Chaotic behavior in a class of delay difference equations. (English) Zbl 1380.34112
Summary: In this paper, we rigorously prove the existence of chaos in a class of delay difference equations, which can be viewed as a discrete analogue of a one-dimensional delay differential equation by using the Euler discretization. We first transform this class of delay difference equations into a high-dimensional discrete dynamical system. Then we prove that the map of the system is chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Finally, we give some computer simulations to illustrate the theoretical result.
MSC:
| 34K23 | Complex (chaotic) behavior of solutions to functional-differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
Keywords:
chaos; delay difference equation; snap-back repeller; chaos in the sense of Devaney; chaos in the sense of Li-YorkeReferences:
| [1] | Busenbrg S, Martelli M: Delay Differential Equations and Dynamical Systems. Springer, Berlin; 1991. · Zbl 0727.00007 · doi:10.1007/BFb0083474 |
| [2] | Jiang M, Shen Y, Jian J, Liao X: Stability, bifurcation and a new chaos in the logistic differential equation with delay.Phys. Lett. A 2006, 350:221-227. · Zbl 1195.34117 · doi:10.1016/j.physleta.2005.10.019 |
| [3] | Peng MS, Uçar A: The use of the Euler method in identification of multiple bifurcations and chaotic behavior in numerical approximations of delay differential equations.Chaos Solitons Fractals 2004, 21:883-891. · Zbl 1054.65126 · doi:10.1016/j.chaos.2003.12.044 |
| [4] | Peng MS, Yuan Y: Stability, symmetry-breaking bifurcation and chaos in discrete delayed models.Int. J. Bifurc. Chaos 2008, 18:1477-1501. · Zbl 1147.39301 · doi:10.1142/S0218127408021117 |
| [5] | He ZM, Lai X: Bifurcation and chaotic behavior of a discrete-time predator-prey system.Nonlinear Anal., Real World Appl. 2011, 12:403-417. · Zbl 1202.93038 · doi:10.1016/j.nonrwa.2010.06.026 |
| [6] | Fan DJ, Wei JJ: Bifurcation analysis of discrete survival red blood cells model.Commun. Nonlinear Sci. Numer. Simul. 2009, 14:3358-3368. · Zbl 1221.37181 · doi:10.1016/j.cnsns.2009.01.015 |
| [7] | Kaplan JL, Yorke JA: Ordinary differential equations which yield periodic solutions of differential delay equations.J. Math. Anal. Appl. 1974, 48:317-324. · Zbl 0293.34102 · doi:10.1016/0022-247X(74)90162-0 |
| [8] | Dormayer P: The stability of special symmetric solutions of[InlineEquation not available: see fulltext.]with small amplitudes.Nonlinear Anal., Theory Methods Appl. 1990, 14:701-715. · Zbl 0704.34086 · doi:10.1016/0362-546X(90)90045-I |
| [9] | Furomochi T: Existence of periodic solutions of one-dimensional differential-delay equations.Tohoku Math. J. 1978, 30:13-35. · Zbl 0376.34057 · doi:10.2748/tmj/1178230094 |
| [10] | Gurney WSC, Blythe SP, Nisbeth RM: Nicholson’s blowflies revisited.Nature 1978, 287:17-21. · doi:10.1038/287017a0 |
| [11] | Heiden U: Delays in physiological systems.J. Math. Biol. 1979, 8:345-364. · Zbl 0429.92009 · doi:10.1007/BF00275831 |
| [12] | Mackey MC, Glass L: Oscillation and chaos in physiological control systems.Science 1977, 197:287-289. · Zbl 1383.92036 · doi:10.1126/science.267326 |
| [13] | Mackey MC, Heiden U: The dynamics of recurrent inhibition.J. Math. Biol. 1984, 19:211-225. · Zbl 0538.92007 · doi:10.1007/BF00277747 |
| [14] | Mackey, MC, Commodity price fluctuations: price dependent delays and nonlinearities as explanatory factors, No. 48 (1990) |
| [15] | Heiden U, Mackey MC: The dynamics of production and destruction: analytic insight into complex behavior.J. Math. Biol. 1982, 16:75-101. · Zbl 0523.93038 · doi:10.1007/BF00275162 |
| [16] | Heiden U, Walther HO: Existence of chaos in control systems with delayed feedback.J. Differ. Equ. 1983, 47:273-295. · Zbl 0477.93040 · doi:10.1016/0022-0396(83)90037-2 |
| [17] | Lani-Wayda B, Walther HO: Chaotic motion generated by delayed negative feedback, Part I: a transversality criterion.Differ. Integral Equ. 1995, 8:1407-1452. · Zbl 0827.34059 |
| [18] | Lani-Wayda B, Walther HO: Chaotic motion generated by delayed negative feedback, Part II: construction of nonlinearities.Math. Nachr. 1996, 180:141-211. · Zbl 0853.34065 · doi:10.1002/mana.3211800109 |
| [19] | Li TY, Yorke JA: Period three implies chaos.Am. Math. Mon. 1975, 82:985-992. · Zbl 0351.92021 · doi:10.2307/2318254 |
| [20] | Devaney RL: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, New York; 1987. |
| [21] | Banks J, Brooks J, Cairns G, Davis G, Stacey P: On Devaney’s definition of chaos.Am. Math. Mon. 1992, 99:332-334. · Zbl 0758.58019 · doi:10.2307/2324899 |
| [22] | Huang W, Ye XD: Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos.Topol. Appl. 2002, 117:259-272. · Zbl 0997.54061 · doi:10.1016/S0166-8641(01)00025-6 |
| [23] | Wiggins S: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York; 1990. · Zbl 0701.58001 · doi:10.1007/978-1-4757-4067-7 |
| [24] | Shi YM, Chen GR: Chaos of discrete dynamical systems in complete metric spaces.Chaos Solitons Fractals 2004, 22:555-571. · Zbl 1067.37047 · doi:10.1016/j.chaos.2004.02.015 |
| [25] | Marotto FR: Snap-back repellers imply chaos in[InlineEquation not available: see fulltext.] . J. Math. Anal. Appl. 1978, 63:199-223. · Zbl 0381.58004 · doi:10.1016/0022-247X(78)90115-4 |
| [26] | Huang Y, Zou XF: Co-existence of chaos and stable periodic orbits in a simple discrete neural network.J. Nonlinear Sci. 2005, 15:291-303. · Zbl 1098.37069 · doi:10.1007/s00332-005-0647-z |
| [27] | Shi YM, Chen GR: Discrete chaos in Banach spaces.Sci. China Ser. A 2004, 34:595-609. (English version 48, 222-238 (2005)) · Zbl 1170.37306 |
| [28] | Shi YM, Yu P: Chaos induced by regular snap-back repellers.J. Math. Anal. Appl. 2008, 337:1480-1494. · Zbl 1131.37023 · doi:10.1016/j.jmaa.2007.05.005 |
| [29] | Shi YM, Yu P: Study on chaos by turbulent maps in noncompact sets.Chaos Solitons Fractals 2006, 28:1165-1180. · Zbl 1106.37008 · doi:10.1016/j.chaos.2005.08.162 |
| [30] | Shi YM, Yu P, Chen GR: Chaotification of dynamical systems in Banach spaces.Int. J. Bifurc. Chaos 2006, 16:2615-2636. · Zbl 1185.37084 · doi:10.1142/S021812740601629X |
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