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We investigate a singularly perturbed, non-convex variational problem arising in material science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan Müller, where it is proven that the solutions of the variational problem are periodic and exhibit a complicated multi-scale structure. In order to get more insight into the rich solution structure, we transform the corresponding Euler-Lagrange equation into a Hamiltonian system of first order ODEs and then use geometric singular perturbation theory to study its periodic solutions. Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters. This allows us to observe the influence of the parameters on the behavior of the orbits and to study their interplay in the minimization process. Our results confirm previous analytical results such as the asymptotics of the period of minimizers predicted by Müller. Furthermore, we find several new structures in the entire space of admissible periodic orbits. [ABSTRACT FROM AUTHOR]

Mathematical models of ordinary differential equations are used to describe and understand biological phenomena. These models are dynamical systems that often describe the time evolution of more than three variables, i.e., their dynamics take place in a multi‐dimensional space, called the phase space. Currently, mathematical domain scientists use plots of typical trajectories in the phase space to analyze the qualitative behavior of dynamical systems. These plots are called phase portraits and they perform well for 2D and 3D dynamical systems. However, for 4D, the visual exploration of trajectories becomes challenging, as simple subspace juxtaposition is not sufficient. We propose ManyLands to support mathematical domain scientists in analyzing 4D models of biological systems. By describing the subspaces as Lands, we accompany domain scientists along a continuous journey through 4D HyperLand, 3D SpaceLand, and 2D FlatLand, using seamless transitions. The Lands are also linked to 1D TimeLines. We offer an additional dissected view of trajectories that relies on small‐multiple compass‐alike pictograms for easy navigation across subspaces and trajectory segments of interest. We show three use cases of 4D dynamical systems from cell biology and biochemistry. An informal evaluation with mathematical experts confirmed that ManyLands helps them to visualize and analyze complex 4D dynamics, while facilitating mathematical experiments and simulations. [ABSTRACT FROM AUTHOR]

Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter \lambda. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called touchdown phenomenon, where the membrane establishes contact with the ground plate. ''Touchdown"" is known to imply the non-existence of steady-state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small ''regularization"" parameter varepsilon. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as ''blow-up"", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters varepsilon and \lambda is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero. [ABSTRACT FROM AUTHOR]

We study the existence of profiles for Chapman-Jouguet detonation and deflagration waves in the Navier-Stokes equations for a reacting gas. In the limit of small viscosity, heat conductivity, and diffusion, the profiles correspond to heteroclinic orbits of a system of singularly perturbed ordinary differential equations. The burned end state of the waves, however, is a nonhyperbolic equilibrium of the associated, purely gas dynamic layer problem, and hence standard methods from geometric singular perturbation theory fail. We show how to resolve this degeneracy by combining a center manifold reduction with the blow-up method. The main result is the existence of viscous profiles for various types of Chapman-Jouguet processes. In addition, we obtain results on the spatial decay rates of these waves which are expected to be relevant for the stability analysis of the waves. [ABSTRACT FROM AUTHOR]

A minimal model describing the embryonic cell division cycle at the molecular level in eukaryotes is analyzed mathematically. It is known from numerical simulations that the corresponding three-dimensional system of ODEs has periodic solutions in certain parameter regimes. We prove the existence of a stable limit cycle and provide a detailed description on how the limit cycle is generated. The limit cycle corresponds to a relaxation oscillation of an auxiliary system, which is singularly perturbed and has the same orbits as the original model. The singular perturbation character of the auxiliary problem is caused by the occurrence of small Michaelis constants in the model. Essential pieces of the limit cycle of the auxiliary problem consist of segments of slow motion close to several branches of a two dimensional critical manifold which are connected by fast jumps. In addition, a new phenomenon of exchange of stability occurs at lines, where the branches of the two-dimensional critical manifold intersect. This novel type of relaxation oscillations is studied by combining standard results from geometric singular perturbation with several suitable blow-up transformations. [ABSTRACT FROM AUTHOR]

We study the Olsen model for the peroxidase-oxidase reaction. The dynamics is analyzed using a geometric decomposition based on multiple timescales. The Olsen model is four-dimensional, not in a standard form required by geometric singular perturbation theory and contains multiple small parameters. These three obstacles are the main challenges we resolve by our analysis. Scaling and the blow-up method are used to identify several subsystems. The results presented here provide a rigorous analysis for two oscillatory modes. In particular, we prove the existence of non-classical relaxation oscillations in two cases. The analysis is based on desingularization of lines of transcritical and submanifolds of fold singularities in combination with an integrable relaxation phase. In this context, our analysis also explains an assumption that has been utilized, based purely on numerical reasoning, in a previous bifurcation analysis by Desroches et al. (Discret Contin Dyn Syst S 2(4):807-827, ). Furthermore, the geometric decomposition we develop forms the basis to prove the existence of mixed-mode and chaotic oscillations in the Olsen model, which will be discussed in more detail in future work. [ABSTRACT FROM AUTHOR]

A detailed geometric analysis of the Goldbeter-Lefever model of glycolytic oscillations is given. In suitably scaled variables the governing equations are a planar system of ordinary differential equations depending singularly on two small parameters ε and δ. In [L. Segel and A. Goldbeter, J. Math. Biol., 32 (1994), pp. 147-160] it was argued that a limit cycle of relaxation type exists for ε « δ « 1. The existence of this limit cycle is proved by analyzing the problem in the spirit of geometric singular perturbation theory. The degeneracies of the limiting problem corresponding to (ε, δ) = (0, 0) are resolved by a novel variant of the blow-up method. It is shown that repeated blowups lead to a clear geometric picture of this fairly complicated two-parameter multiscale problem. [ABSTRACT FROM AUTHOR]

A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues correspond to zeros of an essentially analytic function $${\mathcal{E}(\rho\lambda,\rho\omega)}$$ which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of $${\mathcal{E}(\rho\lambda,\rho\omega)}$$ and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ, ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished that relate the small strength $${\epsilon}$$ of the shock wave to appropriate ranges of values of the spectral parameters ( ρλ, ρ ω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality. [ABSTRACT FROM AUTHOR]

We construct self-similar solutions of the Dafermos regularization of a system of conservation laws near structurally stable Riemann solutions composed of Lax shocks and rarefactions, with all waves possibly large. The construction requires blowing up a manifold of gain-of-stability turning points in a geometric singular perturbation problem as well as a new exchange lemma to deal with the remaining hyperbolic directions. [ABSTRACT FROM AUTHOR]

A traveling waves analysis for a polyatomic gas with n internal energy storage modes is performed. In both the frozen flow and the equilibrium limit the existence and qualitative behavior of traveling waves are investigated. Methods from geometric singular perturbation theory are used to study these limits. [ABSTRACT FROM AUTHOR]