To understand chaos we must first understand a bit about linear and non-linear dynamic systems. Linear dynamics describes those systems in which forces increase linearly with parameters such as position and velocity. Perhaps the best known linear system is a mass oscillating on a spring, or the "simple harmonic oscillator." In this situation, force on the mass increases linearly with displacement by a factor of "k," the spring constant. A graph of the potential energy of this system is parabolic, since F(x) = - dU / dx. A particle of mass m oscillating in the potential well of this system will have an angular frequency of w = (k/m)^(1/2). The system becomes more complex when damping or a driving force are added. Damping alone will cause the particle to sit down in the potential well or "attractor." When the system is forced, the system will oscillate at one frequency determined by the relative strengths of the forcing and damping. If the forcing is too weak or too strong, the system may oscillate at the forcing frequency. In this case, the free movement of the system is essentially drowned by the forcing and damping and the amplitude of oscillation is weak. At some freqency, however, the system will resonate where maximum oscillating amplitude occurs. Again, in the linear system, this resonant frequency is unique and is determined by the forcing, damping, and natural freqencies of the system. In this driven and damped linear system, periodic inputs result in periodic outputs. If there is error in a measurement, this error will increase linearly as the system progresses.
In non-linear systems, however, error in measurement may increase exponentially as the system progresses. A system is non-linear because one or more "forcing elements" does not vary linearly with space parameters. For example, if the spring coefficient in the aforementioned spring system varied with displacement, then the spring force would vary with the square of displacement. Although linear systems make for pretty equations and a neat summary of behavior, non-linear systems seem to pervade real natural systems. Friction forces, damping elements, resistive elements in circuits-- these and many other factors often vary in a non-linear fashion. As a result, the differenial equations describing these systems involve very messy solutions that can only be solved numerically. Even if there were analytic solutions to these systems the behavior in some cases would be difficult or impossible to predict due to the exponential increase in error.
A non-linear system may oscillate at multiple amplitudes for a particular driving frequency, unlike the linear system which has one unique maximum resonant amplitude. Consider the so-called Duffing equation which describes a non-linear spring oscillator with spring constant k(x) = 1+ Bx˛. The solution to the equation yields a resonance curve that looks like this: *****
Notice that for certain values of the driving frequency, the system may be stuck oscillating at a few different amplitudes. The presence of these multiple oscillation amplitudes results from their being multiple non-parabolic potential wells in the system. The system may oscillate in one well or it may oscillate across two or more wells at that same frequency. We observe this effect in our system.
Our pendulum setup provides an inexpensive and understandable system for investigating chaos. The restoring force of the pendula is -mgSin(theta) where theta is the angle of displacement. For small angles, the Sin(theta) term may be approximated with simply "theta." For angles greater than 15° or so, this approximation is inaccurate and the system must be thought of as non-linear. This Sin term in our system is the only non-linear element. Consequently, a graph of the potential for our system is saddle-shaped, not parabolic:
This potential is a combination of the gravitational potential (which includes the sin(theta) term) and the spring potential. (The derivation of this potential is in the Math Modeling section) Near the bottom of each of the two wells, we might expect to see stable, harmonic motion since the well is nearly parabolic like a linear system. Indeed, the system will resonate at the bottom of the well at a particular frequency (call it F*). For a parabolic potential well, the period of oscillation is constant for any total system energy. For our pendulum's potential wells, however, the period of oscillation in the potential wells actually increases as the energy increases. We can't really say that the well has a unique resonant frequency because the period of oscillation in the well depends on the total energy of the system. If we drive the system at a frequency F* (resonant at the bottom of the well) and increase the system energy by increasing the forcing term, we will find that this driving frequency will be out of phase with the system's oscillations. This is one way of seeing how periodic inputs into a non-linear system MAY NOT produce periodic outputs.
This effect leads to chaos: the future of the system depends so much on the initial conditions that we can not know its distant future.
PHASE DIAGRAMS, graphing position and phase, clearly display important information about the history of a system. Because a linear system will resonate at one frequency, the phase diagram of a linear system will have one loop or possibly a spiral toward an attractor after it has progressed for some time. Loops in phase space must close, indicating that phase history repeats. Due to the deterministic nature of the system, NO line in phase space may cross at any time. If two lines were to cross, then this would imply that two systems with identical initial conditions travelled different paths through space-time. Thus, physical laws would not have determined the system.
Phase diagrams of chaotic systems should look like scribble, because the phase history NEVER repeats itself. Loops in phase space will never close and will never cross. Remember, chaos means unpredictable but not probabilistic. A chaotic system is determined by physical laws just as the simple-harmonic oscillator is determined.
POINCARÉ SECTIONS are simply snapshots of the phase diagram taken at certain intervals. Usually, we take points out of the phase diagram once every driving cycle. Poincaré section for non-chaotic systems will have and infinite number of points because phase history repeats. Because the history of the chaotic system never repeats, phase diagrams for chaotic systems must contain an infinite number of curves and Poincaré sections should have an infinite number of points. In fact, chaotic systems yield Poincaré sections with fascinating mathematical patterns with fractal dimension. More on this in the Math Modeling section.