By W J Cunningham
Vintage 1950/1960 period , good written textual content. a bit dated in that a few of the numerical & graphical strategies performed 'by hand' in these days at the moment are automated. worthy examining if only for the numerous reliable examples.
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These values exist when there is no further change, so that dx/dt = 0 and a steady state exists. This construction is commonly used in studying electronic circuits. The time-varying part of the solution is found in step-by-step fashion. At some point in the course of the solution, variable x has the value x('), GRAPHICAL METHODS 51 and the succeeding increment in x is Ax(') occurring in time At. The equation relating these quantities is, from Eq. 28) Here the derivative is replaced by the ratio of finite increments, and the second and third terms are evaluated at the center of the increment in x.
10 indicate the junction of successive arcs used in obtaining the complete solution curve. The effect of the nonlinear term in the differential equation of this example is to make the phase-plane curve have a somewhat elliptical shape. A linear oscillator, described by Eq. 5), leads to the circular phase-plane curve of Fig. 3. 5. Nonoscillatory Problem While the S method is most immediately applicable to equations with oscillatory solutions, it applies to other types of equations as well. The equation of this example has a solution which does not oscillate.
The latter waveform, characterized by skewed flattened peaks and abrupt transitions, is known as a relaxation oscilla- INTRODUCTION TO NONLINEAR ANALYSIS 48 tion. For e small, the period in T units is not far from 27r, its value for e = 0. large, the period increases continuously as a is increased. For e It is worth noting that, if a is very small, the limit cycles for both the Rayleigh equation and the van der Pol equation are almost circular in shape. If a is made larger, the limit cycle for the Rayleigh equation has the simpler shape.
Introduction to Nonlinear Analysis by W J Cunningham