Localised states in an extended Swift-Hohenberg equation

Recent work on the behaviour of localised states in pattern forming partial differential equations has focused on the traditional model Swift-Hohenberg equation which, as a result of its simplicity, has additional structure --- it is variational in time and conservative in space. In this paper we investigate an extended Swift-Hohenberg equation in which non-variational and non-conservative effects play a key role. Our work concentrates on aspects of this much more complicated problem. Firstly we carry out the normal form analysis of the initial pattern forming instability that leads to small-amplitude localised states. Next we examine the bifurcation structure of the large-amplitude localised states. Finally we investigate the temporal stability of one-peak localised states. Throughout, we compare the localised states in the extended Swift-Hohenberg equation with the analogous solutions to the usual Swift-Hohenberg equation

The Swift-Hohenberg equation with a nonlocal nonlinearity

It is well known that aspects of the formation of localised states in a one-dimensional Swift--Hohenberg equation can be described by Ginzburg--Landau-type envelope equations. This paper extends these multiple scales analyses to cases where an additional nonlinear integral term, in the form of a convolution, is present. The presence of a kernel function introduces a new lengthscale into the problem, and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure. When the kernel is short-range, weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term. Nevertheless, these computations can be formulated quite generally in terms of properties of the Fourier transform of the kernel function. When the lengthscale associated with the kernel is longer, our method leads naturally to the derivation of two different, novel, envelope equations that describe aspects of the dynamics in these new regimes. The first of these contains additional bifurcations, and unexpected loops in the bifurcation diagram. The second of these captures the stretched-out nature of the homoclinic snaking curves that arises due to the nonlocal term.Comment: 28 pages, 14 figures. To appear in Physica

Are older people putting themselves at risk when using their walking frames?

Background Walking aids are issued to older adults to prevent falls, however, paradoxically their use has been identified as a risk factor for falling. To prevent falls, walking aids must be used in a stable manner, but it remains unknown to what extent associated clinical guidance is adhered to at home, and whether following guidance facilitates a stable walking pattern. It was the aim of this study to investigate adherence to guidance on walking frame use, and to quantify user stability whilst using walking frames. Additionally, we explored the views of users and healthcare professionals on walking aid use, and regarding the instrumented walking frames (‘Smart Walkers’) utilized in this study. Methods This observational study used Smart Walkers and pressure-sensing insoles to investigate usage patterns of 17 older people in their home environment; corresponding video captured contextual information. Additionally, stability when following, or not, clinical guidance was quantified for a subset of users during walking in an Activities of Daily Living Flat and in a gait laboratory. Two focus groups (users, healthcare professionals) shared their experiences with walking aids and provided feedback on the Smart Walkers. Results Incorrect use was observed for 16% of single support periods and for 29% of dual support periods, and was associated with environmental constraints and a specific frame design feature. Incorrect use was associated with reduced stability. Participants and healthcare professionals perceived the Smart Walker technology positively. Conclusions Clinical guidance cannot easily be adhered to and self-selected strategies reduce stability, hence are placing the user at risk. Current guidance needs to be improved to address environmental constraints whilst facilitating stable walking. The research is highly relevant considering the rising number of walking aid users, their increased falls-risk, and the costs of falls. Trial Registration Not applicable

Turbulent transition in a truncated one-dimensional model for shear flow

We present a reduced model for the transition to turbulence in shear flow that is simple enough to admit a thorough numerical investigation while allowing spatio-temporal dynamics that are substantially more complex than those allowed in previous modal truncations. Our model allows a comparison of the dynamics resulting from initial perturbations that are localised in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localised initial conditions the subcritical transition to a `turbulent' state (i) takes place more abruptly, with a boundary between laminar and `turbulent' flow that is appears to be much less `structured' and (ii) results in a spatiotemporally chaotic regime within which the lifetimes of spatiotemporally complicated transients are longer, and are even more sensitive to initial conditions. The minimum initial energy $E_0$ required for a spanwise-localised initial perturbation to excite a chaotic transient has a power-law scaling with Reynolds number $E_0 \sim Re^p$ with $p \approx -4.3$. The exponent $p$ depends only weakly on the width of the localised perturbation and is lower than that commonly observed in previous low-dimensional models where typically $p \approx -2$. The distributions of lifetimes of chaotic transients at fixed Reynolds number are found to be consistent with exponential distributions.Comment: 22 pages. 11 figures. To appear in Proc. Roy. Soc.