Homoclinic tangles for noninvertible maps

Nonlinear Analysis, Theory, Methods and Applications

Volumn 41, Issue 1, 2000, Pages 259-276

  Sander, Evelyn ^a  

^a GEORGE MASON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CHAOS THEORY; ITERATIVE METHODS; MATHEMATICAL MODELS; MATHEMATICAL MORPHOLOGY; THEOREM PROVING;

DIFFEOMORPHISMS; HOMOCLINIC TANGLES; NONINVERTIBLE MAPS; TRANSVERSE HOMOCLINIC POINTS;

NONLINEAR EQUATIONS;

EID: 0034230845     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0362-546X(98)00277-6     Document Type: Article

References (26)

1. R. Adomaitis, I. Kevrekidis, Noninvertibility and the structure of basins of attraction in a model adaptive control system, J. Nonlinear Sci. 1 (1991) 95-105.
2. D.G. Aronson, M.A. Chory, G.R. Hall, R.P. McGehee, Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Communications in Mathematical Physics, vol. 83, Springer, Berlin, 1982, pp. 303-354.
3. F. Batteli, C. Lazzari, On the bifurcation from critical homoclinic orbits in n-dimensional maps, Discrete Continuous Dyn. Systems 3 (2) (1997) 289-303.
4. R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Reading, MA, 1989.
5. C.E. Frouzakis, R.A. Adomaitis, I.G. Kevrekidis, M.P. Golden, B.E. Ydstie, The structure of basin boundaries in a simple adaptive control system, in: T. Bountis (Ed.), Chaotic Dynamics: Theory and Practice, Plenum Press, New York, 1992, pp. 195-210.
6. C.E. Frouzakis, R.A. Adomaitis, I.G. Kevrekidis, On the dynamics and global stability characteristics of an adaptive control system, Comp. Chem. Eng. 1996.
7. C.E. Frouzakis, L. Gardini, I.G. Kevrekidis, G. Millerioux, C. Mira, On some properties of invariant sets of two-dimensional noninvertible maps, Int. J. Bifur. Chaos Appl. Sci. Eng. 7 (6) (1997) 1167-1194.
8. L. Gardini, Bifurcations and multistability in a class of two-dimensional endomorphisms, Nonlinear Anal. 28 (1) (1997) 61-85.
9. L. Gardini, Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding periodic points, Nonlinear Anal. 23 (8) (1994) 1039-1089.
10. L. Gardini, Homoclinic orbits of saddles in two-dimensional endomorphisms, Iteration Theory (Batschuns, 1992), World Sci. Publishing, River Edge, NJ, 1996, pp. 81-99.
11. S.V. Gonchenko, L.P. Shil'nikov, D.V. Turaev, On models with non-rough Poincaré homoclinic curves, Physica D 62 (1993) 1-14.
12. I. Gumowski, C. Mira, Recurrences and Discrete Dynamic Systems, Springer, Berlin, 1980.
13. L.M. Lerman, L.P. Shil'nikov, Homoclinic structures in infinite-dimensional systems, Sibirskii Matematicheskii Zhurnal 29 (3) (1988) 92-103.
14. E.N. Lorenz, Computational chaos - A prelude to computational instability, Physica D 25 (1989) 299-317.
15. n, J. Math. Anal. Appl. 63 (1978) 199-223.
16. R. Rico-Martínez, I.G. Kevrekidis, R.A. Adomaitis, Noninvertible Dynamics in Neural Network Models, Preprint.
17. R. McGehee, E. Sander, A new proof of the stable manifold theorem, Z. Angew. Math. Phys. 47 (1996) 497-513.
18. S. Newhouse, J. Palis, Bifurcations of Morse-Smale dynamical systems, in: M.M. Peixoto (Ed.), Dynamical Systems, Sec. 3. Academic Press, New York, 1973, pp. 312-320.
19. J. Palis, F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993.
20. C. Robinson, Dynamical Systems, CRC Press, Boca Raton, 1995.
21. E. Sander, Hyperbolic Sets for Noninvertible Maps and Relations, Ph.D. Thesis, University of Minnesota, June, 1996.
22. E. Sander, Hyperbolic Sets for Noninvertible Maps and Relations, CDSNS96 Preprint 254, Georgia Tech., 1996, submitted.
23. M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969) 175-199.
24. H. Steinlein, H.-O. Walther, Hyperbolic sets and shadowing for noninvertible maps, Advanced Topics in the Theory of Dynamical Systems, Academic Press, New York, 1989.
25. 1-maps in banach spaces, J. Dyn. Differential Equations 2 (3) (1990) 325-365.
26. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Texts in Applied Mathematics 2, Springer, Berlin, 1990.