I'll get straight to the point because I can't seem to be able to write a decent sentence today (sorry about that).
I know that the difference between a linear differential equation and a nonlinear differential equations is that the Linear differential equations have both the additive and homogeneity property and that the nonlinear differential equation doesn't.
What I want to know is how not having the property of linearity affects the set of solutions, like what can a nonlinear differential equations do to the set of solutions that a linear differential equation can't to make it "so complicated" ? (This question is a bit vague so don't hesitate to ask for clarification.)