We Can Always Reduce a Non-linear Dynamical System to Linear—At Least Locally—But Does It Help?

Many real-life phenomena are described by dynamical systems. Sometimes, these dynamical systems are linear. For such systems, solutions are well known. In some cases, it is possible to transform a nonlinear system into a linear one by appropriately transforming its variables, and this helps to solve the original nonlinear system. For other nonlinear systems—even for the simplest ones—such transformation is not known. A natural question is: which nonlinear systems allow such transformations? In this paper, we show that we can always reduce a nonlinear system to a linear one—but, in general, it does not help, since the complexity of finding such a reduction is exactly the same as the complexity of solving the original nonlinear system.