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Unformatted text preview: 18.385j/2.036j, MIT Weakly Nonlinear Things: Oscillators. Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts MA 02139 Abstract When nonlinearities are “small” there are various ways one can exploit this fact — and the fact that the linearized problem can be solved exactly 1 — to produce useful approximations to the solutions. We illustrate two of these techniques here , with examples from phase plane analysis: The Poincar´ e–Lindstedt method and the (more ﬂexible) Two Timing method. This second method is a particular case of the Multiple Scales approxima tion technique, which is useful whenever the solution of a problem involves effects that occur on very different scales. In the particular examples we consider, the different scales arise from the basic vibration frequency induced by the linear terms (fast scale) and from the (slow) scale over which the small nonlinear effects accumulate. The material in these notes is intended to amplify the topics covered in section 7.6 and problems 7.6.13–7.6.22 of the book “Nonlinear Dynamics and Chaos” by S. Strogatz. 1 Actually, one can also use these ideas when one has a nonlinear problem with known solution, and wishes to solve a slightly different one. But we will not talk about this here. 1 18.385j/2.036j, MIT Weakly Nonlinear Things: Oscillators . 2 Contents 1 Poincar´ eLindstedt Method (PLM). 3 General ideas behind the method. 1.1 Duﬃng Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Periodic solutions and amplitude dependence of their periods. 1.2 van der Pol equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Calculation of the limit cycle. 2 Two Timing, Multiple Scales method (TTMS) for the van der Pol equation. 8 2.1 Calculation of the limit cycle and stability. . . . . . . . . . . . . . . . . . . . 8 2.2 Higher orders and limitations of TTMS. . . . . . . . . . . . . . . . . . . . . 11 This topic is fairly technical. † 2.3 Generalization of TTMS to extend the range of validity. . . . . . . . . . . . . 14 This topic is fairly technical. † A Appendix. 16 A.1 Some details regarding section 1.1. . . . . . . . . . . . . . . . . . . . . . . . 16 A.2 More details regarding section 1.1. . . . . . . . . . . . . . . . . . . . . . . . . 17 This topic is fairly technical. † A.3 Some details regarding section 1.2. . . . . . . . . . . . . . . . . . . . . . . . 17 † The material here is for completeness, but not actually needed to get a ”basic” understanding. 3 18.385j/2.036j, MIT Weakly Nonlinear Things: Oscillators . 1 Poincar´ eLindstedt Method (PLM). PLM is a technique for calculating periodic solutions . The idea is that , if the linearized equations have periodic solutions and < 1 is a measure of the size of the nonlinear terms then: I . For any finite time period t t ≤ t + T f ( T f > 0), the trajectories for the full ≤ system will remain pretty close to those of the linearized...
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.036J taught by Professor Rodolfor.rosales during the Fall '04 term at MIT.
 Fall '04
 RodolfoR.Rosales

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