Homoclinic twisting bifurcations and cusp horseshoe maps

Afraimovich, V. S., Bykov, V., and ?ilnikov, L. P. (1983). On structurally unstable attracting limit sets of Lorenz attractor type.Trans. Moscow Math. Soc. 2, 153?216.

Atwater, I., Dawson, C. M., Scott, A., Eddlestone, G., and Rojas, E. (1980). The nature of the oscillatory behavior in electrical activity for the pancreatic beta cell. InBiochemistry and Biophysics of the Pancreatic Beta Cell, Georg Thieme, New York, pp. 100?107.

Chay, T. R., and Keizer, J. (1983). Minimal model for membrane oscillations in the pancreatic ?-cell.Biophys. J. 42, 181?190.

Chay, T. R., and Rinzel, J. (1985). Bursting, beating, and chaos in an excitable membrane model.Biophys. J. 47, 357?366.

Chow, S.-N., Deng, B., and Fiedler, B. (1990). Homoclinic bifurcation at resonant eigenvalues.J. Dynam. Diff. Eg. 2, 177?244.

Chow, S.-N., and Hale, J. K. (1982).Methods of Bifurcation Theory, Springer-Verlag, New York.

Deng, B. (1990a). On ?ilnikov's homoclinic-saddle-focus theorem.J. Diff. Eq. 102, 305?329.

Deng, B. (199b). Symbolic dynamics for chaotic systems. Preprint, University of Nebraska-Lincoln.

Deng, B. (1991a). The bifurcations of countable connections from a twisted heteroclinic loop.SIAM J. Math. Anal. 22, 653?679.

Deng, B. (1991b). The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations.SIAM J. Math. Anal. 22, 1631?1650.

Deng, B. (1991c). Constructing homoclinic orbits and chaos. Preprint, University of Nebraska-Lincoln.

Ermentrout, B. (1988).Phase Plane, Version 3.0, Brooks/Cole.

Kokubu, H., Nishiura, Y., and Oka, H. (1988). Heteroclinic and homoclinic bifurcation in bistable reaction diffusion systems. Preprint KSU/ICS, 88-08.

Moser, J. (1973).Stable and Random Motions in Dynamical Systems, Ann. Math. Stud. 77, Princeton University Press, Princeton, NJ.

Munkres, J. R. (1975).Topology, a First Course, Prentice-Hall, Englewood Cliffs, NJ.

Neimark, Y. I. (1972).The Method of Point-to-Point Maps in the Theory of Nonlinear Oscillations, Nauka, Moscow, (in Russian).

Rinzel, J. (1987). A formal classification of bursting mechanisms in excitable systems. In Cleason, A. M. (ed.),Proceedings of the International Congress of Mathematics, AMS, pp. 1578?1593.

Rössler, O. E. (1979). Continuous chaos?four prototype equations. In Gurel, O., and Rössler, O. E. (eds.),Bifurcation Theory and Applications in Scientific Disciplines, Annals of the New York Academy of Science No. 316, pp. 376?392.

Royden, H. L. (1988).Real Analysis, 3rd ed., Macmillan, New York.

?ilnikov, L. P. (1965). A case of the existence of a countable number of periodic motions.Soviet Math. Dokl. 6, 163?166.

?ilnikov, L. P. (1967). On a Poincaré-Berkhoff problem.Math. USSR Sb. 3, 353?371.

Silnikov, L. P. (1968). On the generation of a periodic motion from a trajectories doubly asymptotic to an equilibrium state of saddle type.Math. USSR Sbornik 10, 91?102.

Silnikov, L. P. (1969). On a new type of bifurcation of multidimensional dynamical systems.Soviet Math. Dokl. 10, 1368?1371.

Smale, S. (1967). Differentiable dynamical systems.Bull. AMS 73, 747?817.

Terman, D. (1991). The transition from bursting to continuous spiking in excitable membrane models. Preprint, Ohio State University.

Yanagida, E. (1987). Branching of double pulse solutions from single pulse solutions in nerve axon equations.J.D.E. 66, 243?262.