The Hopf Bifurcation and Its Applications. by J. E. Marsden, M. McCracken

By J. E. Marsden, M. McCracken

The aim of those notes is to provide a reasonahly com­ plete, even though now not exhaustive, dialogue of what's as a rule often called the Hopf bifurcation with functions to spe­ cific difficulties, together with balance calculations. historic­ ly, the topic had its origins within the works of Poincare [1] round 1892 and was once broadly mentioned by means of Andronov and Witt [1] and their co-workers beginning round 1930. Hopf's simple paper [1] seemed in 1942. even if the time period "Poincare­ Andronov-Hopf bifurcation" is extra actual (sometimes Friedrichs is additionally included), the identify "Hopf Bifurcation" turns out extra universal, so we've got used it. Hopf's the most important contribution was once the extension from dimensions to raised dimensions. The relevant procedure hired within the physique of the textual content is that of invariant manifolds. the strategy of Ruelle­ Takens [1] is undefined, with info, examples and proofs additional. a number of components of the exposition by and large textual content come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we're thankful. the overall approach to invariant manifolds is usual in dynamical structures and in traditional differential equations: see for instance, Hale [1,2] and Hartman [1]. in fact, different tools also are on hand. In an try to preserve the image balanced, we've got incorporated samples of other ways. particularly, we have now integrated a translation (by L. Howard and N. Kopell) of Hopf's unique (and in most cases unavailable) paper.

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The Center Manifold Theorem is not Remark. 12) for all ~n(x) ~ M. 11) Ft of has dimension E M. (b) n V Y aO' such that: (a) ~(x) denote the belonging to the part of the spectrum on the unit circle; assume that d < Y ~ in the following sense: since k, we get a sequence of center manifolds but their intersection may be empty. 6 ~, THE HOPF BIFURCATION AND ITS APPLICATIONS 20 regarding the differentiability of M. We will be particularly interested in the case in which bifurcation to stable closed orbits occurs.

10) (xu (y) = X*(u(y) ,y». 12) < 1, we get 11D%(y) II < 1 for all y. We shall carry the estimates just one step further. 13) we have IID2~(y)11 <1 for all y. At this point it should be plausible by imposing a sequence of stronger and stronger conditions on I I Djy(y) II < 1 for all y, Y,A, that we can arrange j = 3,4, •.. ,k+1. * * THE HOPF BIFURCATION AND ITS APPLICATIONS 37 The verification that this is in fact possible is left to the reader. e. u = 0, Du yeo) o 0) IL%. (0) o D% (assuming we note that since 0 is a solution of AU(O) + X(u(O) ,0) = 0 ~u(O) 0, By + Y(u(y) ,y) 0 and [A Du(O) + D1X(0,0)Du(0) + D2X(0,0)] ·Dy(O) [A· 0 + 0 + 0]· Dy (0) = O.

Keep the notation and assumptions of the A is SUfficiently small (and if center manifold theorem. If liB-II I is close enough to one), there exists a function defined and k u*, Y, times continuously differentiable on all of with a second-order zero at the origin, such that a) invariant for b) lim n+ oo I Ix n ru* The manifold "xii If As with n u*(y),y E Y} {(x,y) Ix is in the strict sense. 0/ - u * (y = ) II = I IB-li < 0 I, 1, and y is arbitrary then (where (xn'Yn) = we shall treat 0/ n (x,y)).

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