

1. Matthew D. Johnston, Topics in Chemical Reaction Network Theory, Ph.D. Thesis, University of Waterloo, 2011.