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By A. M., Worthington

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Extra info for Dynamics of Rotation: An Elementary Introduction to Rigid Dynamics, 6th Edition

Example text

0, such that for all t > Trec . tI x/j1 < p 2 x2H for any t > 1 is proved. Note that this property implies that the polynomial nonlinearity becomes uniformly Lipschitz in finite time. 7. 4) has two stable stationary solutions, which we shall denote throughout by C and . 1 ˙ g; and the separatrix by S WD H n D C [ D : We use the symbol ˙ whenever we can choose simultaneously for all those symbols either C or . In this sense we define the reshifted domains by D0˙ WD D ˙ ˙ : Due to the Morse–Smale property the separatrix is a closed C 1 -manifold without boundary in H of codimension 1 separating D C from D .

13. t/ Yi ; ! H /. H /. / Yi . // ; t > 0; ! 14 (Itˆo isometry for simple processes). 12(2). H / is then the unique Hilbert–Schmidt operator such that Q 1=2 Q 1=2 DQ : Proof. s/k2 D E 4 1Ai Yi . ti ^ t// 5 i D0 2 D E4 n 1 X 1Ai 1Ak hYi . ti C1 ^ t/ i;kD0 3 Yk . tk ^ t//i5 3 1Ai 1Ak Ji kl 5 ; i;kD0 lD1 where Jikl WD E ŒhYi . ti ^ t// ; el ihYk .

6 ıN Proof. z/; z 2 H C , be a point on the graph of ˚u . tI z// D T . sI z// ds; t 6 0: 0 So that it is a solution of the truncated Chafee–Infante equation living on the graph of ˚u . This shows that the graph of ˚u is negatively invariant. 0 2 t kzk ! 0; t! 0 2 t kzk ! 0; t! 1: This shows that the graphs of ˚u is a subset of the unstable manifold. Claim 2. xu /k; so that the graph of ˚u is exponentially attracting. tI xu //. Proof. Let z 2 H C . tI x/. In the next calculation we omit the arguments for convenience.

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