This contribution investigates the computational complexity of simulating linear ordinary differential equations (ODEs) on digital computers. We provide an exact characterization of the complexity blowup for a class of ODEs of arbitrary order based on their algebraic properties, extending previous characterization of first order ODEs. Complexity blowup indeed arises in most ODEs (except for certain degenerate cases) and means that there exists a low complexity input signal, which can be generated on a Turing machine in polynomial time, leading to a corresponding high complexity output signal of the system in the sense that the computation time for determining an approximation up to $n$ significant digits grows faster than any polynomial in $n$. Similarly, we derive an analogous blowup criterion for a subclass of first-order systems of linear ODEs. Finally, we discuss the implications for the simulation of analog systems governed by ODEs and exemplarily apply our framework to a simple model of neuronal dynamics$-$the leaky integrate-and-fire neuron$-$heavily employed in neuroscience.

翻译：本文研究了在数字计算机上模拟线性常微分方程的计算复杂性。我们基于代数性质，对任意阶常微分方程的一类系统给出了复杂性爆发的精确刻画，将先前对一阶常微分方程的表征进行了推广。复杂性爆发实际上出现在大多数常微分方程中（除了某些退化情形），这意味着存在一个低复杂度的输入信号（可在图灵机上以多项式时间生成），但系统对应的输出信号具有高复杂度——具体表现为，要确定该输出信号精确到$n$位有效数字所需的计算时间增长速度快于$n$的任何多项式。类似地，我们为一类一阶线性常微分方程系统推导出相应的爆发判据。最后，我们讨论了这些结果对由常微分方程描述的模拟系统仿真的意义，并以神经动力学中广泛使用的简单模型——泄漏积分激发神经元——为例展示了我们的分析框架。