4.2: Classifications of Model Equations

There are some technical terminologies I need to introduce before moving on to further discussions:

Linear system A dynamical equation whose rules involve just a linear combination of state variables (a constant times a variable, a constant, or their sum).

Nonlinear system Anything else (e.g., equation involving squares, cubes, radicals, trigonometric functions, etc., of state variables).

First-order system A difference equation whose rules involve state variables of the immediate past (at time $t−1$) only^a.

Higher-order system Anything else.

^aNote that the meaning of “order” in this context is different from the order of terms in polynomials.

Autonomous system A dynamical equation whose rules don’t explicitly include time $t$ or any other external variables.

Non-autonomous system A dynamical equation whose rules do include time $t$ or other external variables explicitly.

Exercise $\PageIndex{1}$

Decide whether each of the following examples is (1) linear or nonlinear, (2) first-order or higher-order, and (3) autonomous or non-autonomous

1. $x_{t} = ax_{t−1} + b$
2. $x_{t} = ax_{t−1} + bx_{t−2} + cx_{t−3}$
3. $x_{t} = ax_{t−1}(1−x_{t−1})$
4. $x_{t} = ax_{t−1} + bxt−2^{2} + \sqrt[c]{x_{t−1}x_{t−3}}$
5. $x_{t} = ax_{t−1}x_{t−2} + bx_{t−3} + sin(t)$
6. $x_{t} = ax_{t−1} + by_{t−1}, y_{t} = cx_{t−1} + dy_{t−1}$

Also, there are some useful things that you should know about these classifications:

Non-autonomous, higher-order difference equations can always be converted into autonomous, first-order forms, by introducing additional state variables.

For example, the second-order difference equation

$$x_{t}=x_{t-1}+x_{t-2} \label{(4.5)}$$

(which is called the Fibonacci sequence can be converted into a first-order form by introducing a “memory” variable $y$ as follows:

$$y_{t} = x_{t-1}\label{(4.6)}$$

Using this, $x_{t−2}$ can be rewritten as $y_{t−1}$. Therefore the equation can be rewritten as follows:

$$\begin{align} x_{t} &= x_{t-1}+y_{t-1}\label{(4.7)} \\[4pt] y_{t} &= x_{t-1}\label{(4.8)} \end{align}$$

This is now first-order. This conversion technique works for third-order or any higher-order equations as well, as long as the historical dependency is finite. Similarly, a non-autonomous equation

$$x_{t} = x_{t-1} +t\label{(4.9)}$$

can be converted into an autonomous form by introducing a “clock” variable z as follows:

$$z_{t}= z_{t-1} +1, z_{0} =1\label{(4.10)}$$

This definition guarantees $z_{t−1} = t$. Using this, the equation can be rewritten as

$$x_{t} = x_{t-1}+ z_{t-1},\label{(4.11)}$$

which is now autonomous. These mathematical tricks might look like some kind of cheating, but they really aren’t. The take-home message on this is that autonomous first-order equations can cover all the dynamics of any non-autonomous, higher-order equations. This gives us confidence that we can safely focus on autonomous first-order equations without missing anything fundamental. This is probably why autonomous first-order difference equations are called by a particular name: iterative maps.

Exercise $\PageIndex{2}$

Convert the following difference equations into an autonomous, first-order form.

1. $x_{t} = x_{t-1}(1-x_{t-1})sint$

2. $x_{t} = x_{t-1} +x_{t-2}-x_{t-3}$

Another important thing about dynamical equations is the following distinction between linear and nonlinear systems:

Linear equations are always analytically solvable, while nonlinear equations don’t have analytical solutions in general.

Here, an analytical solution means a solution written in the form of $x_{t} = f(t)$ without using state variables on the right hand side. This kind of solution is also called a closed form solution because the right hand side is “closed,” i.e., it only needs $t$ and doesn’t need $x$. Obtaining a closed-form solution is helpful because it gives you a way to calculate (i.e., predict) the system’s state directly from $t$ at any point in time in the future, without actually simulating the whole history of its behavior. Unfortunately this is not possible for nonlinear systems in most cases.