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Equivariant Sternberg-Chen theorem, Journal of Dynamics and Differential Equations, in press. 6. Nauk 30, 101-171 (1975). 7. , DE LA LLAVE R. , Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem, Journal Geom. Anal. 6, 613-649 (1996). 8. BRONSTEIN I. , Normal forms of vector fields satisfying certain geometric conditions, In: Nonlinear Dynamical Systems and Chaos. Birkhauser, Basel, 79-101 (1996). 28 9. , Stability of C°°-mapping. Math. 87 (1968), 89-104.

Nerve axon equations, IV: The stable and unstable impulse. Indiana Univ. Math. , 24:1169-1190,1975. 11. R. A. Gardner & K. Zumbrun. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51:797855, 1998. 12. M. Grillakis, J. Shatah & W. Strauss. Stability theory of solitary waves in the presence of symmetry, I. J. Fund. , 74:160-197, 1987. 13. M. Grillakis, J. Shatah & W. Strauss. Stability theory of solitary waves in the presence of symmetry, II.

We establish the existence of certain types of heteroclinic cycle in these cases by making use of the concept of a subcycle. We also discuss edge cycles, and a generalisation of heteroclinic cycles which we call a heteroclinic web. We apply our methods to three examples. The work briefly reported here was published in: Dynamics and Stability of Systems 15, 353-385 (2000). A nonlinear dynamical system possesses a heteroclinic cycle if it has a series of equilibria that are connected in the sense that some trajectory links the unstable manifold of any given equilibrium to the stable manifold of the next.

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A. I. Maltsevs problem on operations on groups by Ol'shanskii A. Y.


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