MS03.3: Computational Methods

We describe how some minimal surface bifurcation problems can be treated with the Matlab continuation and bifurcation toolbox pde2path, and present symmetry breaking bifurcation results for Enneper's surface, and Schwarz--P and Schwarz--H families. Some of these results are analytically known and thus are used as benchmarks. Others are new and associated to kernels of higher multiplicity.

We investigate numerically the basins of attraction of a simple model of single-reed musical instruments with a time- varying parameter. More precisely, the main control parameter – the blowing pressure γ – is linearly increasing and saturating to account for attack transients, which are known to be of particular importance in a musical context. In the space of initial conditions of the state variables and initial value of γ, we compute the boundary between the so-called dynamical basins using a machine learning technique, namely an explicit design space decomposition method combined with a support vector machine classifier. This allows for the systematic investigation of the effect of attack transients on the observed regime among the coexisting stable solutions. Our results show that, depending on their duration, attack transients might have a dramatic impact on the playability of the different stable regimes.

We discuss recent developments of the toolbox MatCont for numerical bifurcation analysis. We have added two ways to define models: importing from SBML and with a pseudospectral approximation. The latter allows for investigating systems with delays and renewal equations. Next, we highlight a new interface within MatCont to compute Lyapunov exponents, making it easier for users to check for multi-dimensional tori and chaos, which we demonstrate for a nonlinear forced oscillator. Finally, we discuss homoclinic orbits for a laser model. We highlight the workflow of getting bifurcation curves and how to obtain reproducible data for publications.

In this work, a 5 Degrees of Freedom (5DOF) Lumped Parameter Model (LPM) of the implanted human middle ear is proposed. Frequency response curves obtained from the model as well as the curves from the literature are compared. The proposed system shows bistability, which is not presented before in the literature or in the ASTM standard.

The equivalent linearization method can yield an accurate and computationally efficient estimate of the variance of the response of a nonlinear system subject to random excitation, even in the presence of strong nonlinearity. However, spectra yielded by the method are usually inaccurate, and the method cannot predict any loss in coherence between the excitation and the response. An alternative approach is presented here which is equally efficient and which yields good approximations for response spectra and coherence levels. The method is based on finding the first Wiener kernel of the response and then using detailed balance considerations to express the total response spectrum in terms of this kernel. For simplicity the method is presented here for a single degree of freedom system, although it can also be applied to more complex systems.

The right-hand side in delay differential equations, when considered as an abstract ODE, suffers from a lack of classical continuous differentiability if the delay depends on the state. The right-hand side is only continuously differentiable k times from the space of k+m times continuously differentiable functions into the space of m times continuously differentiable functions. One might expect that this may impede high-order convergence for numerical methods based on piecewise polynomial approximations, since even high-order piecewise polynomial are only continuous at their interval boundaries. We show that this is not the case, arriving at a convergence result similar to those for constant delays.