  This technical note considers the identification of nonlinear discrete-time
systems with additive process noise but without measurement noise. In
particular, we propose a method and its associated algorithm to identify the
system nonlinear functional forms and their associated parameters from a
limited number of time-series data points. For this, we cast this
identification problem as a sparse linear regression problem and take a
Bayesian viewpoint to solve it. As such, this approach typically leads to
nonconvex optimisations. We propose a convexification procedure relying on an
efficient iterative re-weighted $\ell_1$-minimisation algorithm that uses
general sparsity inducing priors on the parameters of the system and marginal
likelihood maximisation. Using this approach, we also show how convex
constraints on the parameters can be easily added to our proposed iterative
re-weighted $\ell_1$-minimisation algorithm. In the supplementary material
\cite{appendix}, we illustrate the effectiveness of the proposed identification
method on two classical systems in biology and physics, namely, a genetic
repressilator network and a large scale network of interconnected Kuramoto
oscillators.
