Fenichel's theorems with applications in dynamical systems.

Riley, Jeremy Warren

doi:10.18297/etd/1208

Cited by 3 publications

(3 citation statements)

References 17 publications

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“…Based on the circumstances, the investigation could emphasize slow or fast motions. Numerous subjects are treated well in this relation, including defining the center manifold and the critical manifold, as well as diving into Fenichel's theorem, a core finding that provides a clear separation between analytical and numerical representations of a slow manifold [9,13,16]. As an outcome, the following are the important purposes of this paper:…”

Section: Introductionmentioning

confidence: 95%

Slow Manifold Analysis of Modified Burst Model in the Saccadic System

Mousavinejad

Nia

2023

Preprint

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‎The saccade is one of the eye movements that resulted in the creation of the saccadic model‎. ‎This work is grounded in the basic principles of the saccadic system‎, ‎which are burst neurons and a resettable integrator model‎. ‎We intend to substitute a new model for the original one so that the equilibrium point of the system becomes a differentiable point in the changed model to eliminate the main shortcoming of the original model at the equilibrium point‎. ‎Our principal objective in this study is to determine the geometry of the slow manifold for the innovative system that has one fast and one slow variable‎. ‎Specifically‎, ‎we examine the dynamics around an equilibrium point and the geometry of a slow manifold by using Fenichel's theorem‎. ‎The current study is to ascertain the effects of geometric singular perturbations on this fast-slow system‎. ‎To conclude the debate‎, ‎the stability or instability at the equilibrium point is found by using the center manifold theory‎.

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Section: Introductionmentioning

confidence: 95%

Slow Manifold Analysis of Modified Burst Model in the Saccadic System

Mousavinejad

Nia

2023

Preprint

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“…• Geometric singular perturbation theory [4][5][6][7][8]; • Asymptotic expansion of the solutions [9,10]; • Lower and upper solutions method [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

(Iq)–Stability and Uniform Convergence of the Solutions of Singularly Perturbed Boundary Value Problems

Vrabel

2023

Mathematics

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In this paper, using the notion of (Iq)–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [a,b] have been established for four different types of boundary conditions for a singularly perturbed differential equation εy″=f(x,y,y′), a≤x≤b. In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f(x,y,y′)=0.

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“…The analysis of the differential equations under consideration is complicated by the fact that all roots of characteristic equations of this differential equation are located on the imaginary axis; that is, the differential equation is not hyperbolic. For the singularly perturbed dynamical systems, the dynamics near a normally hyperbolic critical manifold are well-known; see [1][2][3][4][5] for a geometric approach to the singular perturbation theory, Refs. [6][7][8][9] for the lower and upper solution method and [10] for applications in control theory.…”

Section: Introductionmentioning

confidence: 99%