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# Dependence of the Spectral Relation of Double Stars Upon by Perrine C. D. By Perrine C. D.

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However, we prefer to state the main result in a bit more general framework. Namely, the very well known Lyapunov's Theorem 12 ensures that if we have an n-degrees of freedom analytic Hamiltonian system, with the origin being an equilibrium and eigenvalues ±Aj, ±A2,. • •, ±A n satisfying that • Aj is purely imaginary, • none of the quocients j 2 , . . ^ is an integer, then there exists a one-parameter family of periodic orbits accumulating to the origin. In other words, one can prove the existence of a transformation leading such system into (Birkhoff) normal form with respect to the variables associated to the imaginary eigenvalue Ai In our context, we have the following result, which represents the simplest situation where Lyapunov's Theorem would apply, that is, when the 48 spectrum of the differential of the field at the equilibrium consists on two pairs of eigenvalues.

Proc. Roy. Soc. Lond. A, 344:363-374,1975. 4. L. L. Sachs. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm. Math. , 118:15-29,1988. 37 5. J. Bridges. Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc, 121: 147-190,1997. 6. J. Bridges and G. Derks. Hodge duality and the Evans function. Phys. Lett. A, 251:363-372,1999. 7. J. Bridges and G. Derks. Unstable eigenvalues, and the linearisation about solitary waves and fronts with symmetry.

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. 10. , VARCHENKO A. 1. Birkhauser (1985). 11. , Mathematical Methods in Classical Mechanics, SpringerVerlag, New York (1978). 12. KATOK A. , Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press (1995). it Topological instability of action variables in multidimensional nearly integrable Hamiltonian systems is known as Arnold Diffusion.