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Unformatted text preview: 18.385j/2.036j MIT Hopf Bifurcations. Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts MA 02139 Abstract In two dimensions a Hopf bifurcation occurs as a Spiral Point switches from stable to unstable (or vice versa) and a periodic solution appears. There are, however, more details to the story than this : The fact that a critical point switches from stable to unstable spiral (or vice versa) alone does not guarantee that a periodic solution will arise, 1 though one almost always does. Here we will explore these questions in some detail, using the method of multiple scales to find precise conditions for a limit cycle to occur and to calculate its size. We will use a second order scalar equation to illustrate the situation, but the results and methods are quite general and easy to generalize to any number of dimensions and general dynamical systems. 1 Extra conditions have to be satisfied. For example, in the damped pendulum equation: x + µx ˙ +sin x = 0, ¨ there are no periodic solutions for µ = ! 1 18.385j/2.036j MIT Hopf Bifurcations . 2 Contents 1 Hopf bifurcation for second order scalar equations. 3 1.1 Reduction of general phase plane case to second order scalar. . . . . . . . . . 3 1.2 Equilibrium solution and linearization. . . . . . . . . . . . . . . . . . . . . . 3 1.3 Assumptions on the linear eigenvalues needed for a Hopf bifurcation. . . . . 4 1.4 Weakly Nonlinear things and expansion of the equation near equilibrium. . . 5 1.5 Explanation of the idea behind the calculation. . . . . . . . . . . . . . . . . 5 1.6 Calculation of the limit cycle size. . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 The Two Timing expansion up to O ( 3 ). . . . . . . . . . . . . . . . . . . . . 7 Calculation of the proper scaling for the slow time. . . . . . . . . . . 7 Resonances occur first at third order. Nondegeneracy. . . . . . . . . 8 Asymptotic equations at third order. . . . . . . . . . . . . . . . . . . 8 Supercritical and subcritical Hopf bifurcations. . . . . . . . . . . . . . 9 1.7.1 Remark on the situation at the critical bifurcation value. . . . . . . . 9 1.7.2 Remark on higher orders and two timing validity limits. . . . . . . . . 10 1.7.3 Remark on the problem when the nonlinearity is degenerate. . . . . . 10 18.385j/2.036j MIT Hopf Bifurcations . 3 1 Hopf bifurcation for second order scalar equations. 1.1 Reduction of general phase plane case to second order scalar. We will consider here equations of the form ¨ x + h ( ˙ x, x, µ ) = , (1.1) where h is a smooth and µ is a parameter. Note 1 There is not much loss of generality in studying an equation like (1.1), as opposed to a phase plane general system. For let: x ˙ = f ( x, y, µ ) and y ˙ = g ( x, y, µ ) . (1.2) Then we have ¨ x = f x x ˙ + f y y ˙ = f x f + f y g = F ( x, y, µ ) . (1.3) Now, from x ˙ = f ( x, y, µ ) we can, at least in principle, 2 write y = G ( ˙ x, x, µ ) . (1.4) Substituting then (1.4) into (1.3) we get...
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 Fall '04
 RodolfoR.Rosales
 Hopf bifurcation, Hopf bifurcations, MIT Hopf Bifurcations

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