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By Magri F., Morosi C.

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Extra info for A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds (Quaderno 19-1984, Univ. of Milan)

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Bifurcations of Cycles in Generic Two-Parameter Families with an Additional Simple Degeneracy This section begins with a list of degeneracies that occur in generic twoparameter families of germs of diffeomorphisms at a fixed point, corresponding to isolated values of the parameters. The bifurcations of fixed points with multipliers + 1 or - 1 and with an additional degeneracy in the nonlinear terms remind one of bifurcations of singular points with eigenvalue O. In contrast, in the case of a pair of complex conjugate multipliers with an additional degeneracy in the nonlinear terms, along with the appearance of closed invariant curves, the bifurcations lead to completely new effects.

It is convenient to demonstrate the ways stability is lost by investigating a mapping of an annulus into itself, which for the initial values of the parameters contains a smooth invariant curve. The concrete form of the mapping is immaterial; for example it may be as in Afrajmovich and Shil'nikov (1983) or as in Sect. 5 of Chap. 3. Hence, we show only the geometrical picture (Fig. 20). In this drawing the annulus is shown as a rectangle, the left and right sides of which are identified and which consists of points of the stable manifold of a fixed point on the boundary: a saddle-node in Fig.

But its degree of smoothness tends to infinity as e -+ o. Theorem (Newhouse et al. (1983». If two generic one-parameter deformations of germs of diffeomorphisms (1R2, 0) -+ (1R2, 0) with a pair of complex multipliers on the unit circle are topologically equivalent, then the multipliers of the germs being deformed coincide. This theorem follows from the topological invariance of the rotation number for a diffeomorphism of the circle. 4. Nonlocal Bifurcations in One-Parameter Families of Diffeomorphisms.

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A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds (Quaderno 19-1984, Univ. of Milan) by Magri F., Morosi C.


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