# Dynamics, Bifurcation and Symmetry: New Trends and New Tools by Andrei Afendikov, Alexander Mielke (auth.), Pascal Chossat

By Andrei Afendikov, Alexander Mielke (auth.), Pascal Chossat (eds.)

This ebook collects contributions to the convention" Dynamics, Bifurcation and Symmetry, new traits and new tools", which was once held on the Institut d'Etudes Sci entifiques de Cargese (France), September 3-9, 1993. the 1st objective of this convention was once to collect and summarize the paintings of the ecu Bifurcation idea staff after years of life (the EBTG hyperlinks ecu laboratories in 5 international locations through an EC grant). because of a NATO ARW supply, the convention built into a global assembly on bifurcation idea and dynamical structures, with the partic ipation of prime experts not just from Europe but additionally from abroad nations (Canada, united states, South America). It was once an exceptional delight to note the lively, and relatively enthusiastic participation of many younger scientists. this is often mirrored within the current booklet for which many members are PhD scholars or post-doc researchers. even supposing numerous "big" issues (bifurcation with symmetry, low dimensional dynam ics, dynamics in EDP's, functions, . . . ) are found in those court cases, now we have divided the booklet into corresponding elements. in truth those issues overlap in so much contributions, which turns out to mirror a normal tendancy in nonlinear technology. i'm more than happy to thank for his or her help the NATO foreign alternate medical software in addition to the EEC technological know-how software, which made attainable the suc cess of this conference.

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1). 2) However, the additional Z2 symmetry which we require, associated with period doubling bifurcations, cannot be realised in this framework. 1) and have symmetry Z2 x Z2 generated by 8 1 and 8 2 which act by We use the notation Fix(81 ) to denote the fixed point space associated with the group generated by 8 1 etc. 1) and are shown in Table 1. :: ]. Xl - X2 Yl - Y2 The action of the 8 1 symmetry on U is then unchanged but the 8 2 symmetry acts on U by 82 [ :~ U3 U4 ] = [ :~]. 3). 1). 1) implies that the Jacobian of these equations at a symmetric fixed point is diagonal and this path is then defined by 1; = -1 where the asterisk denotes evaluation at the trivial fixed point.

Note that for any! < € < 0 we have ~ + ~ < ! -1(11~2;:;2 and so there exist values of q for which there exist essentially asymptotically stable heteroclinic connections. 21, using DSTOOL. As can be seen, there is evidence of an attracting heteroclinic cycle even though the linear stability of Fix(Qk) has an expanding transverse eigenvalue. This is typical of essential asymptotically stability. 44 1 4 2 Figure 1: A Cayley graph of Q generated by i and j. Solid arrows correspond to left multiplication with i, dashed arrows to left multiplication with j.

7715. If we look close enough at the Fourier amplitudes for the modes eiz and e iy (Figure 6) we notice that the laminar phases in between bursts very much look like one of the 4 possible modulated traveling waves, described in the last section. To test this idea we perform K-L analyses on the the time series in between bursts. e. we cannot visually detect the coming of a burst. Again we go into a co-moving coordinate frame. 7715 e iy , b) amplitude and 800 24 laminar phases we find that the first 5 eigenfunctions for each dataset map onto each other under the symmetry operation rTe and tT'1,respectively, where the shifts ~ and TJ are determined by phase shifts of the modulated traveling waves due to different initial conditions.