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.


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


Zbl 0381.58004
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.