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On bifurcations in a two-parameter family of conservative mappings close to the Hénon map. (English. Russian original) Zbl 0727.58028

Sel. Math. Sov. 9, No. 3, 273-282 (1990); translation from Methods of the qualitative theory of differential equations, Gor’kij, 10-24 (1987).
The author studies the two-parameter family of area-preserving maps \[ (x,y)\to (a+2x+y-x^ 2,-x)+\mu (f_ 1,f_ 2),\quad \mu \quad small. \] For \(a=\mu =0\) the map has the fixed point (0,0) with eigenvalues \(\lambda_{1,2}=1\). This degenerate fixed point bifurcates to a saddle- type fixed point and to an elliptic-type fixed point \(O^{el}\). The qualitative picture of the neighbourhood of \(O^{el}\) depends on its eigenvalues. If they are nonresonant or the order of resonance k is \(\geq 5\) then \(O^{el}\) is stable in the Lyapunov sense. If \(k=3\) then \(O^{el}\) is unstable (Bryuno). Special attention is devoted to the case \(k=4\). By means of some reductions and normal forms, the analysis is reduced to the study of a two-parameter family of Hamiltonians \[ H(\rho,\phi)=\epsilon_ 1\rho +\rho^ 2[\epsilon_ 2+1+\cos (4\phi)+A\rho +B\rho \cos (4\phi +\alpha)], \] where the \(\epsilon_ i\) are small. This is a new result in this paper, because earlier only one- parameter bifurcations of the Hamiltonians \(\epsilon_ 1\rho +A\rho^ 2+\rho^ 2\cos (4\phi)\), \(| A| \neq 1\), have been investigated (by Bryuno).

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
34D20 Stability of solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37B99 Topological dynamics