Physica D 238 (2009) 449–460 Contents lists available at ScienceDirect Physica D journal homepage: www.elsevier.com/locate/physd Unstable periodic solution controlling collision of localized convection cells in binary fluid mixture Makoto Iima ∗ , Yasumasa Nishiura Research Institute for Electronic Science, Hokkaido University, Sapporo, 001-0020, Japan article info Article history: Received 13 November 2007 Received in revised form 6 November 2008 Accepted 17 November 2008 Available online 30 November 2008 Communicated by B. Sandstede PACS: 05.45.-a 74.20.De 47.55.pb Keywords: Dissipative systems Pulse collision Binary fluid abstract We study the collision processes of spatially localized convection cells (pulses) in a binary fluid mixture by the extended complex Ginzburg–Landau equations. Both counter- and co-propagating pulse collisions are examined numerically. For counter-propagating pulse collision, we found a special class of unstable time-periodic solutions that play a critical role in determining the output after collision. The solution profile right after collision becomes close to such an unstable pattern and then evolves along one of the unstable manifolds before reaching a final destination. The origin of such a class of unstable solutions, called scattors, can be traced back to two-peak bound states which are stable in an appropriate parameter regime. They are destabilized, as the parameter is varied, and become scattors which play the role of separators of different dynamic regimes. Delayed feedback control is useful to detect them. Also, there is another regime where the origin of the scattors is different from that of the above case. For co-propagating pulse collision, it is revealed that the result of pulse collision depends on the phase difference between pulses. Moreover, we found that a coalescent pulse keeps a profile of two-peak bound state, which is not observed in the case of counter-propagating pulse collision. Complicated collision dynamics become transparent to some extent from the viewpoint of those unstable objects. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Thermal convection of a moving fluid known as the Rayleigh–Benard pattern [1,2] is a landmark phenomenon in dissi- pative systems and has become a driving force in the development of nonlinear dynamics, both experimentally and theoretically, like the BZ-reaction in chemical systems [1]. Conventional convecting roll patterns are spatially periodic and do not move in the horizontal direction. In contrast, a binary fluid mixture, like water plus ethanol, presents different types of con- vective motion such as traveling wave convection. The presence of the concentration field allows oscillatory instability as a subcriti- cal bifurcation, which is different from ordinary Rayleigh–Benard convection [1,3–6]. In particular, it presents a completely different type of convective motion, which is spatially localized; i.e., only a finite number of convective cells are involved [7–9], and its enve- lope moves steadily in one direction [6,8–11], although the enve- lope can be steady in the laboratory frame [12,13]. A natural question is what happens when two localized moving waves collide with each other. In fact, there is a series of seminal experiments [10,11] showing a variety of outputs after collision, ∗ Corresponding author. Tel.: +81 11 706 2412; fax: +81 11 706 4966. E-mail addresses: makoto@nsc.es.hokudai.ac.jp (M. Iima), nishiura@nsc.es.hokudai.ac.jp (Y. Nishiura). such as forming a bound state (e.g., Fig. 9 in Ref. [10]), and merging into a single wave (e.g., Fig. 7 in Ref. [10] for counter-propagating pulses, and Fig. 16 in Ref. [11] for co-propagating pulses). Although those colliding phenomena have attracted much attention, to the best of our knowledge, there have been no systematic studies to clarify the collision dynamics due to the inherent difficulties, such as large deformation and the transient nature of the collision dynamics. Therefore, usual perturbative methods do not work for this purpose. One of the necessary ingredients to overcome this difficulty is to change the viewpoint from observable objects to non-observable ones; in other words, we focus on the unstable ordered patterns embedded in the process of collision dynamics, which may play a key role in understanding the whole dynamics. It turns out that a hidden network of unstable ordered patterns called scattors controls the input–output relation at the collisional event. The unstable manifolds of scattors and their heteroclinic connections are separators in order to sort out the orbits along the unstable manifolds as the Rayleigh number is varied. Such an idea has been developed by one of the authors (Y.N.) and his collaborators [14] and applied to various types of collisions of pulses, breathers, and spots [14–16] in the Gray–Scott model [17] and complex Ginzburg–Landau equation with external force [18]. The scattor is a single saddle in a simplest case, but it can be a network of saddles and their connecting manner is changed through a transition as parameters are varied [19]; such a viewpoint of network is critical 0167-2789/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2008.11.010