Computer algebra derives normal forms of stochastic differential equations

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term dynamics from undesirably detailed microscale dynamics. I aim to explore normal forms of stochastic differential equations when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of detailed microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to remove all fast time processes. Sri Namachchivaya, Leng and Lin (1990–1) emphasise the importance of quadratic stochastic effects “in order to capture the stochastic contributions of the stable modes to the drift terms of the critical modes.” I derive such important quadratic effects using the normal form coordinate transform to separate slow and fast modes. The results will help us accurately model multiscale stochastic systems.