Comparing the direct normal form method with harmonic balance and the method of multiple scales

Approximate analytical methods have been used extensively for finding approximate solutions to nonlinear ordinary differential equations. In this paper we compare the recently developed direct normal form transformation with two other very well known and long standing methods, harmonic balance and the method of multiple scales. We will show that the direct normal form method combines some of the key advantages of harmonic balance and multiple scales whilst reducing some of the limitations

An analytical approach for detecting isolated periodic solution branches in weakly nonlinear structures

AbstractThis paper considers isolated responses in nonlinear systems; both in terms of isolas in the forced responses, and isolated backbone curves (i.e. the unforced, undamped responses). As isolated responses are disconnected from other response branches, reliably predicting their existence poses a significant challenge. Firstly, it is shown that breaking the symmetry of a two-mass nonlinear oscillator can lead to the breaking of a bifurcation on the backbone curves, generating an isolated backbone. It is then shown how an energy-based, analytical method may be used to compute the points at which the forced responses cross the backbone curves at resonance, and how this may be used as a tool for finding isolas in the forced responses. This is firstly demonstrated for a symmetric system, where an isola envelops the secondary backbone curves, which emerge from a bifurcation. Next, an asymmetric configuration of the system is considered and it is shown how isolas may envelop a primary backbone curve, i.e. one that is connected directly to the zero-amplitude solution, as well as the isolated backbone curve. This is achieved by using the energy-based method to determine the relationship between the external forcing amplitude and the positions of the crossing points of the forced response. Along with predicting the existence of the isolas, this technique also reveals the nature of the responses, thus simplifying the process of finding isolas using numerical continuation

Flavour Democracy in Strong Unification

We show that the fermion mass spectrum may naturally be understood in terms of flavour democratic fixed points in supersymmetric theories which have a large domain of attraction in the presence of "strong unification". Our approach provides an alternative to the approximate Yukawa texture zeroes of the Froggatt-Nielsen mechanism. We discuss a particular model based on a broken gauged $SU(3)_L\times SU(3)_R$ family symmetry which illustrates our approach.Comment: 21 Pages plain latex; includes 5 eps figure

The Significance of Nonlinear Normal Modes for Forced Responses

Nonlinear normal modes (NNMs) describe the unforced and undamped periodic responses of nonlinear systems. NNMs have proven to be a valuable tool, and are widely used, for understanding the underlying behaviour of nonlinear systems. They provide insight into the types of behaviour that may be observed when a system is subjected to forcing and damping, which is ultimately of primary concern in many engineering applications. The definition of an NNM has seen a number of evolutions, and the contemporary definition encompasses all periodic responses of a conservative system. Such a broad definition is essential, as it allows for the wide variety of responses that nonlinear systems may exhibit. However, it may also lead to misleading results, as some of the NNMs of a system may represent behaviour that will only be observed under very specific forcing conditions, which may not be realisable in any practical scenario. In this paper, we investigate how the significance of NNMs may differ and how this significance may be quantified. This is achieved using an energy-based method, and is validated using numerical simulations

On the stability of the primordial closed string gas

We recast the study of a closed string gas in a toroidal container in the physical situation in which the single string density of states is independent of the volume because energy density is very high. This includes the gas for the well known Brandenberger-Vafa cosmological scenario. We describe the gas in the grandcanonical and microcanonical ensembles. In the microcanonical description, we find a result that clearly confronts the Brandenberger-Vafa calculation to get the specific heat of the system. The important point is that we use the same approach to the problem but a different regularization. By the way, we show that, in the complex temperature formalism, at the Hagedorn singularity, the analytic structure obtained from the so-called F-representation of the free energy coincides with the one computed using the S-representation.Comment: 20 pages and 1 figure. The final version that appeared in JHE

Spin Coulomb drag in the two-dimensional electron liquid

We calculate the spin-drag transresistivity $\rho_{\uparrow \downarrow}(T)$ in a two-dimensional electron gas at temperature $T$ in the random phase approximation. In the low-temperature regime we show that, at variance with the three-dimensional low-temperature result [$\rho_{\uparrow\downarrow}(T) \sim T^2$], the spin transresistivity of a two-dimensional {\it spin unpolarized} electron gas has the form $\rho_{\uparrow\downarrow}(T) \sim T^2 \ln T$. In the spin-polarized case the familiar form $\rho_{\uparrow\downarrow}(T) =A T^2$ is recovered, but the constant of proportionality $A$ diverges logarithmically as the spin-polarization tends to zero. In the high-temperature regime we obtain $\rho_{\uparrow \downarrow}(T) = -(\hbar / e^2) (\pi^2 Ry^* /k_B T)$ (where $Ry^*$ is the effective Rydberg energy) {\it independent} of the density. Again, this differs from the three-dimensional result, which has a logarithmic dependence on the density. Two important differences between the spin-drag transresistivity and the ordinary Coulomb drag transresistivity are pointed out: (i) The $\ln T$ singularity at low temperature is smaller, in the Coulomb drag case, by a factor $e^{-4 k_Fd}$ where $k_F$ is the Fermi wave vector and $d$ is the separation between the layers. (ii) The collective mode contribution to the spin-drag transresistivity is negligible at all temperatures. Moreover the spin drag effect is, for comparable parameters, larger than the ordinary Coulomb drag effect.Comment: 6 figures; various changes; version accepted for publicatio

