On local and global aspects of the 1:4 resonance in the conservative cubic H\'enon maps

We study the 1:4 resonance for the conservative cubic H\'enon maps $\mathbf{C}_\pm$ with positive and negative cubic term. These maps show up different bifurcation structures both for fixed points with eigenvalues $\pm i$ and for 4-periodic orbits. While for $\mathbf{C}_-$ the 1:4 resonance unfolding has the so-called Arnold degeneracy (the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient), the map $\mathbf{C}_+$ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by $\pi/4$. For both maps several bifurcations are detected and illustrated.Comment: 21 pages, 13 figure

On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies

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)

Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits

Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation

We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers (\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma), where $0<\lambda<1<|\gamma|$ and $|\lambda^2\gamma|=1$. We show that in a three-parameter family, g_{\eps}, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near \eps =0 where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional H\'enon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional H\'enon maps occupy in the class of quadratic volume-preserving diffeomorphisms.Comment: laTeX, 25 pages, 6 eps figure

On local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps

We study the 1:4 resonance for the conservative cubic Henon maps C6 with positive and negative cubic terms. These maps show up different bifurcation structures both for fixed points with eigenvalues 6i and for 4-periodic orbits. While for C-, the 1:4 resonance unfolding has the so-called Arnold degeneracy [the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient], the map Cþ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by p/4. For both maps, several bifurcations are detected and illustrated