On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies
- Gonchenko, Sergey V.
- Gonchenko, Vladimir S.
2010 Mathematics Subject Classification
- 58F12 58F13
- homoclinic tangency, invariant curve, Andronov-Hopf bifurcation, strange attractors, Newhouse regions
The bifurcation of the birth of a closed invariant curve in the two-parameter unfolding of a two-dimensional diffeomorphism with a homoclinic tangency of invariant manifolds of a hyperbolic fixed point of neutral type (i.e. such that the Jacobian at the fixed point equals to 1) is studied. The existence of periodic orbits with multipliers e±iψ (0 < ψ < π) is proved and the first Lyapunov value is computed. It is shown that, generically, the first Lyapunov value is non-zero and its sign coincides with the sign of some separatrix value (i.e. a function of coefficients of the return map near the global piece of the homoclinic orbit).