What branch of math deals with nonlinear systems?

[V0ODO0CH1LD]

Like linear algebra goes in depth about linear systems, what should I look for to learn about the extension of linear algebra to nonlinear systems? Is there a name of the field of study? If I go into a book store to buy books about it, what should I be looking for?

Abstract Algebra? Complex Analysis? If not, what are those anyway?

Also, I've realized that is cool to know some calculus before linear algebra to relate some topics, but not necessary. Is multivariable calculus and differential equations something I should know before all of this other things? I'm just asking cause most universities have calculus up to differential equations before any of the stuff I'm asking about, including linear algebra. Is that for a particular reason?

Thanks!

 

[Muphrid]

I think linear algebra occupies a unique position because even arbitrary transformations can be treated as instantaneously linear and then integrated with respect to some parameter to capture the full, nonlinear effect of the operator. So, some calculus would be helpful there.

Abstract algebra goes into stuff about rings, fields (not fields not a vector space, but fields of numbers), and other general structures which admit an algebra but may be more exotic than the algebra of real numbers. Vector spaces are a topic of study under abstract algebra, too.

Complex analysis is the study of functions of a complex-valued variable, just as real analysis is the study of functions of a real-valued variable. Complex numbers in general are just a way of talking about points or vectors on a 2d plane.

 

[V0ODO0CH1LD]

Okay, I've another question related to the subjects we are discussing, but not to my previous question.

If you look at an equation like f(x) = y = x. It has one input and one output, right?

But you can also write it like f(x, y) = 0 = x - y. In which case you have two inputs and one constant output.

In one equation I transform the 1-dimensional vector x into the vector y using the identity function. And in the other case I transform the 2-dimensional vector <x, y> into the the 1-dimensinal vector 0. So <x, y> can be any orthogonal vector to <1, -1>.

Are both ways of looking at the equation valid?