Applications of Centre Manifold Theory by Jack Carr (auth.)

By Jack Carr (auth.)

These notes are according to a sequence of lectures given within the Lefschetz middle for Dynamical structures within the department of utilized arithmetic at Brown collage through the educational yr 1978-79. the aim of the lectures used to be to provide an advent to the purposes of centre manifold conception to differential equations. lots of the fabric is gifted in a casual model, through labored examples within the wish that this clarifies using centre manifold concept. the most software of centre manifold thought given in those notes is to dynamic bifurcation idea. Dynamic bifurcation conception is worried with topological alterations within the nature of the recommendations of differential equations as para­ meters are different. Such an instance is the production of periodic orbits from an equilibrium element as a parameter crosses a serious worth. In sure situations, the appliance of centre manifold concept reduces the measurement of the method lower than research. during this appreciate the centre manifold idea performs a similar position for dynamic difficulties because the Liapunov-Schmitt process performs for the research of static options. Our use of centre manifold concept in bifurcation difficulties follows that of Ruelle and Takens [57) and of Marsden and McCracken [51).

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5) f 3 (w,y,z,E) where fl (W,y,Z,E) • -~w + N(w+xOY+Z'y) 6 -1 0 -1 f 2 (w,y,z,c) • (! ~ xo - 1 - Y1 b O)y + (! ~ - Y2)Z + f 3 (w,y,z,c) • -Yl bOY - YlYz N(e,y) • _~-2e3 + 3~-lxoe2 - yeo In order to apply centre manifold theory we change the time scale by setting respect to s by t = ES. 5) can now be written in the form by w' • g(W,y,Z,E) y' Z• E' Suppose that ~ > O. 6) · o. 6) has one negative eigenvalue and three zero eigenvalues. 6) has a centre mani- w· h(y,z,E). 6) is determined by the equation 3.

V be a neighborhood of the origin in mm and Let let F~: V +mm ~ € mq with be a smooth map depending on a parameter F~(O) • 0 for all ~. If bifurcation is to take place then the linearized problem must be critical. assume that with FO FO has simple complex eigenvalues We A(O). reO) IA(O)I - 1. A(O) ; tl. while all other eigenvalues of are inside the unit circle. By centre manifold theory all bifurcation phenomena take place on a two-dimensional manifold so we can assume m· 2 without loss of generality.

EXAMPLES R(t,a) ~ laK- l I 1 / 2 + O(lal) e(t,a) • wot + O(laI 1/ 2) T(a)· (2n/w O) + O(lal l / 2). with period K < 0 and unstable if tion is stable if The periodic solu- K > Finally, we note that since the value of only on the nonlinear terms evaluated at O. i; ;';"i',;nds a· 0, when apply - ing the above theorem to (3 . 3. 0 non-zero. Hopf Bifurcation in a Singular Perturbation Problem In this section we study a singular perturbation prob- lem which arises from a mathematical model of the immune response to antigen (52).

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