188 research outputs found
Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps
Invariant tori play a fundamental role in the dynamics of symplectic and
volume-preserving maps. Codimension-one tori are particularly important as they
form barriers to transport. Such tori foliate the phase space of integrable,
volume-preserving maps with one action and angles. For the area-preserving
case, Greene's residue criterion is often used to predict the destruction of
tori from the properties of nearby periodic orbits. Even though KAM theory
applies to the three-dimensional case, the robustness of tori in such systems
is still poorly understood. We study a three-dimensional, reversible,
volume-preserving analogue of Chirikov's standard map with one action and two
angles. We investigate the preservation and destruction of tori under
perturbation by computing the "residue" of nearby periodic orbits. We find tori
with Diophantine rotation vectors in the "spiral mean" cubic algebraic field.
The residue is used to generate the critical function of the map and find a
candidate for the most robust torus.Comment: laTeX, 40 pages, 26 figure
Hopf bifurcation with non-semisimple 1:1 resonance
A generalised Hopf bifurcation, corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance), is analysed using a normal form approach. This bifurcation has linear codimension-3, and a centre subspace of dimension 4. The four-dimensional normal form is reduced to a three-dimensional system, which is normal to the group orbits of a phase-shift symmetry. There may exist 0, 1 or 2 small-amplitude periodic solutions. Invariant 2-tori of quasiperiodic solutions bifurcate from these periodic solutions. The authors locate one-dimensional varieties in the parameter space 1223 on which the system has four different codimension-2 singularities: a Bogdanov-Takens bifurcation a 1322 symmetric cusp, a Hopf/Hopf mode interaction without strong resonance, and a steady-state/Hopf mode interaction with eigenvalues (0, i,-i)
Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach
We perform a bifurcation analysis of normal–internal resonances in parametrised families of quasi–periodically forced
Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the ‘backbone’
system; forced, the system is a skew–product flow with a quasi–periodic driving with basic frequencies. The
dynamics of the forced system are simplified by averaging over the orbits of a linearisation of the unforced system. The
averaged system turns out to have the same structure as in the well–known case of periodic forcing ; for a real
analytic system, the non–integrable part can even be made exponentially small in the forcing strength. We investigate
the persistence and the bifurcations of quasi–periodic –dimensional tori in the averaged system, filling normal–internal
resonance ‘gaps’ that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary
resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of ‘gaps within
gaps’ makes the quasi–periodic case more complicated than the periodic case
Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio
We study the exponentially small splitting of invariant manifolds of
whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable
Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a
torus whose frequency ratio is the silver number . We show
that the Poincar\'e-Melnikov method can be applied to establish the existence
of 4 transverse homoclinic orbits to the whiskered torus, and provide
asymptotic estimates for the tranversality of the splitting whose dependence on
the perturbation parameter satisfies a periodicity property. We
also prove the continuation of the transversality of the homoclinic orbits for
all the sufficiently small values of , generalizing the results
previously known for the golden number.Comment: 17 pages, 2 figure
Quasi-periodic stability of normally resonant tori
We study quasi-periodic tori under a normal-internal resonance, possibly with
multiple eigenvalues. Two non-degeneracy conditions play a role. The first of
these generalizes invertibility of the Floquet matrix and prevents drift of the
lower dimensional torus. The second condition involves a Kolmogorov-like
variation of the internal frequencies and simultaneously versality of the
Floquet matrix unfolding. We focus on the reversible setting, but our results
carry over to the Hamiltonian and dissipative contexts
Numerical computation of travelling breathers in Klein-Gordon chains
We numerically study the existence of travelling breathers in Klein-Gordon
chains, which consist of one-dimensional networks of nonlinear oscillators in
an anharmonic on-site potential, linearly coupled to their nearest neighbors.
Travelling breathers are spatially localized solutions having the property of
being exactly translated by sites along the chain after a fixed propagation
time (these solutions generalize the concept of solitary waves for which
). In the case of even on-site potentials, the existence of small
amplitude travelling breathers superposed on a small oscillatory tail has been
proved recently (G. James and Y. Sire, to appear in {\sl Comm. Math. Phys.},
2004), the tail being exponentially small with respect to the central
oscillation size. In this paper we compute these solutions numerically and
continue them into the large amplitude regime for different types of even
potentials. We find that Klein-Gordon chains can support highly localized
travelling breather solutions superposed on an oscillatory tail. We provide
examples where the tail can be made very small and is difficult to detect at
the scale of central oscillations. In addition we numerically observe the
existence of these solutions in the case of non even potentials
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