Granular clustering in a hydrodynamic simulation

We present a numerical simulation of a granular material using hydrodynamic equations. We show that, in the absence of external forces, such a system phase-separates into high density and low density regions. We show that this separation is dependent on the inelasticity of collisions, and comment on the mechanism for this clustering behavior. Our results are compatible with the granular clustering seen in experiments and molecular dynamic simulations of inelastic hard disks.Comment: 4 pages, 5 figure

Magnetized gas in the smith high velocity cloud

We report the first detection of magnetic fields associated with the Smith High Velocity Cloud. We use a catalog of Faraday rotation measures toward extragalactic radio sources behind the Smith Cloud, new H I observations from the Robert C. Byrd Green Bank Telescope, and a spectroscopic map of Hα from the Wisconsin H-Alpha Mapper Northern Sky Survey. There are enhancements in rotation measure (RM) of =100 rad m-2 which are generally well correlated with decelerated Hα emission. We estimate a lower limit on the line-of-sight component of the field of =8 μG along a decelerated filament; this is a lower limit due to our assumptions about the geometry. No RM excess is evident in sightlines dominated by H I or Hα at the velocity of the Smith Cloud. The smooth Hα morphology of the emission at the Smith Cloud velocity suggests photoionization by the Galactic ionizing radiation field as the dominant ionization mechanism, while the filamentary morphology and high (=1 Rayleigh) Hα intensity of the lower-velocity magnetized ionized gas suggests an ionization process associated with shocks due to interaction with the Galactic interstellar medium. The presence of the magnetic field may contribute to the survival of high velocity clouds like the Smith Cloud as they move from the Galactic halo to the disk. We expect these data to provide a test for magnetohydrodynamic simulations of infalling gas

An analytical method for the optimisation of weakly nonlinear systems

In this paper we discuss how backbone curves can be used to guide the design and optimisation of weakly nonlinear systems with multiple degrees-of-freedom. Aft er decomposing the system using the modes of the equivalent linear system (the linear modes), we show how the backbone curves of the unforced, undamped equivalent system can be calculated. These consist of pure responses in each of the linear modes and, in certain parameter regimes, responses which are a combination of two or more linear modes - a feature which can be linked to internal resonance. Using an example system we will investigate how these backbone curves can be used to describe particular characteristics of the response. An energy balancing technique is also employed to relate the backbone curves to the response of the forced and damped system, and anticipate the conditions for which a particular characteristic will be seen. Finally, we discuss how the analytical nature of these techniques enables us to precisely design and optimise characteristics of such systems and how this can be expanded to systems with a greater number of degrees-of-freedom

On bird species diversity and remote sensing – utilizing lidar and hyperspectral data to assess the role of vegetation structure and foliage characteristics as drivers of avian diversity

Avian diversity has long been used as a surrogate for overall diversity. In forest ecosystems, it has been assumed that vegetation structure, composition and condition have a significant impact on avian diversity. Today, these features can be assessed via remote sensing. This study examined how structure metrics from lidar data and narrowband indices from hyperspectral data relate with avian diversity. This was assessed in four deciduous-dominated woods with differing age and structure set in an agricultural matrix in eastern England. The woods were delineated into cells within which metrics of avian diversity and remote sensing based predictors were calculated. Best subset regression was used to obtain best lidar models, hyperspectral models and finally, the best models combining variables from both datasets. The aims were not only to examine the drivers of avian diversity, but to assess the capabilities of the two remote sensing techniques for the task. The amount of understorey vegetation was the best single predictor, followed by Foliage Height Diversity, reflectance at 830 nm, Anthocyanin Reflectance Index 1 and Vogelmann Red Edge Index 2. This showed the significance of the full vertical profile of vegetation, the condition of the upper canopy, and potentially tree species composition. The results thus agree with the role that vegetation structure, condition and floristics are assumed to have for diversity. However, the inclusion of hyperspectral data resulted in such minor improvements to models that its collection for these purposes should be assessed critically