Bifurcation and Chaos of a Discrete-Time Mathematical Model for Tissue Inflammation

Chen, Xianwei; Yuan, Shaoliang; Jing, Zhujun; Fu, Xiangling

doi:10.1007/s10884-014-9413-y

Cited by 2 publications

(1 citation statement)

References 14 publications

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“…The concept of snap-back repellers is not new, but it is still the subject of intensive study. Some works that appeared in the last ten years are [Chen et al, 2016, Peng, 2007, Ren et al, 2017, Salman et al, 2016, Shi and Yu, 2008, Yuan et al, 2011, Zhao and Li, 2016. Note that differentiability of the map is an essential condition for Theorem 1, but see Gardini and Tramontana [2010a,b] for results which do not require smoothness.…”

Section: Snap-back Repellersmentioning

confidence: 99%

Periodicity and Chaos Amidst Twisting and Folding in Two-Dimensional Maps

Garst¹,

Sterk

2018

Int. J. Bifurcation Chaos

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We study the dynamics of three planar, noninvertible maps which rotate and fold the plane. Two maps are inspired by real-world applications whereas the third map is constructed to serve as a toy model for the other two maps. The dynamics of the three maps are remarkably similar. A stable fixed point bifurcates through a Hopf–Neĭmark–Sacker which leads to a countably infinite set of resonance tongues in the parameter plane of the map. Within a resonance tongue a periodic point can bifurcate through a period-doubling cascade. At the end of the cascade we detect Hénon-like attractors which are conjectured to be the closure of the unstable manifold of a saddle periodic point. These attractors have a folded structure which can be explained by means of the concept of critical lines. We also detect snap-back repellers which can either coexist with Hénon-like attractors or which can be formed when the saddle-point of a Hénon-like attractor becomes a source.

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