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### Open Access Journals

DCDS-B

Observational and model based studies provide ample evidence for the presence of
multidecadal variability in the North Atlantic sea-surface temperature known as
the Atlantic Multidecadal Oscillation (AMO). This variability is characterised
by a multidecadal time scale, a westward propagation of temperature anomalies,
and a phase difference between the anomalous meridional and zonal overturning
circulations.

We study the AMO in a low-order model obtained by projecting a model for thermally driven ocean flows onto a 27-dimensional function space. We study bifurcations of attractors by varying the equator-to-pole temperature gradient ($\Delta T$) and a damping parameter ($\gamma$).

For $\Delta T = 20^\circ$C and $\gamma = 0$ the low-order model has a stable equilibrium corresponding to a steady ocean flow. By increasing $\gamma$ to 1 a supercritical Hopf bifurcation gives birth to a periodic attractor with the spatio-temporal signature of the AMO. Through a period doubling cascade this periodic orbit gives birth to Hénon-like strange attractors. Finally, we study the effects of annual modulation by introducing a time-periodic forcing. Then the AMO appears through a Hopf-Neĭmark-Sacker bifurcation. For $\Delta T = 24^\circ$C we detected at least 11 quasi-periodic doublings of the invariant torus. After these doublings we find quasi-periodic Hénon-like strange attractors.

We study the AMO in a low-order model obtained by projecting a model for thermally driven ocean flows onto a 27-dimensional function space. We study bifurcations of attractors by varying the equator-to-pole temperature gradient ($\Delta T$) and a damping parameter ($\gamma$).

For $\Delta T = 20^\circ$C and $\gamma = 0$ the low-order model has a stable equilibrium corresponding to a steady ocean flow. By increasing $\gamma$ to 1 a supercritical Hopf bifurcation gives birth to a periodic attractor with the spatio-temporal signature of the AMO. Through a period doubling cascade this periodic orbit gives birth to Hénon-like strange attractors. Finally, we study the effects of annual modulation by introducing a time-periodic forcing. Then the AMO appears through a Hopf-Neĭmark-Sacker bifurcation. For $\Delta T = 24^\circ$C we detected at least 11 quasi-periodic doublings of the invariant torus. After these doublings we find quasi-periodic Hénon-like strange attractors.

DCDS-B

We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedom
corresponding to fast motion and the other one corresponding to slow motion. The Hamiltonian
function is the sum of potential and kinetic energies, the kinetic energy being a weighted sum
of squared momenta. The ratio of time derivatives of slow and fast variables is of order
$\epsilon $«$ 1$. At frozen values of the slow variables there is a separatrix on the phase
plane of the fast variables and there is a region in the phase space (the domain of
separatrix crossings) where the projections of phase points onto the plane of the fast
variables repeatedly cross the separatrix in the process of evolution of the slow
variables. Under a certain symmetry condition we prove the existence of many, of order
$1/\epsilon$, stable periodic trajectories in the domain of the separatrix crossings. Each of
these trajectories is surrounded by a stability island whose measure is estimated from
below by a value of order $\epsilon$. Thus, the total measure of the stability islands is
estimated from below by a value independent of $\epsilon$. We find the location of stable
periodic trajectories and an asymptotic formula for the number of these trajectories. As
an example, we consider the problem of motion of a charged particle in the parabolic
model of magnetic field in the Earth magnetotail.

DCDS

We study arbitrary generic unfoldings of a Hopf-zero
singularity of codimension two. They can be written in the following
normal form:
\begin{eqnarray*}
\left\{
\begin{array}{l}
x'=-y+\mu x-axz+A(x,y,z,\lambda,\mu)
\\
y'=x+\mu y-ayz+B(x,y,z,\lambda,\mu)
\\
z'=z^2+\lambda+b(x^2+y^2)+C(x,y,z,\lambda,\mu),
\end{array}
\right.
\end{eqnarray*}
with $a>0$, $b>0$ and where $A$, $B$, $C$ are $C^\infty$ or $C^\omega$ functions of order
$O(\|(x,y,z,\lambda,\mu)\|^3)$.

Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopf-zero singularities has been discussed previously in the literature, no result valid for arbitrary generic unfoldings is available. In this paper we present new techniques to study global bifurcations from Hopf-zero singularities. They allow us to obtain a general criterion for the existence of Shilnikov homoclinic bifurcations and also provide a detailed description of the bifurcation set. Criteria for the existence of Bykov cycles are also provided. Main tools are a blow-up method, including a related invariant theory, and a novel approach to the splitting functions of the invariant manifolds. Theoretical results are applied to the Michelson system and also to the so called extended Michelson system. Paper includes thorough numerical explorations of dynamics for both systems.

Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopf-zero singularities has been discussed previously in the literature, no result valid for arbitrary generic unfoldings is available. In this paper we present new techniques to study global bifurcations from Hopf-zero singularities. They allow us to obtain a general criterion for the existence of Shilnikov homoclinic bifurcations and also provide a detailed description of the bifurcation set. Criteria for the existence of Bykov cycles are also provided. Main tools are a blow-up method, including a related invariant theory, and a novel approach to the splitting functions of the invariant manifolds. Theoretical results are applied to the Michelson system and also to the so called extended Michelson system. Paper includes thorough numerical explorations of dynamics for both systems.

DCDS

Non-integrability criteria, based on differential Galois theory and requiring
the use of higher order variational equations (VE

_{k}), are applied to prove the non-integrability of the Swinging Atwood's Machine for values of the parameter which can not be decided using first order variational equations (VE_{1}).
DCDS-B

We study asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view. Using analytic information, we conjecture the basis of functions of an asymptotic expansion and then extract actual values of the coefficients of the asymptotic series numerically. The computations are performed with high-precision arithmetic, which involves up to several thousands of decimal digits. This approach allows us to obtain information which is usually considered to be out of reach of numerical methods. In particular, we use our results to test that the asymptotic series are Gevrey-1 and to study positions and types of singularities of their Borel transform. Our examples are based on generalisations of the standard and Hénon maps.

DCDS-B

We discuss the problem of existence of elliptic periodic trajectories inside lobes bounded by segments of stable and unstable separatrices of a hyperbolic fixed point. We show that such trajectories generically exist in symplectic maps arbitrary close to integrable ones. Elliptic periodic trajectories as a rule, generate stability islands. The area of such an island is of the same order as the lobe area, but the quotient of areas can be very small. Numerical examples are included.

DCDS

The problem of three bodies with equal masses in $\mathbb{S}^2$ is known to have
Lagrangian homographic orbits. We study the linear stability and also a
"practical'' (or effective) stability of these orbits on the unit sphere.

DCDS

We consider some simple Hamiltonian systems, variants or generalizations of the
Hénon-Heiles system, in two and three degrees of freedom, around a positive
definite elliptic point, in resonant and non-resonant cases. After reviewing
some theoretical background, we determine a measure of the domain of chaoticity
by looking at the frequency of positive Lyapunov exponents in a sample of
initial conditions. The question we study is how this measure depends on the
energy and parameters and which are the dynamical objects responsible for the
observed behaviour.

DCDS-B

The purpose of this paper is to develop a numerical procedure for the
determination of frequencies and amplitudes of a quasi--periodic
function, starting from equally-spaced samples of it on a finite time
interval. It is based on a collocation method in frequency domain.
Strategies for the choice of the collocation harmonics are discussed,
in order to ensure good conditioning of the resulting system of
equations. The accuracy and robustness of the procedure is checked
with several examples. The paper is ended with two applications of its
use as a dynamical indicator. The theoretical support for the method
presented here is given in a companion paper [21].

DCDS-B

This paper focuses on the parametric abundance and the 'Cantorial'
persistence under perturbations of a recently discovered class of strange
attractors for diffeomorphisms, the so-called quasi-periodic
Hénon-like. Such attractors were first detected in the Poincaré map of a
periodically driven model of the atmospheric flow: they were characterised
by marked quasi-periodic intermittency and by
$\Lambda_1>0,\Lambda_2\approx0$, where $\Lambda_1$ and $\Lambda_2$ are the
two largest Lyapunov exponents. It was also conjectured that
these attractors coincide with the closure of the unstable manifold of a
hyperbolic invariant circle of saddle-type.

This type of attractor is here investigated in a model map of the solid torus, constructed by a skew coupling of the Hénon family of planar maps with the Arnol$'$d family of circle maps. It is proved that Hénon-like strange attractors occur in certain parameter domains. Numerical evidence is produced, suggesting that quasi-periodic circle attractors and quasi-periodic Hénon-like attractors persist in relatively large subsets of the parameter space. We also discuss two problems in the numerical identification of so-called strange nonchaotic attractors and the persistence of all these classes of attractors under perturbation of the skew product structure.

This type of attractor is here investigated in a model map of the solid torus, constructed by a skew coupling of the Hénon family of planar maps with the Arnol$'$d family of circle maps. It is proved that Hénon-like strange attractors occur in certain parameter domains. Numerical evidence is produced, suggesting that quasi-periodic circle attractors and quasi-periodic Hénon-like attractors persist in relatively large subsets of the parameter space. We also discuss two problems in the numerical identification of so-called strange nonchaotic attractors and the persistence of all these classes of attractors under perturbation of the skew product structure.

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