LABORATORY

The discussion of oscillators in Chapter 3 was limited to linear systems. When pressed to divulge greater detail, however, nature insists of being nonlinear; examples are the flapping of a flag in the wind, the dripping of a leaky water faucet, and the oscillations of a double pendulum. The techniques learned thus far for linear systems may not be useful for nonlinear systems, but a large number of techniques have been developed for nonlinear systems, some of which we address in this chapter. We use numerical techniques to solve some of the nonlinear equations in this chapter.
The equation of motion for the damped and driven oscillator of Chapter 3 moving in only one dimension can be written as
(식 표현이 안되는군요.ㅠㅠ)(4.1)
If or contains powers of or , respectively, higher than linear, then the physical system is nonlinear. Complete solutions are not always available for Equation 4.1, and sometimes special treatment is need to solve such equations. For example, we can learn much about a physical system by considering the deviation of the force from linearity and by examining phase diagrams. Such a system is the simple plane pendulum, a system that is linear only when small oscillations are assumed.
In the beginning of the nineteenth century, the famous French mathematician Pierre Simon de Laplace espoused the view that if we knew the position and velocities of all the particle in the universe, then we would know the future for all time. This is the deterministic view of nature. In recent years, researchers in many disciplines have come to realize that knowing the laws of nature is no enough. Much of nature seems to be chaotic. In this case, we refer to deterministic chaos, as opposed to randomness, to be the motion of system whose time evolution has a sensitive dependence on initial conditions. The deterministic development refers to the way a system develops from the moment to the next, where the present system depends on the one just past in a well-determined way though physical laws. We are not referring to a random process in which the present system has no casual connection to the previous one (e.q., the flipping of a coin).
Measurements made on the state of a system at a given time may not allow us to predict the future situation even moderately far ahead, despite the fact that the governing equations are known exactly, Deterministic chaos is always associated with a nonlinear system; nonlinearity is a necessary condition for chaos but not a sufficient one. Chaos occurs when a system depends in a sensitive way on its previous state. Even a tiny effect, such as a butterfly flying nearby, may be enough to vary the conditions such that the future is entirely different than what it might have been, not just a tiny bit different. The advent of computers has allowed chaos to be studied because we now have the capability of performing calculations of the time evolution of the properties of a system that includes these tiny variations in the initial conditions. Chaotic systems can only be solved numerically, and there are no simple, general ways to predict when a system will exhibit chaos.
Chaotic phenomena have been uncovered in practically all areas of science and engineering - in irregular heartbreaks; the motion of planet in our solar system; water dripping from a tap; electrical circuits; weather pattern; epidemics; changing populations of insects, birds, and animals; and the motion of electrons in atoms. the list goes on and one. Henry Poincare is generally given credit for first recognizing the existence of chaos during his investigation of celestial mechanics at the end of the nineteenth century. He came to the realization that the motion of apparently simple systems, such as the planets in our solar system, can be extremely complicated. Although various investigators also eventually came to understand the existence of chaos, tremendous breakthroughs did not happen until the 1970s, when computers were readily available to calculate the long-time histories required to document the behavior.
The study of chaos has become widespread, and we will only be able to look at the rudimentary aspects of the phenomena. Specialized textbooks on the subject have become abundant for those desiring future study. For example, space does not permit us to discuss the fascinating area of fractals, the complicated patterns that arise from chaotic processes.