Dynamical Systems, Differential doi:10.3934/proc.2015.0954 Equations and Applications AIMS Proceedings, 2015 pp. 954–964 INFINITELY MANY MULTI-PULSES NEAR A BIFOCAL CYCLE Alexandre A. P. Rodrigues Centro de Matem´ atica da Universidade do Porto and Faculdade de Ciˆ encias da Universidade do Porto Rua do Campo Alegre 687, 4169–007 Porto, Portugal Abstract. The entire dynamics in a neighbourhood of a reversible heteroclinic cycle involving a bifocus is far from being understood. In this article, using the well known theory of reversing symmetries, we prove that there are infinitely many pulses near a cycle involving two symmetric equilibria, a real saddle and a bifocus, giving rise to a complex network. We also conjecture that suspended blenders might appear in the neighbourhood of the network. 1. Introduction. In three-dimensions, heteroclinic cycles and networks involving saddle- foci have been considered by several authors in different settings [8, 11, 12, 16, 17, 19]. Nowadays, the big challenge is the analysis of heteroclinic cycles involving a bifocus in four dimensions. These cycles are usually known as bifocal cycles. Shilnikov [18, 19] was the first who studied the dynamics associated to them and proved the existence of a countable set of periodic solutions in their neighbourhood. Subsequent works [5, 13] considered the formation and bifurcations of these solutions by studying the Poincar´ e map associated to a cross-section in a neighbourhood of the bifocus. In any transverse section to the bifocal cycle, for any N ∈ N, there is a compact invariant hyperbolic set on which the Poincar´ e map is topologically conjugate to the Bernoulli shift on N symbols [9]. The existence of periodic solutions near heteroclinic cycles in reversible systems has been explored in [4, 20]. A bifurcation analysis of cycles in Hamiltonian systems can be found in [15] – here, sequences of parameter values have been detected for which homoclinic fold bifurcations occur. Note that the symbolic dynamics, which has been proved to occur near the cycle in the Hamiltonian case, are not expected in the reversible case because of the lack of level sets in the latter case. In [10], the authors studied the dynamics around a heteroclinic cycle associated to two saddles, in four dimensions. In the unfolding of the cycle, using the Lin’s method, they showed that homoclinic snaking occurs if and only if at least one of the saddles is a bifocus. In the present paper, in the spirit of the works of [1, 3, 7], we study the behaviour of a four dimensional reversible vector field whose flow has a heteroclinic cycle involving a bifocus and a saddle, for which we give a description in Section 3. Without breaking the bifocal cycle, we prove the existence of infinitely many periodic solutions and homoclinic cycles coexisting with the original cycle, which themselves are accompanied by complex dynamics. At the end of this article, we conjecture about the existence of heterodimensional cycles and blenders near the bifocal cycle. An example of a bifocal cycle associated to a saddle and a bifocus has been found in [10, Ch. 3], in the context of Bussinesq equations. 2010 Mathematics Subject Classification. Primary: 37C29; Secondary: 34C28, 37C27, 37C20. Key words and phrases. Bifocal cycles, Hyperchaos, Multi-pulses, Blenders. The author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by Funda¸c˜ ao para a Ciˆ encia (FCT), Portugal, with national (MEC) and European structural funds through the pro- grams FEDER, under the partnership agreement PT2020. Alexandre Rodrigues is also supported by grant SFRH/BPD/84709/2012 of FCT. 954