The Method of Multiple Scales for Ordinary Differential Equations
Various physical problems are characterized by the presence of a small disturbance which, because it is active over a long time, has a non-negligible cumulative effect. For example, the effect of a small damping force over many periods of oscillation is to produce a decay in the amplitude of an oscillator. A more interesting example having the same physical and mathematical features is that of the motion of a satellite around Earth. Here the dominant force is a spherically symmetric gravitational field. If this were the only force acting on the satellite, the motion would be periodic (for sufficiently low energies). The presence of a thin atmosphere, a slightly nonspherical Earth, a small moon, a distant sun, and so on, all produce small but cumulative effects which, after a sufficient number of orbits, drastically alter the nature of the motion.
KeywordsOrdinary Differential Equation Periodic Solution Multiple Scale Nonlinear Oscillator Scale Expansion
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