UNSPECIFIED. (2003) Secondary bifurcations in systems with all-to-all coupling. PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 459 (2036). pp. 1969-1986. ISSN 1364-5021

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Abstract

We study the existence, branching geometry and stability of secondary branches of equilibria in all-to-all coupled systems of differential equations, that is, equations that are equivariant under the permutation action of the symmetric group S-N. Specifically, we consider the most general cubic-order system of this type, which arises in models of polymorphism in evolutionary biology. Primary branches in such systems correspond to partitions of N into two parts, and secondary branches correspond to partitions of N into three parts of sizes a, b, c, respectively. If a = b = c, then the cubic-order system is too degenerate to provide secondary branches. The cases when one of a, b, c is equal to N/3 are special, and are not treated here. In all other cases, secondary branches exist, and the secondary branch corresponding to (a, b, c) intersects the primary branches corresponding to (a + b, c), (a, b + c) and (a + c, b). All such secondary branches are globally unstable in the cubic-order system. Abstract considerations suggest that such secondary branches are locally unstable, which would explain the common occurrence of jump bifurcations between primary branches in numerical simulations of the cubic-order system. However such considerations do not prove instability due to the possible existence of hidden symmetries. In this paper, we carry out the calculations required to verify that the secondary branches are unstable, and we show, moreover, that these branches are globally unstable.

Item Type:	Journal Article
Subjects:	Q Science
Journal or Publication Title:	PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Publisher:	ROYAL SOC LONDON
ISSN:	1364-5021
Date:	8 August 2003
Volume:	459
Number:	2036
Number of Pages:	18
Page Range:	pp. 1969-1986
Identification Number:	10.1098/rspa.2002.1103
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/9423

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