4.2: Classifications of Model Equations
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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) onlya.
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.
Decide whether each of the following examples is (1) linear or nonlinear, (2) first-order or higher-order, and (3) autonomous or non-autonomous
- x_{t} = ax_{t−1} + b
- x_{t} = ax_{t−1} + bx_{t−2} + cx_{t−3}
- x_{t} = ax_{t−1}(1−x_{t−1})
- x_{t} = ax_{t−1} + bxt−2^{2} + \sqrt[c]{x_{t−1}x_{t−3}}
- x_{t} = ax_{t−1}x_{t−2} + bx_{t−3} + sin(t)
- 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.
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.