Dynamical Systems and Turbulence, Warwick 1980 Proceedings of a Symp
- PDF / 19,622,033 Bytes
- 400 Pages / 461 x 684 pts Page_size
- 56 Downloads / 182 Views
l. 729: Ergodic Theory. Proceedings, 1978. Edited by M. Denker and K. Jacobs. XII, 209 pages. 1979.
Vol. 701: Functional Analysis Methods in Numerical Analysis. Pro· ceed1ngs, 1977. Edited by M. Zuhair Nashed. VII, 333 pages. 1979.
Vol. 730: Functional Differential Equations and Appro11mation of Fixed Points. Proceedings, 1978. Edited by H.-0. Pe1tgen and H.-0. Walther. XV, 503 pages. 1979.
Vol. 702: Yun N. Bib1kov, Local Theory of Nonlinear Analytic Ordinary Differential Equations. IX, 14 7 pages. 1979. Vol. 703: Equadiff IV. Proceedings, 1977. Edited by J. Fabera. XIX. 441 pages. 1979. Vol. 704: Computing Methods in Applied Sciences and Engineering, 1977, I. Proceedings. 1977. Edited by R. Glowinski ar>d J. L Lions. VI, 391 pages. 1979. Vol. 705: 0. Forster und K. Knorr, Konstruktion verseller Familien kompakter komplexer Rtiume. VII, 141 Seiten.1979. Vol. 706: Probability Measures on Groups, Proceedings, 1978. Edited by H. Heyer. XIII. 348 pages. 1979 Vol. 707: R. Zielke, Discontinuous CebySev Systems. VI. 111 pages. 1979. Vol. 708: J. P. Jouanolou, Equations de Pfaff algebnques. V, 255 pages. 1979. Vol. 709: Probability in Banach Spaces II. Proceedings, 1978. Edited by A Beck. V. 205 pages. 1979. Vol. 710: Seminaire Bourbaki vol. 1977178, Exposes 507-524. IV. 328 pages. 1979. Vol. 711: Asymptotic Analysis. Edited by F. Verhulst V, 240 pages. 1979. Vol. 712: Equations Differentielles et Systemes de Pfaff dans le Champ Complexe. Ed1te par R. Gerard et J.-P. Ramis. V, 364 pages. 1979. Vol. 713: Seminaire de Theorie du Potentiel, Paris No. 4. Edite par F. Hirsch et G. Mokobodzk1. VII. 281 pages. 1979. Vol. 714: J. Jacod, Calcul Stochaatique et Problemee de Martingales X. 539 pages. 1979. Vol. 715: lnder Bir S. Passi, Group Rings and Their Augmentahon Ideals. VI, 137 pages. 1g79, Vol. 716: M. A Scheunert. The Theory of Lie Superalgebras. X, 271 pages. 1979. Vol. 717: Grosser, S.dualrtiume und Ve O.
To express this precisely consider the linearisation
of (3) dv/dt = f (t,lJ.!v)
-
(4)
-
This is to be thought of as a complex linear equation (with real coefficients) on H , the complexification of H.
Associated with (4) is a linear operator on the space
of
l' -periodic vector fields on
J
= -d/dt + f (t, IJ.I.) (5) IJ. -u are called Floguet exponents. The orbit u = 0 is stable if all Floquet
Eigenvalues of J IJ exponents have negative real part, and unstable if any has positive real part.
The loss
of stability at IJ. = 0 is assumed to occur in the simplest way :
Bifurcation Assumptions : There is a Floquet exponent a(lJ.) (i) a(O)
=
iW
= 2;r
0
s;
=
+ i7)(IJ.) such that
r < l,
O (it) a(lJ.) and a(lJ.) are isolated algebraically simple eigenvalues of JIJ. .
(iii)
> 0 .
(iv) All eigenvalues of J
O
other than a(O) and (](O) have negative real part.
The type of bifurcation that occurs depends on the value of r , (i) Strong Resonance: if r
= min
and n
= 1,2,3,
or n
=4
and a certain inequality
holds then nT-periodic solutions bifurcate. (Ii) Y.H. Wan [6J has shown that there is an invariant
Data Loading...
