ISSN:
1420-9039
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract We consider the dynamics of the Ginzburg-Landau equation in a small neighborhood of a known pulse solution by studying a Poincaré map,P: π T →π T , where π T is a section which is transverse to the pulse. Due to the fact that the Ginzburg-Landau equation possesses both a rotational symmetry and a spatial symmetry, we are able to conduct a detailed analytical study of this map in neighborhoods arbitrarily close to the pulse solution. Thus, we are able to complement the work of Holmes [8], who conducted an analytical study of the Poincaré map in a punctured neighborhood of the pulse. We find that the Poincaré map contains an invariant set Ω⊃πitT, where Ω is not necessarily a Cantor set of points, such thatP: Ω→Ω is homeomorphic to a shift map on (at least) two symbols. Furthermore, we find that for eachm≥ 1 the mapP itm possesses a fixed point. Since Ω is not necessarily a Cantor set, this is not immediately clear. Finally, we find that when the pulse solution is broken, for eachm≥1 there exist parameter values such that pulses possessingm maxima appear.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00916827
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