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.