New Elements of Analysis of a Degenerate Chenciner Bifurcation

A new transformation of parameters for generic discrete-time dynamical systems with two independent parameters is defined, for when the degeneracy occurs. Here the classical transformation of parameters (α1,α2)→(β1,β2) is not longer regular at (0,0); therefore, implicit function theorem (IFT) cannot be applied around the origin, and a new transformation is necessary. The approach in this article to a case of Chenciner bifurcation is theoretical, but it can provide an answer for a number of applications of dynamical systems. We studied the bifurcation scenario and found out that, by this transformation, four different bifurcation diagrams are obtained, and the non-degenerate Chenciner bifurcation can be described by two bifurcation diagrams.

Populations facing a nonlinear environmental gradient: Steady states and pulsating fronts

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.

Electrostatic-Elastic MEMS with Fringing Field: A Problem of Global Existence

In this paper, we prove the existence and uniqueness of solutions for a nonlocal, fourth-order integro-differential equation that models electrostatic MEMS with parallel metallic plates by exploiting a well-known implicit function theorem on the topological space framework. As the diameter of the domain is fairly small (similar to the length of the device wafer, which is comparable to the distance between the plates), the fringing field phenomenon can arise. Therefore, based on the Pelesko–Driscoll theory, a term for the fringing field has been considered. The nonlocal model obtained admits solutions, making these devices attractive for industrial applications whose intended uses require reduced external voltages.

Implicit Solitary Waves for One of the Generalized Nonlinear Schrödinger Equations

Application of transformations for dependent and independent variables is used for finding solitary wave solutions of the generalized Schrödinger equations. This new form of equation can be considered as the model for the description of propagation pulse in a nonlinear optics. The method for finding solutions of equation is given in the general case. Solitary waves of equation are obtained as implicit function taking into account the transformation of variables.

Novel Investigation of Multivariable Conformable Calculus for Modeling Scientific Phenomena

New investigation on the conformable version (CoV) of multivariable calculus is proposed. The conformable derivative (CoD) of a real-valued function (RVF) of several variables (SVs) and all related properties are investigated. An extension to vector-valued functions (VVFs) of several real variables (SRVs) is studied in this work. The CoV of chain rule (CR) for functions of SVs is also introduced. At the end, the CoV of implicit function theorem (IFThm) for SVs is established. All results in this work can be potentially applied in studying various modeling scenarios in physical oceanography such as Stommel’s box model of thermohaline circulation and other related models where all our results can provide a new analysis and computational tool to investigate these models or their modified formulations.