Read e-book online Abstract convex analysis PDF

By Singer I.

ISBN-10: 0471160156

ISBN-13: 9780471160151

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Sample text

Using Eq. 5), we obtain -2f"'( -2) - 3[f"( -2)] 2 = -12 < 0. 15 then declares that the equilibrium point -2 is asymptotically stable. 10 illustrates the stair step diagram of the equation. Remark: One may generalize the result in the preceding example to a general quadratic map Q(x) = ax 2 + bx + c, a =1 0. 15. 1. [x\n)+x(n)]. 2. x(n+1)=x 2 (n)+~. 3. x(n + 1) = tan- 1 x(n). 4. x(n + 1) = x 2 (n). 5. x(n + 1) = x 3 (n) + x(n). B > 0. 30 I. Dynamics of First Order Difference Equations 7. x(n + 1) = -x 3(n)- x(n).

2, and (b) m, = 1, md = 2, bd = 14, and bs = 2. 9. 12) is a solution ofEq. 10). 10. 12) to show that (a) if -1

Let b be a k period point of f. Then b is (i) stable if it is a stable fixed point of fk, (ii) asymptotically stable (attracting) if it is an attracting fixed point of fk, (iii) repelling if it is a repelling fixed point of fk. Notice that if b possesses a stability property then so does every point in its k cycle {x(O) = b, x(l) = f(b), x(2) = f 2 (b), ... , x(k- 1) = fk- 1(b)}. Hence we often speak of the stability of a k cycle or a periodic orbit. 8 is not stable as a fixed point of T 2 , while the 2 cycle in the logistic map is asymptotically stable.

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Abstract convex analysis by Singer I.


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