Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Chaos theory is an often-misunderstood field of mathematics. Many people associate chaos mathematics with the famous "butterfly flapping its wings in China and causing a tornado in Texas" metaphor. This example is well-meaning in that it shows the dependence of large, complicated systems on small changes in initial circumstances. This metaphor is not terribly illuminating regarding chaos theory, however, because the earth's atmosphere is immensely complicated, with many variables, and it is not too surprising that it behaves in strange ways. Mathematical chaos is most remarkable not because it arises in huge, complicated systems, such as that connected with our planet's weather, but rather because it appears to be a governing factor even in simple systems, systems that one would think should be fairly predictable but that instead turn out to be chaotic. So in order to observe and study chaos, we do not need a large, complicated system; our only requirement is that our system be nonlinear.
In high school, we learned that a linear equation is any expression of the form y = mx + b, with m and b representing constants (such as 3 and -7) and x and y representing variables, generally called the independent and dependent variables, respectively. The equation is "linear" because its graph (all the "x,y" points on the coordinate plane that satisfy the equation) is a straight line, and also because a small change in the value of x effects a proportional, constant change in y. A nonlinear equation is something that doesn't have just a first power of the independent variable and consequently can't be graphed as a simple straight line. One such example is a quadratic equation, ax^{2} + bx + c = 0.
In our study of chaos, we will need to expand the definitions of linear and nonlinear to include differential equations. Recall from our discussion in the preceding chapter on spontaneous synchronization that a differential equation is an equation that contains both variables and derivatives, or instantaneous rates of change. A linear differential equation is an equation in which dependent variables and their derivatives appear only to the first power. For example:
This equation is linear because y, the dependent variable (it depends on t), occurs only to the first power, as does its derivative. This differential equation is also linear:
Note that this equation contains a second derivative of the dependent variable, but only to the first power. The following equation also is linear:
Although this equation involves higher powers, they apply only to t, which is the independent variable. The dependent variable, y, and its derivative both appear only to the first power, which is what determines whether or not a differential equation is linear.
Consider this differential equation:
This equation contains a derivative raised to the second power, so it is classified as nonlinear. An equation, containing derivatives or not, can be nonlinear in other ways besides containing powers greater than one of dependent variables or derivatives. For example, the following equations are nonlinear:
A linear system, then, is a set of equations that express a certain physical situation without involving terms that include a dependent variable or the derivatives of that variable to a power greater than one. A nonlinear system is like a linear one, except that one or more terms are nonlinear.
The distinction between linear and nonlinear systems in mathematics defines the boundary between the relatively knowable, and the frustratingly elusive. Both types of systems can describe the dynamics of many different processes, such as planets orbiting each other, fluctuations in animal populations, the behavior of electrical circuits, and so on. The difference between linear and nonlinear lies in the details of the equations that govern how these systems interact. For systems that behave linearly, it is relatively easy to find exact solutions that we can use to predict future behavior within the system. For nonlinear systems, we are lucky to find any such solution. Indeed, in nonlinear dynamics, we often have to redefine what we consider to be a solution. Before we get to this new view of solutions, however, let's take a closer look at the older, linear view.
If we attached a weight of mass m to the free end of a spring of strength k that is suspended vertically from a board or the ceiling and allowed the mass to bounce up and down, we would have what is known as a harmonic oscillator. Given an initial displacement (either lifting the mass above or pulling the mass below its resting position), the weight would bounce up and down until the friction of the air, the inelasticity in the spring, and the force of gravity combine to slow the oscillations to a stop. The position of the mass is a dynamical system and is easily defined with this well-known differential equation:
This equation represents the balance of forces acting upon the mass. We know that, given time, the mass will return to rest at its original position; in other words, the forces acting to cause the oscillations must balance out to a zero sum. The first term in the equation comes from Newton's second law of motion F = ma (force equals mass times acceleration). In our equation, the mass is represented by m and the acceleration is represented by . The second term is the product of the velocity of the mass, ,and some constant, b, that represents the effect of air resistance. The final term represents the force contributed by the contraction of the spring. This contribution is proportional to how far the spring has been stretched—the more the stretching, the greater the contribution. To find this contribution, we simply multiply the strength of the spring, k, by the amount by which it is stretched, x. We add all these contributions together and set them equal to zero in accordance with Newton's third law of motion, which states that every action has an equal and opposite reaction.
