NLS Bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

We consider the bifurcations of standing wave solutions to the nonlinear Schr\"odinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called dumbbell, recently studied by Marzuola and Pelinovsky. The authors of that study found the ground state undergoes two bifurcations, first a symmetry-breaking, and the second which they call a symmetry-preserving bifurcation. We clarify the type of the symmetry-preserving bifurcation, showing it to be transcritical. We then reduce the question, and show that the phenomena described in that paper can be reproduced in a simple discrete self-trapping equation on a combinatorial graph of bowtie shape. This allows for complete analysis both by geometric methods and by parameterizing the full solution space. We then expand the question, and describe the bifurcations of all the standing waves of this system, which can be classified into three families, and of which there exists a countably infinite set

Standing wave instabilities in a chain of nonlinear coupled oscillators

We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schroedinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with nontrivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than pi/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes.Comment: 57 pages, 21 figures, to be published in Physica D. Revised version: Figs. 5 and 12 (f) replaced, some new results added to Sec. 5, Sec.7 (Conclusions) extended, 3 references adde

Deformation of geometry and bifurcation of vortex rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.Comment: 26 page

Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.Comment: 36 pages, 10 figures, accepted on International Journal of Bifurcation and Chaos. arXiv admin note: substantial text overlap with arXiv:1401.285

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 $p$ sites along the chain after a fixed propagation time $T$ (these solutions generalize the concept of solitary waves for which $p=1$). 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

An energy-momentum map for the time-reversal symmetric 1:1 resonance with Z_2 X Z_2 symmetry

We present a general analysis of the bifurcation sequences of periodic orbits in general position of a family of reversible 1:1 resonant Hamiltonian normal forms invariant under $\Z_2\times\Z_2$ symmetry. The rich structure of these classical systems is investigated both with a singularity theory approach and geometric methods. The geometric approach readily allows to find an energy-momentum map describing the phase space structure of each member of the family and a catastrophe map that captures its global features. Quadrature formulas for the actions, periods and rotation number are also provided.Comment: 22 pages, 3 figures, 1 tabl

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe