Discrete chaos in Banach spaces

Science in China, Series A: Mathematics

Volumn 48, Issue 2, 2005, Pages 222-238

  Shi, Yuming ^a   Chen, Guanrong ^b  

^a SHANDONG UNIVERSITY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Banach space; Chaos; Discrete dynamical system; Marotto's theorem

Indexed keywords

EID: 15044354626     PISSN: 10069283     EISSN: None     Source Type: Journal    
DOI: 10.1360/03ys0183     Document Type: Article

References (27)

1. Li, T., Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 1975, 82: 985-992.
2. Banks, J., Brooks, J., Cairns, G. et al., On Devaney's definition of chaos, Amer. Math. Monthly, 1992, 99: 332-334.
3. Devaney, R. L., An Introduction to Chaotic Dynamical Systems, Redwood City: Addison-Wesley Publishing Company, 1987; 2nd ed., 1989.
4. Martelli, M., Dang, M., Seph, T., Defining chaos, Math. Magazine, 1998, 71(2): 112-122.
5. Robinson, C., Dynamical Systems: Stability, Symbolic Dynamics and Chaos, Florida: CRC Press, 1995.
6. Huang, W., Ye, X., Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology and Its Applications, 2002, 117: 259-272.
7. Chen, G., Chaotification via feedback: The discrete case, in Chaos Control: Theory and Applications (eds. Chen, G., Yu, X.), Heidelberg: Springer-Verlag, 2003, 159-177.
8. 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. Physics, 1998, 39(12): 6459-6489.
9. Keener, J. P., Chaotic behavior in piecewise continuous difference equations, Trans. Amer. Math. Soc., 1980, 261: 589-604.
10. n, J. Math. Anal. Appl., 1978, 63: 199-223.
11. Wang, X. F., Chen, G., Chaotifying a stable map via smooth small-amplitude high-frequency feedback control, Int. J. Circ. Theory Appl., 2000, 28: 305-312.
12. Lin, W., Ruan, J., Zhao, W., On the mathematical clarification of the snap-back repeller in higher-dimensional systems and chaos in a discrete neural network model, Int. J. of Bifurcation and Chaos, 2002, 12(5): 1129-1139.
13. Albeverio, S., Nizhnik, I. L., Spatial chaos in a fourth-order nonlinear parabolic equation, Physics Letters A, 2001, 288: 299-304.
14. Kovacic, G., Wiggins, S., Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation, Physica D, 1992, 57: 185-225.
15. Krishnan, J., Kevrekidis, I. G., Or-Guil, M. et al., Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction-diffusion medium, Computer Methods in Applied Mechanics and Engineering, 1999, 170: 253-275.
16. Li, C., Mclaughlin, D., Morse and Melnikov functions for NLS Pde's, Comm. Math. Phys., 1994, 162: 175-214.
17. Li, Y., Mclaughlin, D. W., Homoclinic orbits and chaos in discretized perturbed NLS systems: Part I. Homoclinic orbits, J. Nonlinear Sci., 1997, 7: 211-269.
18. Li, Y., Wiggins, S., Homoclinic orbits and chaos in discretized perturbed NLS systems: Part II. Symbolic dynamics, J. Nonlinear Sci., 1997, 7: 315-370.
19. Shibata, H., Lyapunov exponent of partial differential equations, Physica A, 1999, 264: 226-233.
20. Torkelsson, U., Brandenburg, A., Chaos in accretion disk dynamos? Chaos, Solitons and Fractals, 1995, 5: 1975-1984.
21. Boyarsky, A., Góra, P., Lioubimov, V., Snap-back repellers and scrambled sets in general topological spaces, Nonlinear Analysis, 2001, 43: 591-604.
22. Shi, Y., Chen, G., Chaos for discrete dynamical systems in complete metric spaces, Chaos, Solitons and Fractals, 2004, 22: 555-571.
23. Pugachev, V. S., Sinitsyn, I. N., Lectures on Functional Analysis and Applications, Singapore: World Scientific Publishing, 1999.
24. Rudin, W., Functional Analysis, New York: McGraw-Hill, 1973.
25. Li, C., Chen, G., An improved version of the Marotto Theorem, Chaos, Solitons and Fractals, 2003, 18: 69-77; Erratum, same journal, 2003, in press.
26. Lang, S., Real and Functional Analysis, New York: Springer-Verlag, 1993.
27. Zhang, Z., Principles of Differential Dynamical Systems (in Chinese), 2nd ed., Beijing: Science Press, 1997.