Complex systems, almost certainly, exhibit non-linear dynamics.

If so, how do we mathematically analyze such non-linear dynamic systems? This is a challenge since in mathematics we are well equipped to solve linear equations.

Before we embark on an approach let us pause for a moment to consider a nuanced behavior of complex system, which researchers till date, seemed to have either overlooked or ignored.

The fact is that it is not necessary for complex systems to always behave non-linearly as we might suppose. For most of its time, I dare say, all complex systems would behave more-or-less linearly, possibly within the limits of usual variations. Only at some critical juncture does a complex system show non-linear behavior. If it weren’t so it would have been nearly impossible to live anyway.

So, the answer to the question – ‘How do we mathematically analyze such non-linear dynamic system?’ is surprisingly simple — use mathematics for solving linear systems. We shall in a moment, see why that is a perfectly legitimate way of mathematically analyzing any complex non-linear system.

To do so, we start with the question on — why understanding linear systems is so very important?

The neat answer is: because we can solve them! Most of the time, it is enough to solve linear problems. However the more important reason that we do so is simply because fundamental laws of physics are often linear. For example, Maxwell equations for the laws of electricity are linear. Even the great laws of Quantum Mechanics, as we know of today, turn out to be linear equations. So, it is worthwhile spending time and effort on linear equations since if we do so, we are ready, in principle, to understand a lot of things including the non-linear behavior of complex systems.

But what do we do when complex systems actually start behaving non-linearly? Well, a non-linear equation cannot be solved in any other way but numerically — that is by numerical analysis as well as graphical analysis.

Hence, to sum up, any complex system can be understood by solving linear equations and by numerical/graphical methods. It turns out that for fifty percent of the time we would be happy solving linear equations for complex systems in physics and engineering and the rest fifty percent time we would be happy solving non-linear behavior with numerical and graphical analysis.

That seems to be rather fair. But more than that it clearly shows a path ahead to deal with complex systems involved in engineering and physics.