4. Measuring Chaos
4.1 Harmonic Oscillator
A large portion of the first two chapters of this book dealt with the behavior of discrete dynamical systems, in particular, the family of quadratic mappings. Three basic types of behavior were observed: fixed, periodic, and chaotic (or ergodic). The first two kinds of behavior are present in many continuous dynamical systems of the kind described by differential equations with exact solutions. In this chapter, a comparison will be made between the best known continuous periodic system, the harmonic oscillator, and the best known discrete periodic system, the logistic equation. Given the analogous nature of the two systems, it should be possible to perform a parallel analysis of the simple harmonic oscillator and the logistic equation. From such a study, a new topic in discrete dynamical analysis will arise, namely, an analog to the driven harmonic oscillator: what I call the driven logistic equation. In the process, a device for quantifying the behavior of iterated mappings will be introduced, the Lyapunov exponent, with the indirect result of producing ever more fascinating images.
The simple harmonic oscillator (SHO) is a mass connected to some elastic object of negligible mass that is fixed at the other end and constrained so that it may only move in one dimension. This simplified model approximates many systems that vibrate or oscillate: drum heads, guitar strings, the quantum mechanical descriptions of an atom, etc. The importance of this problem, however, lies in the fact that equations of a similar form arise when a particle moves through any region whose potential has one or more local minima: planetary and satellite motion, the classical description of an electron in orbit around a nucleus, pendulums, etc. Similar equations also arise in the study of LCR circuits: the type used in analog communications devices and electric power transmission. (The letters L, C, and R refer to the symbols used to identify the electrical quantities of inductance, capacitance, and resistance respectively.)
When dissipative forces such as friction and air resistance are ignored, the net force will be directly proportional to the displacement of the mass from the system's equilibrium position and pointing in the opposite direction; a condition known as Hooke's law. Beginning with Newton's second law of motion, we can derive a second order linear differential equation whose solution gives us the displacement of the mass as a function of time.
Given…
x =  displacement 
k =  spring constant 
m =  mass 
A =  ±x_{max} = amplitude 
φ =  phase 
ω_{0} =  natural frequency 
Then Newton's second law becomes a second order differential equation…
F_{net} =  ma 
−kx =  mẍ 
mẍ + kx =  0 
with the following solution…
x = A cos(ω_{0}t + φ)  
Where…
ω_{0} = √  k 
m 
The motion is periodic with a frequency that depends on the nature of the mass and the elastic object (here assumed to be a spring). Amplitude "A" and phase "φ" [phi] are constants determined by the initial displacement and velocity of the system.
A more realistic physical model is one that includes dissipative forces — the damped harmonic oscillator. For the sake of simplicity, assume that any dissipative force is directly proportional to the velocity of the mass and in the opposite direction. This is a good approximation of the behavior of air resistance and produces another differential equation with an exact solution. In fact, it is the only type of dissipative force for which the differential equation of motion has an exact solution.
Given…
b =  drag coefficient 
γ =  damping factor 
ω_{1} =  damped frequency 
Again, Newton's second law becomes a second order differential equation but with an extra term.
F_{net} =  ma 
−bẋ − kx =  mẍ 
mẍ + bẋ + kx =  0 
The solution is now…
x = A e^{−γt} cos(ω_{1}t + φ) 
Where…
ω_{0} = √  k 
m 
γ =  b 
2m 
ω_{1} = √(ω_{0}^{2} − γ^{2})  
We now have an equation that yields different behavior for different parameter values. When the damping factor equals zero the system reduces to the case of the simple harmonic oscillator: continuous oscillation at the natural frequency with constant amplitude. When the damping factor is greater than zero the system may or may not oscillate, depending of the relation between the damping factor "γ" [gamma] and the natural frequency ω_{0} [omega nought].
ω_{0} > γ  The system is said to be under damped and exhibits transient behavior, oscillating at the damped frequency with an amplitude that decays exponentially. If we wait long enough the system will settle into its equilibrium position. 
ω_{0} = γ  The critically damped case. The system will return quickly and smoothly to its equilibrium position. There is no oscillatory behavior at all this time. The motion is described entirely by exponential decay. 
ω_{0} < γ  The over damped case. The solution is now the sum of two exponential decay terms, one slower than the other, and is of the form… 
x = A_{1}exp(−γ_{1}t) + A_{2}exp(−γ_{2}t)  
Where…  
γ_{1} = γ + √(γ^{2} − ω_{0}^{2}) γ_{2} = γ − √(γ^{2} − ω_{0}^{2}) 

The motion approaches a steady state, but more slowly than in the critically damped case. 
Another common mechanical problem arises when a damped harmonic oscillator is driven by some timedependent external applied force — the driven harmonic oscillator. The most important case is that of a force that oscillates in a sinusoidal manner. If the driving force is of the form…
F(t) = F_{0}cos(ωt + φ_{0})
then the differential equation has an exact solution.
Given…
F_{0} =  maximum driving force 
φ_{0} =  driving phase 
ω =  driving frequency 
We get a bigger, more complicated second order differential equation.
F_{net} =  ma 
F(t) − bẋ − kx =  mẍ 
mẍ + bẋ + kx =  F_{0}cos(ωt + φ_{0}) 
Which has the solution…
x = Ae^{−γt}cos(ω_{1}t 
F_{0} / m  sin(ωt 
√[(ω_{0}^{2} − ω^{2})^{2} 
Where…
ω_{0} = √  k 
m 
γ =  b 
2m 
ω_{1} = √(ω_{0}^{2} − γ^{2})  
β = tan−^{1}  ⎡ ⎣ 
ω_{0}^{2} − ω^{2}  ⎤ ⎦ 
2γω 
The solution has two parts: transient and steady state. The transient portion, which has the same solution as the damped harmonic oscillator, dies out exponentially and depends on the initial conditions. The steady state portion has an amplitude that remains constant and does not depend on the initial conditions. Thus no mater what initial conditions the oscillator had, it will eventually acquire behavior that is wholly dependent upon the driving force.
The amplitude that the oscillator eventually acquires depends on the relation of the driving frequency to the natural frequency of the oscillator and on the damping factor. It is a maximum when…
(ω_{0}^{2} − ω^{2})^{2} + 4γ^{2}ω^{2}
is a minimum. This occurs when the ratio of the two frequencies is equal to…
√(1 + γ^{2}).
This condition is known as resonance and results in a large amplitude of oscillation. When the driving frequency equals the natural frequency, the amplitude of the steady state portion of the solution…
F_{0} / m 
√[(ω_{0}^{2} − ω^{2})^{2} + 4γ^{2}ω^{2}] 
reduces to…
F_{0} / m 
2γω 
As the damping factor approaches zero, the steady state amplitude approaches infinity. This illustrates the importance of damping in structures susceptible to vibration such as suspension bridges and steel framed buildings. Without damping, a structure could shake itself to pieces from a tiny external force with just the right frequency.
In summary, we have seen how a second order linear differential equation, the simple harmonic oscillator, can generate a variety of behaviors. In the damped harmonic oscillator we saw exponential decay to an equilibrium position with natural periodicity as a limiting case. The determining factor that described the system was the relation between the natural frequency and the damping factor. In the driven harmonic oscillator we saw transience leading to some steady state periodicity. The final behavior of the system depended on the relation between the driving frequency and the natural frequency (and to a lesser extent the damping factor). The behaviors described above are also found in first order nonlinear difference equations — the quadratic mapping and the related logistic equation. I will review the latter of these and present it in a manner similar to what has appeared so far.