Computational complexity of solving polynomial differential equations over unbounded domains

Abstract

In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) $y^{\prime}=p(y)$ over \emph{unbounded time domains}, where $p$ is a vector of polynomials. Contrarily to the bounded (compact) time case, this problem has not been well-studied, apparently due to the "intuition" that it can always be reduced to the bounded case by using rescaling techniques. However, as we show in this paper, rescaling techniques do not seem to provide meaningful insights on the complexity of this problem, since the use of such techniques introduces a dependence on parameters which are hard to compute. We present algorithms which numerically solve these ODEs over unbounded time domains. These algorithms have guaranteed accuracy, i.e. given some arbitrarily large time $t$ and error bound $\varepsilon$ as input, they will output a value $\tilde{y}$ which satisfies $\|y(t)-\tilde{y}\|\leq\varepsilon$. We analyze the complexity of these algorithms and show that they compute $\tilde{y}$ in time polynomial in several quantities including the time $t$, the accuracy of the output $\varepsilon$ and the length of the curve $y$ from $0$ to $t$, assuming it exists until time $t$. We consider both algebraic complexity and bit complexity.