Recursive expression of gradient for dynamical sytem

Question

Let $g:\mathbb{R}^d \to \mathbb{R}^d$ and let $x \in \mathbb{R}^d$. Consider the following discrete-time dynamical system:

$$G_{t+1} = g(G_t) ,\qquad G_0 = x.$$

Is there a recursive formula/expression for $\nabla_x G_{t+1}$ in terms of $G_t$ and $\nabla G_t$?

Answer 1

By the chain rule $$ ∇_xG_{t+1}(x)=∇_Gg(G_t(x))\,∇_xG_t(x),~~~~ ∇_xG_0(x)=∇_xx=I. $$