in my opinion after looking into this to some degree, you will probably not do any better than the paper you've cited for background and references which are extensive. the authors specifically indicate that they have devised a novel approach that does not seem to have any direct precedent in the literature, and this seems correct. in other words, as they state themselves:
Here we present a novel continuous-time dynamical system for k-SAT, with a dynamics that
is rather different from previous approaches.
so their formulation (in the introduction, eg eqs.1,2) can be seen/regarded as a seminal new "bridge theorem" between two previously mainly different fields of dynamical systems theory and the theory of NP completeness.
this has happened many times in the past with NP completeness, and an early notice of this came in 1988 by Papadimitriou, in NP Completeness, a retrospective, which can be seen as a kind of TCS sequel/analog to the famous essay Unreasonable effectiveness of mathematics in the physical sciences, Wigner.
they cite the following as nearest references to their work:
Although the theoretical possibility of efﬁcient computation via chaotic dynamical systems has been shown previously , nonlinear dynamical systems theory has not been
exploited for NP-complete problems in spite of the fact that, as shown by Gu et al., Nagamatu
et al.  and Wah et al. , k-SAT can be formulated as a continuous global optimization
problem, and even cast as an analog dynamical system [20, 21].
in SAT literature Mezard was a pioneer in introducing statistical physics analogies which have some strong connections to their new framework, but again they cite him.
here are two other sideways angles outside the paper references. Linear programming algorithms have a concept of continous variables, iteration and convergence somewhat analogous to a constrained dynamical systems trajectory. there is a significant amount of research in this vein under the heading of Maslov's method. here is another early reference on using linear programming to solve SAT:
another area that seems loosely similar is the use of artificial neural networks to solve SAT. again there is a concept of an algorithm with continuous variables that converges to a solution under update rules expressed as differential equations. see eg