Homoclinic orbits in degenerate reversible-equivariant systems inR6

Existence of Chaos in the Chen System with Linear Time-Delay Feedback

“…Therefore, it has chaos in a mathematical sense [Shil'nikov, 1965;Silva, 1993;Huang & Yang, 2005]. In recent years, research on homoclinic and heteroclinic orbits in dynamical systems has attracted much attention [Ren & Li, 2010;Lv & Tang, 2013;Chen, 2013;Li et al, 2013;Costa & Tehrani, 2014;Balasuriya & Padberg-Gehle, 2014;Lima & Teixeira, 2013;Zhou et al, 2004;Wang et al, 2007;EI-Dessoky et al, 2012;Zheng & Chen, 2006]. The study about the homoclinic and heteroclinic orbits in second-order differential systems has received major development [Lv & Tang, 2013;Chen, 2013;Li et al, 2013;Costa & Tehrani, 2014;Balasuriya & Padberg-Gehle, 2014;Lima & Teixeira, 2013].…”

“…In recent years, research on homoclinic and heteroclinic orbits in dynamical systems has attracted much attention [Ren & Li, 2010;Lv & Tang, 2013;Chen, 2013;Li et al, 2013;Costa & Tehrani, 2014;Balasuriya & Padberg-Gehle, 2014;Lima & Teixeira, 2013;Zhou et al, 2004;Wang et al, 2007;EI-Dessoky et al, 2012;Zheng & Chen, 2006]. The study about the homoclinic and heteroclinic orbits in second-order differential systems has received major development [Lv & Tang, 2013;Chen, 2013;Li et al, 2013;Costa & Tehrani, 2014;Balasuriya & Padberg-Gehle, 2014;Lima & Teixeira, 2013]. Reference [Zhou et al, 2004] determined the homoclinic and heteroclinic orbits in the Chen system by using the undetermined coefficient method, which was also applied to the Lorenz-family system [Wang et al, 2007], Lü system, Zhou's system [EI-Dessoky et al, 2012], a class of 3D quadratic autonomous chaotic systems [Zheng & Chen, 2006] and the Chen system with time-delays [Ren & Li, 2010].…”

Complete transversals of symmetric vector fields

“…In particular, in [3] a relationship between purely equivariant systems (without reversing symmetries) and a class of reversible equivariant systems is established. The normal form of a Γ-reversible-equivariant system inherits the symmetries and reversing symmetries if the changes of coordinates are equivariant under the group Γ. Belitskii normal form has been used by many authors in different aspects; for example, in the analysis of occurrence of limit cycles or families of periodic orbits either in purely reversible vector fields or in reversible equivariant ones (see [19] and references therein). Motivated by these works, in [4] we have established an algebraic result related to those by Belitskii [5] and Elphick [12] in the reversible equivariant context using tools from invariant theory.…”

Complete transversals of reversible equivariant singularities of vector fields

“…Purely reversible systems have been studied for a longer time, for example [10,12,21,20,23,25]. In more recent years, reversible equivariant systems have become an object of study by many authors, see for example [1,2,3,9,24,26,28]. In particular, in [3] a relationship between purely equivariant systems (without reversing symmetries) and a class of reversible equivariant systems is established.…”

“…In particular, in [3] a relationship between purely equivariant systems (without reversing symmetries) and a class of reversible equivariant systems is established. The normal form of a Γ−reversible-equivariant system inherits the symmetries and reversing symmetries if the changes of coordinates are equivariant under the group Γ. Belitskii normal form has been used by many authors in different aspects; for example, in the analysis of occurrence of limit cycles or families of periodic orbits either in purely reversible vector fields or in reversible equivariant ones (see [27,26,28,31]). Motived by these works, in [4] we have established an algebraic result related to those by Belitskii [5] and Elphick [15] in the reversible equivariant context using tools from invariant theory.…”