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We study order-preserving $\mathcal{C}^1$-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.

Mixed-mode oscillations (MMOs) as complex firing patterns with both relaxation oscillations and sub-threshold oscillations have been found in many neural models such as the stellate neuron model, HH model, and so on. Based on the work, we discuss mixed-mode oscillations in the Av-Ron-Parnas-Segel model which can govern the behavior of the neuron in the lobster cardiac ganglion. By using the geometric singular perturbation theory we first explain why the MMOs exist in the reduced Av-Ron-Parnas-Segel model. Then the mixed-mode oscillatory phenomenon and aperiodic mixed-mode behaviors in the model have been analyzed numerically. Finally, we illustrate the influence of certain parameters on the model.

This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.

In this paper, we study the limiting behavior of solutions to a 1D two-point boundary value problem for viscous conservation laws with genuinely-nonlinear fluxes as $\varepsilon$ goes to zero. We here discuss different types of non-characteristic boundary layers occurring on both sides. We first construct formally the three-term approximate solutions by using the method of matched asymptotic expansions. Next, by energy method we prove that the boundary layers are nonlinearly stable and thus it is proved the boundary layer effects are just localized near both boundaries. Consequently, the viscous solutions converge to the smooth inviscid solution uniformly away from the boundaries. The rate of convergence in viscosity is optimal.

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