# Finite escape time

A trajectory of a dynamical system is said to have **finite escape time** if for some . 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

corresponding to the reactions and , with both rate constants equal to one and a positive initial concentration ^{[1]}. This system has solution which satisfies for . 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

corresponding to the reaction with associated rate constant and a strictly positive initial concentration. The solution blows up but there is no point such that . Consequently, no trajectory of this mechanism exhibits *finite* escape time.

## References

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