Anti-Controlling Codimension-2 Bifurcation of Discrete Dynamical Systems in 1 ∶ 2 Resonance

YANG Yujiao; XU Huidong; ZHANG Jianwen

doi:10.21656/1000-0887.420118

Volume 43 Issue 2

Feb. 2022

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Article Navigation > Applied Mathematics and Mechanics > 2022 > 43(2): 142-155

YANG Yujiao, XU Huidong, ZHANG Jianwen. Anti-Controlling Codimension-2 Bifurcation of Discrete Dynamical Systems in 1 ∶ 2 Resonance[J]. Applied Mathematics and Mechanics, 2022, 43(2): 142-155. doi: 10.21656/1000-0887.420118

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Anti-Controlling Codimension-2 Bifurcation of Discrete Dynamical Systems in 1 ∶ 2 Resonance

doi: 10.21656/1000-0887.420118

1.
College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P.R.China
2.
College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, P.R.China

Received Date: 2021-04-30
Accepted Date: 2021-04-30
Rev Recd Date: 2021-06-19

Available Online: 2022-01-08

Publish Date: 2022-02-01

Abstract

Abstract

A set of nonlinear feedback control strategies were designed to realize the bifurcation solutions of codimensional bifurcations in discrete dynamical systems with 1∶2 resonance from the perspective of bifurcation anti-controlling. Firstly, aimed at the limitation of traditional bifurcation criteria for determination of high codimensional bifurcation points, a new explicit criterion for codimension-2 bifurcation in 1∶2 resonance was proposed. Based on this explicit criterion, the linear control gain was designed to ensure the existence of such codimension-2 bifurcation. Then, the central manifold of 1∶2 resonance was derived. Based on the normal form method, the types and stability of codimension-2 bifurcation solutions in 1∶2 resonance were analyzed through design of nonlinear control gain. Finally, an Arneodo-Coullet-Tresser mapping was taken as an example, and various bifurcation solutions with 1∶2 resonance bifurcation properties were realized by control at the specified parameter points, to further validate the theoretical analysis.
- codimension-2 bifurcation in 1∶2 resonance,
- explicit criterion,
- anti-controlling bifurcation,
- discrete dynamical system