As this equation is written above, it incorporates both first and second derivatives, making it somewhat difficult to solve directly. We can transform the equation to one without a second derivative and, hence, one more easily solved by performing a change of variables. To do this, we must first recognize that is just the first derivative of . If we let x = x_{1}, and = x_{2}, then becomes . With the second derivative conveniently eliminated, we can now write a system of equations to model our oscillator:
We can rewrite this as:
This is a linear system because all of its terms are single, first-degree variables with constant coefficients. We need not work through the details of the solution to this system. It is important to realize, however, that it would be some function x(t) that describes where the mass would be at any time, t, that we choose. The solution is an equation that can be used to determine the exact location of the mass at any time during its oscillation.
Because this system is linear, we could use the principle of superposition to solve it. This principle enables us to break a system of equations into pieces that are more easily solved, solve them, and then combine the partial solutions to find a solution of the entire system. This is a case of the whole solution being exactly the sum of the partial solutions. Because of the applicability of this principle of superposition, it is relatively easy to get exact, predictive solutions for linear systems.
One nice thing about linear systems is that, because they are exactly solvable, we can categorize the types of behavior that they can exhibit. When we refer to the behavior of a system, what we are really concerned with is the behavior of the variables that describe the state of the system. For our oscillating spring, the pertinent variables are the position of the mass, x, and its velocity, . Given these two values, we know exactly what the system is doing at any moment. In general, the variables that describe the state of linear systems can:
All four of these behaviors are nice and predictable in the linear view. Unfortunately, most real-life systems are not so well behaved and do not fit well into a linear model.
Let's look at a slightly different type of oscillator, a pendulum. This is a very common nonlinear system. To make things easier on ourselves, let's say our pendulum is just a mass, m, at the end of a string (considered to have no mass) of length, L, moving under the acceleration due to gravity, g. Such a pendulum exists only in the mind of a physicist; the arm of a real pendulum has mass and is affected by air resistance, even when it is only a string or thread. However, this simplified, ideal model is good for our present purposes.
The force on the pendulum mass is a balance of the tension in the string and the acceleration due to gravity. These forces vary, depending on the angle of the pendulum. For instance, at the bottom of the swing, gravity is completely mitigated by the tension in the string. At the top of the swing, the tension in the string acts in the same direction as gravity. To model these varying forces, we need a sinusoidal function.
The acceleration in terms of the angle the pendulum makes with the vertical is then given by:
The sine term of the dependent variable makes this a nonlinear equation. To solve this, we can make our lives easier, as we did before in the example using a spring, by performing a change of variables. To do this, we let θ = θ_{1}, and θ_{2} = . then becomes . Our system then becomes:
This eliminated the second derivative, but the sine term is still there, so this system remains nonlinear.
These so-called nonlinear systems can exhibit some wild behaviors, behaviors that might be considered surprising, behaviors that don't fit so nicely into equations. For example, our simple pendulum behaves very smoothly and predictably as long as it doesn't swing too high.
For larger and larger angles, the range of possible behaviors is more varied than the simple cycling back and forth. For example, if the pendulum has sufficient momentum, it will swing past the horizontal line of the pivot and go all the way around, over the top. If it has a little less momentum than this, it might stall near the vertical position above the pivot, lose the tension of the string, and drop almost straight down under the influence of gravity. Both of these behaviors are examples of nonlinearities. It's worth noting that for a pendulum to swing higher than its pivot, the mass must have some initial velocity. Velocity due to gravity alone will not suffice. Since we are only concerned with general methods and qualitative behavior, we can ignore this.
Some nonlinear systems do behave nicely and predictably, while others do not. The range of nonlinear behaviors is vast, with chaos being just one type. It's the type that we understand the best. As we will see in the next section, our understanding of chaos does not mean that we can make exact predictions in a chaotic system, as we can with linear systems. In fact, to go any further in our exploration of chaos we will have to redefine what we even mean by the term "solution."
Next: 13.3 Limits of Predictability