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trky_asmp_fxdpt - 18.385j/2.036j MIT Tricky Asymptotics...

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Tricky Asymptotics Fixed Point Notes. 18.385j/2.036j , MIT. Contents 1 Introduction. 2 2 Qualitative analysis. 2 3 Quantitative analysis, and failure for n = 2 . 6 4 Resolution of the diﬃculty in the case n = 2 . 9 5 Exact solution of the orbit equation. 14 6 Commented Bibliography. 15 List of Figures 1.1 Phase plane portrait for the Dipole Fixed Point system ( n = 1.) . . . . . . . . . . . 3 3.1 Phase plane portrait for the Dipole Fixed Point system ( n = 5.) . . . . . . . . . . . 10 Abstract In this notes we analyze an example of a linearly degenerate critical point, illustrating some of the standard techniques one must use when dealing with nonlinear systems near a critical point. For a particular value of a parameter, these techniques fail and we show how to get around them. For ODE’s the situations where standard approximations fail are reasonably well understood, but this is not the case for more general systems. Thus we do the exposition here trying to emphasize generic ideas and techniques, useful beyond the context of ODE’s. MIT, Department of Mathematics, Cambridge, MA 02139. 1

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Tricky asymptotics fixed point. Notes: 18.385j/2.036j ,MIT. Fall 2004 . 2 1 Introduction. Here we consider some subtle issues that arise while analyzing the behavior of the orbits near the (single, thus isolated) critical point at the origin of the Dipole Fixed Point system (see problem 6.1.9 in Strogatz book) dx 2 dy 2 = xy , and = y 2 x , (1.1) dt n dt where 0 < n 2 is a constant. Our objective is to illustrate how one can analyze the behavior of the orbits near this linearly degenerate critical point and arrive at a qualitatively 1 correct description of the phase portrait. We will use for this “standard” asymptotic analysis techniques. The case n = 2 is of particular interest, because then the standard techniques fail , and some extra tricks are needed to make things work. Just so we know what we are dealing with, a computer made phase portrait for the system 2 (case n = 1) is shown in figure 1.1. Other values of 0 < n 2 give qualitatively similar pictures. However, for n > 2 there is a qualitative change in the picture. We will not deal with the case n > 2 here, but the analysis will show how it is that things change then. The threshold between the two behaviors is precisely the tricky case where “standard” asymptotic analysis techniques do not work. 2 Qualitative analysis. We begin by searching for invariant curves, symmetries, nullclines, and general “orbit shape” prop- erties for the system in (1.1). A. Symmetries. The equations in (1.1) are invariant under the transformations: 3 A1 and A2 show that we need only study the behav- A1. x −→ x ior of the equation in the quadrant x 0 , y 0 . A2. y −→ y and t −→ t A3. x −→ ax , y −→ at , and t −→ t/a , for any constant a > 0. 1 With quantitative extra information.
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