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Chaos induced by regular snap-back repellers. (English) Zbl 1131.37023

Summary: This paper is concerned with chaos induced by regular snap-back repellers. One new criterion of chaos induced by strictly coupled-expanding maps in compact sets of metric spaces is established. By employing this criterion, the nondegenerateness assumption in the Marotto theorem established in 1978 [F. R. Marotto, ibid. 63, 199–223 (1978; Zbl 0381.58004)] is weakened. In addition, it is proved that a regular snap-back repeller and a regular homoclinic orbit to a regular expanding fixed point in finite-dimensional spaces imply chaos in the sense of Li-Yorke. An illustrative example is provided with computer simulations.

MSC:

37B25 Stability of topological dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Citations:

Zbl 0381.58004
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References:

[1] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On Devaney’s definition of chaos, Amer. math. monthly, 99, 332-334, (1992) · Zbl 0758.58019
[2] Blanchard, F.; Glasner, E.; Kolyada, S.; Maass, A., On li – yorke pairs, J. reine angew. math., 547, 51-68, (2002) · Zbl 1059.37006
[3] Block, L.S.; Coppel, W.A., Dynamics in one dimension, Lecture notes in math., vol. 1513, (1992), Springer-Verlag Berlin
[4] Chen, G.; Hsu, S.; Zhou, J., Snap-back repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span, J. math. phys., 39, 6459-6489, (1998) · Zbl 0959.37027
[5] Chen, L.; Aihara, K., Strange attractors in chaotic neural networks, IEEE trans. circuits syst. I, 47, 1455-1468, (2000) · Zbl 0994.37016
[6] Chen, S.; Shih, C., Transversal homoclinic orbits in a transiently chaotic neural network, Chaos, 12, 654-671, (2002) · Zbl 1080.37611
[7] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[8] Devaney, R.L., An introduction to chaotic dynamical systems, (1989), Addison-Wesley Publishing Company · Zbl 0695.58002
[9] Dohtani, A., Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. appl. math., 52, 1707-1721, (1992) · Zbl 0774.93049
[10] Elaydi, S.N., Discrete chaos, (2000), Chapman and Hall/CRC · Zbl 0945.37010
[11] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. math. sci., vol. 42, (1983), Springer-Verlag New York · Zbl 0515.34001
[12] Huang, W.; Ye, X., Devaney’s chaos or 2-scattering implies li – yorke’s chaos, Topology appl., 117, 259-272, (2002) · Zbl 0997.54061
[13] Kaizoji, T., Speculative price dynamics in a heterogeneous agent model, Nonlinear dyn. psychol. life sci., 6, 217-229, (2002) · Zbl 1201.91237
[14] Kennedy, J.; Yorke, J.A., Topological horseshoes, Trans. amer. math. soc., 353, 2513-2530, (2001) · Zbl 0972.37011
[15] Li, T.; Yorke, J.A., Period three implies chaos, Amer. math. monthly, 82, 985-992, (1975) · Zbl 0351.92021
[16] Marotto, F.R., Snap-back repellers imply chaos in \(\mathbf{R}^n\), J. math. anal. appl., 63, 199-223, (1978) · Zbl 0381.58004
[17] Marotto, F.R., On redefining a snap-back repeller, Chaos solitons fractals, 25, 25-28, (2005) · Zbl 1077.37027
[18] Martelli, M.; Dang, M.; Seph, T., Defining chaos, Math. mag., 71, 112-122, (1998) · Zbl 1008.37014
[19] Morris, H.C.; Ryan, E.E.; Dodd, R.K., Snap-back repellers and chaos in a discrete population model with delayed recruitment, Nonlinear anal., 7, 571-621, (1983) · Zbl 0558.92010
[20] Robinson, C., Dynamical systems: stability, symbolic dynamics and chaos, (1999), CRC Press, Inc. Florida · Zbl 0914.58021
[21] Shi, Y.; Chen, G., Chaos of discrete dynamical systems in complete metric spaces, Chaos solitons fractals, 22, 555-571, (2004) · Zbl 1067.37047
[22] Shi, Y.; Chen, G., Discrete chaos in Banach spaces, Sci. China ser. A, Sci. China ser. A, 48, 222-238, (2005), (Chinese version); English version: · Zbl 1170.37306
[23] Shi, Y.; Chen, G., Chaotification of discrete dynamical systems governed by continuous maps, Internat. J. bifur. chaos, 15, 547-556, (2005) · Zbl 1082.37031
[24] Y. Shi, G. Chen, Some new criteria of chaos induced by coupled-expanding maps, in: Proc. 1st IFAC Conference on Analysis and Control of Chaotic Systems, Reims, France, June 28-30, 2006, pp. 157-162
[25] Shi, Y.; Yu, P.; Chen, G., Chaotification of discrete dynamical systems in Banach spaces, Internat. J. bifur. chaos, 16, 2615-2636, (2006) · Zbl 1185.37084
[26] Shi, Y.; Yu, P., Study on chaos induced by turbulent maps in noncompact sets, Chaos solitons fractals, 28, 1165-1180, (2006) · Zbl 1106.37008
[27] Y. Shi, P. Yu, Chaos induced by snap-back repellers and its applications to anti-control of chaos, in: Proc. 4th DCDIS International Conference on Engineering Applications and Computational Algorithms, Guelph, Ontario, Canada, July 27-29, 2005, pp. 364-369
[28] Y. Shi, Chaos in first-order partial difference equations, submitted for publication
[29] Wang, X.F.; Chen, G., Chaotification via arbitrary small feedback controls, Internat. J. bifur. chaos, 10, 549-570, (2000) · Zbl 1090.37532
[30] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (1990), Springer-Verlag New York · Zbl 0701.58001
[31] Yang, X.; Tang, Y., Horseshoes in piecewise continuous maps, Chaos solitons fractals, 19, 841-845, (2004) · Zbl 1053.37006
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