approaches. Moreover, as long as $P\neq NP$ (where $P$ is the complexity class of problems that can be solved in poylnomial time on a deterministic Turing machine), even efficient heuristics for some of these problems will remain elusive. See Fig. \[fig:complexity\_map\] for the hypothesized relationship between the most popular classes.

![The computational complexity map for the most popular complexity classes.[]{data-label="fig:complexity_map"}](ComplexityMap2){width="65.00000%"}

In this work, we start by surveying the use of dynamical systems in the context of constructing state-of-the-art algorithms for NP-hard problems. In particular, we will cover the use of dynamical systems theory for constructing decentralized graph clustering algorithms [@Cit:cluster1; @Cit:cluster2], solutions for the TSP [@Cit:TSP_sahai], and quantum-inspired networks of Duffing oscillators for solving the MAX-CUT problem [@Cit:quantum_net]. We then switch to the use of dynamical systems theory for analysis of algorithms [@Cit:koopman] and the underlying problems [@Cit:zoltan; @Cit:zoltan2].

The goal of this survey paper is to highlight the potential application of dynamical systems theory for optimization of complex functions and analysis of computational complexity theory. This is a nascent field which presents the possibility of tremendous impact. Additionally, we expect this area to lead to new theoretical developments in nonlinear dynamics theory and novel algorithms for computationally intractable problems.

