By Florent.

ISBN-10: 2711720365

ISBN-13: 9782711720361

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**Example text**

4. an analogous l There ~s emma'~ f o r < . a( A,) . 3) operator. Theorem. Let Let T: E a(T) C {zlRe z < O} + E be a bounded linear (resp. a(T) ~ {zlRe z > O}), then the origin is an attracting (resp. repelling) fixed point for the flow ¢t = e tT of T. TION AND ITS APPLICATIONS Proof. 5 for if a}, is compact. there is an Thus r < e rt letAI with < 0 0 ->- cr(T) as t ->- < r, as cr(T):J cr(T) +"". 4 from Section l. 4) Let Theorem. P Banach manifold X and be such that C {zEclRez < O}, cr(dX(PO» cl be a then vector field on a X(PO) = O.

Denote striction of D (0) to D~(O) (I-P)Z, D~(O) Y. Then to Z X = Y denote and X $ Y B A denote the res- denote the restriction of and (Ax+X*(x,y), ~(x,y) PZ, By + Y*(x,y)), where A is bounded linear operator on X with spectral radius strictly less than one. B is a bounded operator on the unit circle. e. DX(O,O) with spectrum on (All we actually need is that the spectral radius of X* is a Y 0, and X with a second-order X(O,O) = ° and mapping of a neighborhood of the origin into Y with a second-order zero at the THE HOPF BIFURCATION AND ITS APPLICATIONS 30 origin.

D(x,y) dt Exercise. (nonuniqueness of the center manifold X (x,y) = (-x,y 2 ). Let = X(x,y) Solve the equation and draw the integral curves. 7 with y X satisfies the conditions of the y-axis. center manifold for the flow. Show that the y-axis is a Show that each integral curve in the lower half plane goes toward the origin as that the curve becomes parallel to the y-axis as t t ->- ~ and 00 Show -00. that the curve which is the union of the positive y-axis with any integral curve in the lower half plane is a center manifold x.

### Algebre lineaire by Florent.

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