By Rachkovskii D.A., Slipchenko S.V.

Difficulties (inherent in compound hierarchical representations) of keeping details at the constitution of coded information and techniques of fixing them through "binding" of data substructures are mentioned. one of those dispensed representations is taken into account during which info are coded by means of multidimensional binary sparse vectors. the restrictions imposed on a binding approach are formulated and its implementation is equipped. a few examples of illustration of dependent facts are thought of.

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1 Overview The starting point is a Hamiltonian H 0 with an equilibrium at the origin. It is supposed to be close to some resonance of the form p : q, and to depend on several coeﬃcients ai . Optionally, the system may be invariant (or reversing) under some symmetry group Γ , which is supposed to respect the symplectic structure. ) The ﬁrst step is to apply the Birkhoﬀ procedure around the resonance, resulting in a system H n which has acquired a (formal) S1 -symmetry. This step singles out a detuning parameter denoted by b1 , which measures the deviation from the resonance around which the Birkhoﬀ procedure is performed.

G. [AM78, Bro79, CS85, CB97, Mee85, Tak74b]. The idea is to divide out the symmetry, and regard the associated conserved quantities as parameters (also called integrals), a procedure known as orbit space reduction. Sometimes this reduction is done on the entire phase space, and sometimes on each leaf of the foliation deﬁned by the levels of the integrals. , when some points have nontrivial isotropy group); see [AM78, CS85] for details. The result is a reduced Hamiltonian system with one degree of freedom, and whose dynamics coincides with the projection of the dynamics of the original system onto the orbit space.

In Sect. 1 we give a necessary and suﬃcient condition for a deformation to be versal. It amounts to solvability of the well-known inﬁnitesimal stability equation2 adapted to our equivariant context. In the case of the 1 : 2 resonance, the central singularity is isomorphic to x(x2 + y 2 ), with a symmetry group Z2 acting on R2 via (x, y) → (x, −y). 4) g(x, y) = α1 (x, y)x ∂f ∂f ∂f + α2 (x, y)y 2 + α3 (x, y)y + u1 x + u2 y 2 . ∂x ∂x ∂y Here f = x(x2 + y 2 ) is the central singularity. For this f the condition is indeed satisﬁed; see Sect.