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This master thesis concerns the implementation of a novel, computationally efficient bifurcation numerical analysis interface in the Julia compiled programming language. The interface involves the use of the well-known bisection or Newton-Raphson methods in order to locate bifurcations in the neuron models, as well as the use of numerical approximation methods of Jacobian matrices through forward numerical differentiation of the system's equations.
The interface that is built aims at the identification of the bifurcations in neuron models in order to determine their excitability type. A recent paper-motivated canonical model is chosen as an example to which the interface can be applied as a proof of concept. This numerical analysis of the example model outputs results that highlight the importance of dynamical analysis of neuron models, i.e. analysis over a range of time-scale parameters, versus the more common static analysis of models through the visual inspection of their phase plane representation. 
Normal form identification based on visual inspection only is at considerable risk that the original system is identified to may not be the correct one. The results obtained through the use of this interface on a two-dimensional therefore motivate the need for extensive numerical analysis of original high-dimensional neuron models for various values of time-scale separation in order to reliably identify the bifurcation normal form that they can be reduced to.

Neuroscience --- Neuron mathematical modelling --- Non-linear systems --- Bifurcation analysis --- Julia --- Scientific computing --- Ingénierie, informatique & technologie > Multidisciplinaire, généralités & autres

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This book brings together all 16 articles published in the Special Issue "Applied Mathematics to Mechanisms and Machines" of the MDPI Mathematics journal, in the section “Engineering Mathematics”. The subject matter covered by these works is varied, but they all have mechanisms as the object of study and mathematics as the basis of the methodology used. In fact, the synthesis, design and optimization of mechanisms, robotics, automotives, maintenance 4.0, machine vibrations, control, biomechanics and medical devices are among the topics covered in this book. This volume may be of interest to all who work in the field of mechanism and machine science and we hope that it will contribute to the development of both mechanical engineering and applied mathematics.

Technology: general issues --- History of engineering & technology --- path generation --- dimensional synthesis --- hybrid optimization --- slider-crank mechanism --- hybrid compliant mechanisms --- path analysis --- numerical methods --- elliptic integrals --- kinematics --- Levenberg–Marquardt algorithm --- convergence --- neural networks --- local minima --- optimization --- three-legged parallel mechanism --- compliant mechanism --- flexure-based mechanism --- flexure --- compliant joint --- decoupled motion --- coupled motion --- stiffness --- compliance --- suspension algorithm --- predictive suspension --- land vehicle --- sprung mass --- autoencoder --- aeroengine --- denoising --- GRU --- rotordynamic --- adaptive rejection control --- sinusoidal disturbance --- flexible rotor --- hydrodynamic journal bearing --- six-bar mechanism --- dynamic balancing --- fully Cartesian coordinates --- multiobjective optimization --- differential evolution --- Pareto front --- serial robotic manipulators --- singularity identification --- geometric algebra --- rotor group --- distance to a singularity --- air flow medical sensor --- emergency air flow sensor --- low-cost air flow sensor --- nanostructures --- COVID-19 --- FEM analysis --- biomechanics --- effort mathematical models --- NMR reconstruction --- reverse engineering --- elbow joint --- zipper-coupled tubes --- Miura-ori pattern --- deployable mechanism --- origami inspired design --- smooth sheet attachment --- relative rotation --- jump --- safe basin --- fractal --- heteroclinic bifurcation --- delayed feedback --- helical spring --- vibration --- Frenet trihedral --- dispersion relation --- natural frequency --- approximate entropy --- non-linear systems --- phase space reconstruction --- fault classification --- random forest --- support vector machines --- lateral dynamics --- track parameters --- vehicle mathematical modeling --- safety vehicle index