# Discussion of Oscillation and Resonance in Linear and Non-Linear Systems

#### Linear Systems:

A linear oscillatory system has a restoring force that varies linearly with position.  Such a system obeys Hooke's Law,  F(x) = k*x, where k is some constant.  However, in almost all natural oscillatory systems there is always some damping involved and thus the oscillatory system loses energy to friction or dispersion.  The damping of oscillations resulting from friction can be counteracted if some mechanism supplies the system with energy from an external source at a rate equal to that absorbed by the damping medium.  This counteracting mechanism is usually referred to as driven or forced oscillation.  Resonance occurs when the amplitude of a driven and damped oscillatory system is at a maximum.  This resonant frequency is a function of the driving frequency, w0, and the damping coefficient, B, of the system.  The resonant frequency in a driven and damped oscillatory system is described by the following equation: wr = (w02+2B2)1/2

 Also it is customary to describe the degree of damping in an oscillating system in terms of a "quality factor" Q of the system.  The "quality factor" is described by the following equation: Q = wr / 2B.  If only a very little damping occurs in the system then the Q value is very large and the amplitude of oscillation at resonance is illustrated by a sharply defined peak.  If the damping is larger, then the Q value is much smaller so that the amplitude of oscillation at resonance is much lower and described by more of a flattened curve.  The actual values of Q in real physical situations vary greatly.  Tuning forks and quartz crystal oscillations may have Q values of about 104.  Since the classical understanding of atomic radiation essentially just involves the concept of linear oscillation, we may also define Q values for some atomic systems.  According to the classical picture of an atom, the oscillation of electrons within atoms leads to radiation.  The sharpness of emitted spectral lines is then limited by the damping due to the loss of energy by radiation.  The Q value of such an atomic oscillator can be calculated to be approximately 107.  We find resonance with the highest Q values to occur in the radiation from gas lasers, such as Helium-Neon and Carbon-Dioxide lasers, which yields Q values of about 1014.
 Linear oscillations apply to a wide variety of physical systems such as pendulums, vibrating membranes and plates, masses on springs, and electrical circuits.  Atomic systems can also be represented classically as linear oscillators.  When light, which consists of high frequency electromagnetic radiation, falls on matter, it causes the atoms and molecules to vibrate.  When some of the incident light has one of the resonant frequencies of the atomic or molecular systems, the energy is absorbed and the atoms and molecules vibrate at maximum amplitudes.  In this fashion quantum mechanics uses linear oscillatory theory to explain many of the phenomena associated with light absorption, dispersion, and radiation.  It is often especially demonstrative to describe the dynamics of atomic systems with the classical mechanical analog.  This paper will examine the effects of resonance in three linear oscillatory systems: the mechanical system of vibrating plates and the atomic system of the Helium-Neon laser and the Carbon-Dioxide laser.

#### Non-Linear Systems:

Many real physical systems are actually non-linear in nature.  For example, the dripping of a leaky water faucet, the flapping of a flag in the wind, and the oscillation of a double pendulum are all non-linear systems.  In a non-linear oscillatory system the driving force does not vary linearly with position and therefore solutions to the non-linear equations that describe the system must be solved using numerical techniques.  An example of a non-linear driving force of an oscillatory system is: F(x) = -k*x + b*x3

 When the equation of motion for a driven and damped non-linear oscillator is solved there is not always a unique solution.  That is, for some values of the driving frequency, many different amplitudes of oscillation are possible.  Such a result is perplexing and seemingly contrary to the deterministic view of nature.  This behavior in non-linear systems however is not random or purely chaotic.  Such behavior in non-linear systems is referred to as deterministic chaos because though the conditions of the present system are impossible to predict they are still determined by the past conditions of the system.  Thus deterministic chaos is the motion of a non-linear system whose time evolution has a very sensitive dependence on initial conditions.   While the chaotic behavior is determined by earlier conditions, we are unable to predict the future conditions of the system because of the complex non-linear equations involved.  It is important to note that while non-linearity is a necessary condition for chaos it is not wholly sufficient to produce chaotic behavior.  Chaos occurs when a system depends in a very sensitive way on its previous state, to the meteorological extent that if a butterfly flapped its wings the weather would be entirely altered on the other side of the world.

Chaotic behavior has been uncovered in many areas of science including atomic physics, biology and meteorology.  In our discussion we will focus on the non-linear ideas that at resonance there is not just one amplitude possible and that a periodic driving force does not always yield a periodic output.  We will address these ideas in the study of a pendulum and an RLC circuit and also comment on the presence of chaotic behavior in these systems.