Turbulence Seminar Berkeley 1976/77
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Voi. 431: Sâminaire Bourbaki -voi. 1973/74. Exposes 436-452. IV, 347 pages. 1975.
Voi. 401: M. F. Atiyah, Elliptic Operators and Compact Groups. V, 93 pages. 1974.
Voi. 432: R. P. Pflug, Holomorphiegebiete, pseudokonvexe Gebiete und das Levi·Problem. VI, 210 Seiten. 1975.
Voi. 402: M. Waldschmidt, Nombres Transcendants. VIII, 277 · pages. 1974.
Voi. 433: W. G. Faris, Self-Adjoint Operators. VII, 115 pages. 1975.
Voi. 403: Combinatorial Mathematics. Proceedings 1972. Edited by O. A. Holton. VIII, 148 pages. 1974.
Voi. 434: P. Brenner, V. Thomee, and L. B. Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problema. 11, 154 pages. 1975.
Voi. 404: Theorie du Potential el Analyse Harmonique. Edite par J. Faraut. V, 245 pages. 1974. Voi. 405: K. J. Devlin and H. Johnsbrâten, The Souslin Problem. VIII, 132 pages. 1974. Voi. 406: Graphs and Combinatorics. Proceedings 1973. Edited by R. A Bari and F. Harary. VIII, 355 pages. 1974.
Voi. 435: C. F. Dunkl and O. E. Ramirez, Representations of Corn. mutative Semitopological Semigroups. VI, 181 pages. 1975. Voi. 436: L. Auslanderand R. Tolimieri, Abelian HarmonicAnalysis, Theta Functiona and Function Algebras on a Nilmanifold. V, 99 pages. 1975.
Voi. 407: P. Berthelot, Cohomologie Cristalline des Schemas de Caracteristique p > o. 11, 604 pages. 1974.
Voi. 437: O. W. Masser, Elliptic Functiona and Transcendence. XIV, 143 pages. 1975.
Voi. 408: J. Wermer, Potential Theory. VIII, 146 pages. 1974.
Voi. 438: Geometric Topology. Proceedings 1974. Edited by L. C. Glaser and T. B. Rushing. X, 459 pages. 1975.
Voi. 409: Fonctions de Plusieurs Variables Complexes, Sâminaire Fran r > (40-(0+1)2/ 40
r
1
and
occurs.
1+£ :
i.e. 1 > r > -2.025). At this value the first bifurcation
One real eigenvalue for the linearization at zero
crosses the imaginary axis travelling at nonzero speed on the real axis, for the origin a fixed point. points branch off.
Two stable fixed
hey are at C±,!b(r-l) ,±,!bCr-l), r--L) .
This is a standard and elementary bifurcation resulting in a loss of stability by the origin.
10
As
r
increases the two stahle fixed points develop two
complex conjugate and one negative real eigenvalues.
The
picture now looks like (z-axis is oriented upwards and the plane is the
x 0 z
plane):
unstable manifold of the origin manifold of the origin
As
r
increases, the "snails" become more and more
inflated.
r
13.926;
At around this value (found only by numerical
methods) the "snails" are so big that they will enter the stable manifold of the origin.
Stable and unstable
manifold become identical; the origin is a homoclinic point.
Another bifurcation now takes place.
The
picture is, looking in along the x-axis.
(The pair of fixed points do not lie in the yz-plane; they are stable)
11
The two orbits with infinite period "starting" and "ending" in the origin "cross over".
The "snails"
still inflate and by doing this, the homoclinic orbits leave behind unstable closed periodic orbits.
The
picture of the right han
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