Overview

This application-oriented monograph presents a comprehensive theoretical and numerical investigation of all types of oscillators and bifurcations (such as Hopf, Bogdanov-Takens, Bautin, and homoclinic) generated by the FitzHugh-Nagumo model. The wide diversity of the oscillators as used in electrophysiology, biology, and engineering is emphasised. Various asymptotic behaviours are revealed. The dramatic changes in oscillations connected with the emergence or disappearance of concave limit cycles are investigated. Codimension of bifurcations is minutely analysed. New types of codimension one and two bifurcations of planar systems were found. A detailed global bifurcation diagram concludes the work. Audience: This volume will be of interest to researchers and graduate students whose work involves the mathematics of biology, ordinary differential equations, approximations and expansions, cardiac electrophysiology, biological transport, and cell membranes.

Full Product Details

Author:   C. Rocsoreanu ,  A. Georgescu ,  N. Giurgiteanu
Publisher:   Springer
Imprint:   Springer
Edition:   1st ed. Softcover of orig. ed. 2000
Volume:   10
Dimensions:   Width: 17.00cm , Height: 1.30cm , Length: 24.40cm
Weight:   0.454kg
ISBN:  

9789048155125

ISBN 10:   9048155126
Pages:   238
Publication Date:   08 December 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

1 Models and Dynamics.- 1.1 Models of the Heart Functioning.- 1.2 Elements of Finite-Dimensional Dynamics.- 1.3 Bifurcation.- 1.4 Regular and Singular Perturbations.- 2 Static Bifurcation and Linearization of the FitzHugh-Nagumo Model.- 2.1 Geometric Properties of Phase Trajectories.- 2.2 Equilibria.- 2.3 Eigenvalues of the Linearized System. Eigenvectors and Eigen-Directions.- 2.4 Static Bifurcation Diagrams: Partial Dynamical Characterization.- 2.5 Asymptotic Behaviour of the Static Bifurcation Diagrams as c ? ?.- 2.6 Types of Hyperbolic Equilibria.- 2.7 The Center Manifold and the Saddle-Node Bifurcation.- 3 Dynamic Bifurcation for the FitzHugh-Nagumo Model.- 3.1 Hopf Bifurcation.- 3.2 Bogdanov-Takens Bifurcation.- 3.3 Homoclinic Bifurcation.- 3.4 Breaking Saddle Connection Bifurcation.- 3.5 Bautin Bifurcation. Non-Hyperbolic Limit Cycle Bifurcation.- 4 Models of Asymptotic Approximation for the FitzHugh-Nagumo System as c ? ?.- 4.1 Types of Asymptotic Behaviour of the Solution of the F-N Model.- 4.2 First Order Asymptotic Approximations as ? ? 0.- 4.3 Higher Order Asymptotic Approximations as ? ? 0.- 4.4 Some Particular Cases.- 4.5 Asymptotic Results on Ducks (French Canards) and Related Objects.- 5 Global Bifurcation Diagram and Phase Dynamics for the FitzHugh-Nagumo Model.- 5.1 Global Bifurcation Diagram for the FitzHugh-Nagumo Model.- 5.2 Basins of Attraction.- 5.3 Transient Regimes and Non-Periodic Oscillations.- 5.4 Limit Cycles and Periodic Oscillations.- 5.5 The Initiation of Heart Beats.- 5.6 Concluding Remarks of Interest to Physiologists.- 5.7 Open Mathematical Problems.- A Liapunov Coefficients.- B Brief Description of the Soft Diecbi.- References.

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