Complex dynamics in simple delayed two-parameterized models

In this paper, some geometrical aspects of root distributions in a special polynomial of the form λτ(λ−(1−α))−βλτ(λ−(1−α))−β are discussed. Equivariant structures are explored in the corresponding systems. Some sufficient and necessary conditions for a pair of complex conjugate roots of the polynomial with τ=3τ=3 lying on the unit circle. A comparison is made between two simple delayed discrete models, where one can be viewed as its perturbation of the other with delayed feedback. There exist rich dynamics in the perturbed system, such as chaotic, or even hyperchaotic behavior whereas only regular oscillation modes can be observed in the perturbed system. The introduction of delayed feedback can break or increase the special symmetrical/topological structure of the original system, which leads to complexity. Rich dynamics near equivariant bifurcations under the Z4/Z8Z4/Z8 cyclic group action is explored, including multiple bifurcations, multistability, chaos and hyperchaos etc. As the applications, one can find that there exist higher-codimensional bifurcations with 1:1 strong resonance and 1:2 strong resonance in those models with/without special equivariant systems, which were not discussed in Guo et al. (2008) [15], Guo et al. (2008) [16], Kuruklis (1994) [17], Peng et al. (2008) [18], Peng and Yang (2010) [19], Peng and Yuan (2008) [20], Peng and Yuan (2007) [21], Wang and Peng (2009) [22] etc.