Introduction to the Modeling and Analysis of Complex Systems

6.2. CLASSIFICATIONS OF MODEL EQUATIONS 101This conversion technique works for third-order or any higher-order equations as well, aslong as the highest order remains finite.Here is another example. This time it is a non-autonomous equation:d 2 θdt = − g sin θ + k sin(2πft + φ) (6.7)2 LThis is a differential equation of the behavior of a driven pendulum. The second termon the right hand side represents a periodically varying force applied to the pendulumby, e.g., an externally controlled electromagnet embedded in the floor. As we discussedbefore, this equation can be converted to the following first-order form:dθdt = ω (6.8)dωdt = − g sin θ + k sin(2πft + φ)L(6.9)Now we need to eliminate t inside the sin function. Just like we did for the discrete-timecases, we can introduce a “clock” variable, say τ, as follows:dτdt= 1, τ(0) = 0 (6.10)This definition guarantees τ(t) = t. Using this, the full model can be rewritten as follows:dθdt = ω (6.11)dωdt = − g sin θ + k sin(2πfτ + φ)L(6.12)dτ= 1,dtτ(0) = 0 (6.13)This is now made of just autonomous, first-order differential equations. This conversiontechnique always works, assuring us that autonomous, first-order equations can cover allthe dynamics of any non-autonomous, higher-order equations.Exercise 6.1Convert the following differential equation into first-order form.d 2 xdt 2 − xdx dt + x2 = 0 (6.14)