By Peter J. Olver
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Extra info for Applications of Lie groups to differential equations MCde
It is known that f is a Li-Yorke chaos provided that h(f) > 0. The converse is not true, however, and the existence of a Li-Yorke chaos f with vanishing h(f) is known. 3): 'f has a period 2"(2rn + 1) orbit with n 2 0 and ni 2 1'. The necessary and sufficient condition for h(f) > 0 may take various different forms, one of which is concerned with the magnifying property of a map f : 'there exist disjoint closed intervals I and J and a natural number n such that f"(I) n f"(J) 2 I and f " ( Z ) n f " ( J ) 3 J ' .
14(a). 14(b). That is, the solution x = xk becomes unstable and there appear new stable solutions on both sides leading to a pitchfork bifurcation. What is important, therefore, is the inequality g"'(xk) c 0. What has been explained so far may be expressed mathematically as follows. The equation giving the fixed points for R > R, is where y = x - x k . 15) where R = R, + 6 R . Since ag;"(xrC) > aR in most cases, g"'(xk) < 0 implies D > 0 leading to two real solutions. Therefore the necessary condition for the infinite sequence of pitchfork birfurcations is gf(xd < 0 at R = R, (n = 1 , , 2, 3 , .
This covering of I is clearly finer than CY or #?. Take I = [0, 11 and open coverings ar = [ [0, 11) and #? = [ [ 0 , f ) , (f, 1])6 for example. Then CY v fi is the set of the intersections of the open intervals, given by [ [0, , ($, , ($, , :) , 11). However, the interval [0, 11 is covered i ) ,(a, i) i) 4) (i, (i, i), (i, ' Semi-open intervals such as [0, ft) or (i, I ] are employed when they contains the point 0 or 1 , since the outside of the interval [O. I] is not considered here. Top0log ical entropy 27 i) (i, with only three intervals, [0, f ) , ( f , and 13, and one obtains N ( a v f I ) = 3 as the smallest number of covering intervals.
Applications of Lie groups to differential equations MCde by Peter J. Olver