Dynamical systems (Math 118) by Sternberg S. PDF

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5034 . . and plot the successive 199 iterates of Lµ applied to x we obtain the upper graph in Fig. 9. The lower graph gives x(j + 3) − x(j) for j = 1 to 197. 002 so that µ = 1 + 8 is the parameter giving the onset of period three. 1599288 . . 514355 . . 9563180 . .. ) Since the graph of P is tangent to the diagonal at the double roots, P (x) = 1 at these points, so the period three orbit is not strictly speaking stable. But using the same initial seed as above, we do get slow convergence to the period three orbit, as is indicated by Fig.

So we have verified our claim in all cases. Many interesting properties of a transformation are preserved under conjugation by a homeomorphism. ) For example, if p is a periodic point of period n of f , so that f ◦n (p) = p, then g ◦n (h(p)) = h ◦ f ◦n (p) = h(p) if h ◦ f = g ◦ h. So periodic points are carried into periodic points of the same period under a conjugacy. We will consider several other important properties of a transformation as we go along, and will prove that they are invariant under conjugacy.

449499... the points of period two become unstable and (stable) points of period four appear. ) and bifurcate into period eight points, initially stable. 7 Reprise. 55, is as follows: For µ < b1 := 1, there is no non-zero fixed point. Past the first bifurcation point, b1 = 1, the non-zero fixed point has appeared close to zero. 5 and is superattractive. As µ increases, the fixed point continues to move to the right. 5. 23606797... 5 is a period two point, and so the period two points √ are superattractive.

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Dynamical systems (Math 118) by Sternberg S.

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