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References

[1]	AGLIARI A, NAIMZADA A K, PECORA N. Nonlinear dynamics of a Cournot duopoly game with differentiated products[J]. Applied Mathematics and Computation, 2016, 281: 1-15. doi: 10.1016/j.amc.2016.01.045
[2]	AGLIARI A, NAIMZADA A, PECORA N. Bifurcation structures of a cobweb model with memory and competing technologies[J]. Communications in Nonlinear Science and Numerical Simulation, 2018, 58: 78-91. doi: 10.1016/j.cnsns.2017.04.008
[3]	ZHANG L M, ZHANG C F, HE Z R. Codimension-one and codimension-two bifurcations of a discrete predator-prey system with strong Allee effect[J]. Mathematics and Computers in Simulation, 2019, 162: 155-178. doi: 10.1016/j.matcom.2019.01.006
[4]	CHEN Q L, TENG Z D. Codimension-two bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response[J]. Journal of Difference Equations and Applications, 2017, 23(12): 2093-2115. doi: 10.1080/10236198.2017.1395418
[5]	YUAN L G, YANG Q G. Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system[J]. Applied Mathematical Modelling, 2015, 39(8): 2345-2362. doi: 10.1016/j.apm.2014.10.040
[6]	PECORA N. Analysis of 1∶4 resonance in a monopoly model with memory[J]. Chaos, Solitons and Fractals, 2018, 110: 95-104. doi: 10.1016/j.chaos.2018.03.005
[7]	REN J L, YU L P. Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model[J]. Journal of Nonlinear Science, 2016, 26(6): 1895-1931. doi: 10.1007/s00332-016-9323-8
[8]	LI B, HE Q Z. Bifurcation analysis of a two-dimensional discrete Hindmarsh-Rose type model[J]. Advances in Difference Equations, 2019(1): 124. doi: 10.1186/s13662-019-2062-z
[9]	CHEN D S, WANG H O, CHEN G R. Anti-control of Hopf bifurcations[J]. IEEE Transactions on Circuits and Systems I, 2001, 48(6): 661-672. doi: 10.1109/81.928149
[10]	WEN G L, WANG Q G, CHIU M S. Delay feedback control for interaction of Hopf and period doubling bifurcations in discrete-time systems[J]. International Journal of Bifurcation and Chaos, 2006, 16(1): 101-112. doi: 10.1142/S0218127406014617
[11]	伍新, 文桂林, 徐慧东, 等. 惯性式冲击振动落砂机周期倍化分岔的反控制[J]. 固体力学学报, 2015, 36(1): 28-34. (WU Xin, WEN Guilin, XU Huidong, et al. Anti-controlling period-doubling bifurcation of an inertial impact shaker system[J]. Chinese Journal of Solid Mechanics, 2015, 36(1): 28-34.(in Chinese)
[12]	张玲梅, 张建文, 吴润衡. 具有对应分段系统和指数系统的新混沌系统的Hopf分岔控制研究[J]. 物理学报, 2014, 63(16): 160505. (ZHANG Lingmei, ZHANG Jianwen, WU Runheng. Anti-control of Hopf bifurcation in the new chaotic system with piecewise system and exponential system[J]. Acta Physica Sinica, 2014, 63(16): 160505.(in Chinese) doi: 10.7498/aps.63.160505
[13]	刘素华, 唐驾时. 四维Qi系统零平衡点的Hopf分岔反控制[J]. 物理学报, 2008, 57(10): 6162-6168. (LIU Suhua, TANG Jiashi. Anti-control of Hopf bifurcation at zero equilibrium of 4D Qi system[J]. Acta Physica Sinica, 2008, 57(10): 6162-6168.(in Chinese) doi: 10.3321/j.issn:1000-3290.2008.10.018
[14]	WEN G L, XU D L, XIE J H. Controlling Hopf bifurcations of discrete-time systems in resonance[J]. Chaos, Solitons and Fractals, 2005, 23(5): 1865-1877. doi: 10.1016/S0960-0779(04)00451-5
[15]	伍新, 文桂林, 徐慧东, 等. 三自由度含间隙碰撞振动系统Neimark-Sacker分岔的反控制[J]. 物理学报, 2015, 64(20): 200504. (WU Xin, WEN Guilin, XU Huidong, et al. Anti-controlling Neimark-Sacker bifurcation of a three-degree-of-freedom vibration system with clear-ance[J]. Acta Physica Sinica, 2015, 64(20): 200504.(in Chinese) doi: 10.7498/aps.64.200504
[16]	徐慧东, 文桂林, 伍新, 等. 三自由度含间隙碰撞振动系统Poincaré映射Hopf-Hopf交互分岔的反控制[J]. 振动工程学报, 2015, 28(6): 952-959. (XU Huidong, WEN Guilin, WU Xin, et al. Anti-controlling Hopf-Hopf interaction bifurcations on Poincaré map of a three-degree-of-freedom vibro-impact system with clearance[J]. Journal of Vibration Engineering, 2015, 28(6): 952-959.(in Chinese)
[17]	WEN G L. Criterion to identify Hopf bifurcations in maps of arbitrary dimension[J]. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2005, 72(2): 026201. doi: 10.1103/PhysRevE.72.026201
[18]	XU H D, WEN G L. Alternative criterion for investigation of pitchfork bifurcations of limit cycle in relay feedback systems[J]. Journal of Computational and Nonlinear Dynamics, 2014, 9(3): 031004. doi: 10.1115/1.4025744
[19]	YAO S J. New Bifurcation critical criterion of Flip-Neimark-Sacker bifurcations for two-parameterized family of n-dimensional discrete systems[J]. Discrete Dynamics in Nature and Society, 2012, 2012: 264526. doi: 10.1155/2012/264526
[20]	KUZNETSOV Y A. Elements of Applied Bifurcation Theory[M]. 3rd ed. New York: Springer-Verlag, 2004.
[21]	WEN G L, CHEN S J, JIN Q T. A new criterion of period-doubling bifurcation in maps and its application to an inertial impact shaker[J]. Journal of Sound and Vibration, 2008, 311(1/2): 212-223.
[22]	LASALLE J P. The Stability and Control of Discrete Processes[M]. Berlin: Springer-Verlag, 1986.
[23]	JURY E I, PAVLIDIS T. Stability and aperiodicity constraints for system design[J]. IEEE Transactions on Circuit Theory, 1963, 10(1): 137-141. doi: 10.1109/TCT.1963.1082100
[24]	KUZNETSOV Y A, MEIJER H G E, VAN VEEN L. The fold-flip bifurcation[J]. International Journal of Bifurcation and Chaos, 2004, 14(7): 2253-2282. doi: 10.1142/S0218127404010576
[25]	DU B S, LI M C, MALKIN M I. Topological horseshoes for Arneodo-Coullet-Tresser maps[J]. Regular and Chaotic Dynamics, 2006, 11(2): 181-190. doi: 10.1070/RD2006v011n02ABEH000344