# What is meant when mathematicians or engineers say we cannot solve nonlinear systems?

I was watching a video on "system identification" in control theory, in which the creator says that we don't have solutions to nonlinear systems. And I have heard this many times in many contexts, related to control problems or nonlinear odes, etc. I think I am reacting to these kinds of blanket statements, and I would like to understand more precisely what is meant.

But I wanted to understand precisely what is meant that we can't solve nonlinear systems? Indeed, there are probably hundreds of questions on Math SE regarding numerical solutions to nonlinear systems. There are many algorithms for numerically solving different types of nonlinear systems of equations, including Newton's method, sequential quadratic programming, BFGS, Broyden's method, etc. All of these methods have their own limitations, such as positive definiteness, the existence of hessians, and so forth.

Now in a linear ode or linear system of equations, we can get the solution for the system pretty easily, even for large systems. So I can solve large system of equations using numerical linear algebra, etc. I can plot the vector field for a linear system of equations pretty easily, because I have a Jacobian for the system and I can plug in points to plot the corresponding vector field. For a nonlinear system I have to compute those trajectories directly.

So I am hoping that this question is not overly broad. But the issue is that I keep hearing this claim over and over, but I am not sure what the actual specific issue is. I understand that there are difficulties getting solutions for nonlinear equations, it is not like they are all "non-solvable."

• Hi. Can you clarify what you mean by "plot" and "plot pretty easily"? Plotting makes sense to me when you have only two variables x,y, or at most three variables x,y,z, but starting at 4 variables I'm not sure what "plot" means since I can only see in 2 or 3 dimensions.
– Stef
Commented Mar 26 at 10:59
• Note that when control engineers speak about "systems", they mean "dynamical systems" (i.e., systems of ordinary differential equations without explicit time dependence), not "systems of linear/nonlinear equations". Commented Mar 26 at 16:53
• @Stef correct, I was just talking about plotting in 2 or 3 dimensions. What I was getting at, is that for linear systems I can find a formula" for computing the vector field of the system. I can just plug in points to the Jacobian in any number of dimensions, as long as the system is linear, and get the corresponding vectors. In a nonlinear system, I don't have that kind of formula for the vector field. If I have a nonlinear system, I can linearize around any point, compute the Jacobian and get the local vector field. But that is different than having a nice formula. That is all I meant. Commented Mar 26 at 22:17
• @FedericoPoloni yes true. I was being a bit loose with my language. So the systems are dynamical systems, in particular linear time invariant systems--in many/most cases--though control of nonlinear dynamical systems is an important area as well. So what I am talking about: linear/nonlinear odes, is different than just solving a purely nonlinear system of equations $f(x1, x2, ..) = 0$. Commented Mar 26 at 22:32