By George D. Birkhoff
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Additional resources for Dynamical Systems
S? ¢o ' ' ' oi,7 ' ' ' oI. o~ o. ' oI~o ' / ! ~ ' o!. Time maps n i n T 1 : 1 . 8 7 7 9 0 6 7 2 ' oi~ ' oI. ' nax T 1 : 4 . 1 oI. ' o'. ~, 25 26 I. Dirichlet Branches Bifurcating from Zero (iv) f ( u ) = l n ( a u + ~ ) with a > 0 is a C-function on ] - f l / o q o c [ . 1 shows the time maps of f with fl = 1. 4 since f " < 0. 2 PROPOSITION. Let f : R -+ R be a p o l y n o m i a l o f degree n > 2 w i t h a11 zeroes of f being reM. T h e n f is a B-£unction on ~ , an A - f u n c t i o n on all intervals where f ' ~ 0 and an A - B - f u n c t i o n on a11 intervals which do not contain any m u l t i p l e zeroes of f .
_. __----'-------i -:: "~-~I .... I01655 ! 3 thus d/du f(u)/u < 0 on the interval where f' > 0. (iii) There are of course many examples of functions f satisfying (1-3-7). 3 shows the time maps. In this example it looks like Ti(p) goes to 0 as p ~ + , - c ¢ . 6 for how asymptotic properties of time maps like this one can be proved. (iv) In a way the other extreme to example (ii) are all polynomials f having only purely imaginary zeroes except for f ( 0 ) = 0, f'(0) > 0. Those f have a representation n i=1 with a > 0 , n > 2 and ai ~ O.
Neumann Problems, Period Maps and Semilinear Dirichlet Problems The inverted graphs (iT(a), a) thus give the branches of (2-1-1) projected to the (A, u(O)) -plane. By a little observation we are able to apply the results for Dirichlet-time-maps to the Neumann case: First of all we could as well assume that r = O, for via fi := u - r we transform (2-1-3) to ~" + / ( ~ ) (2-1-6) 0 = ~'(0) = 0, ~(0) = a - with /(u) := f(,~ + r). So ] satisfies (1-0-1) on ] 5 - , ~ + [ := ] a - - r , a + - r [ . zeroes, thus (2-1-7) T(a) = ¢(a - ~t' and u' have the same r), if T is the time map of / .
Dynamical Systems by George D. Birkhoff
Categories: Differential Equations