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Complex dynamics in simple delayed two-parameterized models. (English) Zbl 1276.37048
Summary: In this paper, some geometrical aspects of root distributions in a special polynomial of the form $$\lambda ^{\tau }(\lambda - (1 - \alpha )) - \beta$$ 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 $$\tau =3$$ lying on the unit circle. A comparison is made between two simple delayed discrete models, where one can be viewed as the perturbation of the other with a 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 $$\mathbb Z_{4}/\mathbb Z_{8}$$ 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.

##### MSC:
 37M20 Computational methods for bifurcation problems in dynamical systems 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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