All of the differential equations that are solved in a first or even second semester differential equations course have analytic solutions. The differential equations that you solve are carefully crafted for you to do some basic algebraic manipulation and use some basic calculus to solve. However, the vast majority of differential equations have no analytic solutions, so the techniques you learned will not be of any use. So if we can't get a closed form solution for the differential equations that govern the natural world, how do we understand the natural world? This is the focus of the study of nonlinear dynamics, as almost any differential equation that models something complex will be nonlinear, and pretty much any nonlinear ODE has no analytic solution.
Basic Example
One of the simplest examples of a nonlinear ODE, is the simple pendulum.
This is easily derived using Newton's second law. This models a pendulum with no air resistance. If we limit our investigation to small angles (i.e. small values for \(\theta\)), then we can often substitute \(\sin\) for \(\theta\) in the differential equation since for small angles we have
Then our ODE would become
This turns the nonlinear ODE into a linear one with an analytic solution. But this is a limiting view of a real physical model. Instead if we wanted to model the behavior for any \(\theta\), then there is no analytic solution. This shows how limited linear ODEs are, as they cannot even properly model something as basic as the motion of a pendulum. However, this also demonstrates how important Nonlinear theory is.
Again we return to the question of how to understand these equations if we can't find explicit solutions to them. The first step in this process is a technique called state augmentation.
State Augmentation
The primary result here, is that we can break down any higher order ODE, into a sysetm of first order ODEs. This allows us incorporate linear algebra into understanding the system. Consider the example from above.
We can incorporate a new variable to break this equation down. Let
Then
So we can now write
Rearranging the equation we have
Therefore we now have two first order differential equations,
As mentioned, this reduction process is known as state augmentation, and is one of the first steps to understanding the nature of the ODE.
Autonomous and Non-Autonomous ODEs
In later sections we will focus our attention on autonomous systems. These are ODEs of the form,
More specifically, these are systems without an explicit time dependence. A non-autonomous system is of the form
Note the temporal dependence in the second equation as opposed to the first. We say that the time dependence is not explicit in the first example since \(x\) is a function of \(t\) and therefore there is a time dependence, but it is implicit through \(x\).
The reason we will focus on autonomous systems, is that we can decompose any non-autonomous system into an autonomous one of one dimension higher through state augmentation.
Turning Non-Autonomous ODEs into Autonomous ODEs
Consider
Let's introduce a new variable
Then we have that
Since the derivative is with respect to time. Then we can write our 1D non-autonomous system as a 2D autonomous system.
It is because of the results above that our focus will be on autonomous systems of first order ODEs.