6.2: Classifications of Model Equation

Distinctions between linear and nonlinear systems as well as autonomous and non-autonomous systems, which we discussed in Section 4.2, still apply to continuous-time models. But the distinction between first-order and higher-order systems are slightly different, as follows.

First-order system

A differential equation that involves first-order derivatives of state variables ($\dfrac{dx}{ dt})$ only.

Higher-order system

A differential equation that involves higher-order derivatives of state variables ($\dfrac{d^{2}x} {dt^{2}}$, $\dfrac{d^{3}x} {dt^{3}}$, etc.).

Luckily, the following is still the case for continuous-time models as well:

continuous-time models

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

Here is an example:

$$\dfrac{d^{2}\theta}{dt^{2}} =-\dfrac{g}{L}\sin\theta \label{(6.3)}$$

This equation describes the swinging motion of a simple pendulum, which you might have seen in an intro to physics course. $θ$ is the angular position of the pendulum, $g$ is the gravitational acceleration, and $L$ is the length of the string that ties the weight to the pivot. This equation is obviously nonlinear and second-order. While we can’t remove the nonlinearity from the model, we can convert the equation to a first-order form, by introducing the following additional variable:

$$\omega =\dfrac{d\theta}{dt}\label{(6.4)}$$

Using this, the left hand side of Equation $\ref{(6.3)}$ can be written as $\dfrac{d\omega}{dt}$, and therefore, the equation can be turned into the following first-order form:

$$\dfrac{d\theta}{dt}=\omega\label{(6.5)}$$

$$\dfrac{d\omega}{dt}=-\dfrac{g}{L}\sin\theta\label{(6.6)}$$

This conversion technique works for third-order or any higher-order equations as well, as long as the highest order remains finite. Here is another example.

Example $\PageIndex{1}$: driven pendulum

Consider the non-autonomous equation:

$$\dfrac{d^{2}\theta}{dt^{2}} = -\dfrac{g}{L}\sin\theta+k\sin(2\pi{ft}+\phi)\label{6.7}$$

This is a differential equation of the behavior of a driven pendulum. The second term on the right hand side represents a periodically varying force applied to the pendulum by, e.g., an externally controlled electromagnet embedded in the floor. As we discussed before, this equation can be converted to the following first-order form:

$$\begin{align*} \dfrac{d\theta}{dt} &=\omega\label{(6.8)} \\[4pt]\dfrac{d\omega}{dt} &=-\dfrac{g}{L} \sin \theta +k \sin (2\pi f t +\phi)\label{(6.9)} \end{align*}$$

Now we need to eliminate $t$ inside the $\sin$ function. Just like we did for the discrete-time cases, we can introduce a “clock” variable, say $τ$, as follows:

$$\dfrac{d\tau}{dt} =1, \ \tau(0) =0\label{(6.10)}$$

This definition guarantees $τ(t) = t$. Using this, the full model can be rewritten as follows:

$$\begin{align*} \dfrac{d\theta}{dt} &=\omega \label{(6.11)} \\[4pt] \dfrac{d\omega}{dt} &=-\dfrac{g}{L}\sin\theta +k\sin(2\pi{f\tau} +\phi) \label{(6.12)} \\[4pt] \dfrac{d\tau}{dt} &=1, \tau(0) =0 \label{(6.13)} \end{align*}$$

This is now made of just autonomous, first-order differential equations.

This conversion technique always works, assuring us that autonomous, first-order equations can cover all the dynamics of any non-autonomous, higher-order equations.

Exercise $\PageIndex{1}$

Convert the following differential equation into first-order form.

$$\dfrac{d^{2}}{dt^{2}}-x\dfrac{dx}{dt} +x^{2}=0 \label{(6.14)}$$

Exercise $\PageIndex{2}$

Convert the following differential equation into an autonomous, first order form.

$$\dfrac{d^{2}x}{dt^{2}} -a\cos{bt}=0 \label{(6.15)}$$

For your information, the following facts are also applicable to differential equations, as well as to difference equations:

Linear dynamical systems can show only exponential growth/decay, periodic oscillation, stationary states (no change), or their hybrids (e.g., exponentially growing oscillation)^a.

^asometimes they can also show behaviors that are represented by polynomials (or products of polynomials and exponentials) of time. This occurs when their coefficient matrices are non-diagonalizable.

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