Finite escape time

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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.