88 research outputs found
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
with positive and negative cubic term. These maps show up
different bifurcation structures both for fixed points with eigenvalues
and for 4-periodic orbits. While for 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
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 . 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
On bifurcations of symmetric elliptic orbits
We study bifurcations of symmetric elliptic fixed points in the case of
\emph{p}:\emph{q} resonances with odd . We consider the case where the
initial area-preserving map possesses the
central symmetry, i.e. is invariant under the change , .
We construct normal forms for such maps in the case , where and are mutually prime integer numbers,
and is odd, and study local bifurcations of the fixed point in
various settings. We prove the appearance of garlands consisting of four
-periodic orbits, two orbits are elliptic and two orbits are saddle, and
describe the corresponding bifurcation diagrams for one- and two-parameter
families. We also consider the case where the initial map is reversible and
find conditions when non-symmetric periodic orbits of the garlands are
non-conservative (compose symmetric pairs of stable and unstable orbits as well
as area-contracting and area-expanding saddles).Comment: 23 pages, 4 figure
Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation
We study bifurcations of a three-dimensional diffeomorphism, , that has
a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers
(\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma), where
and . We show that in a
three-parameter family, g_{\eps}, of diffeomorphisms close to , 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
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