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 emphasized. 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 analyzed. New types of codimension-one and -two bifurcations of planar systems were found. A detailed global bifurcation diagram concludes the work. This volume should be useful 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:   2000 ed.
Volume:   10
Dimensions:   Width: 15.60cm , Height: 1.50cm , Length: 23.40cm
Weight:   1.190kg
ISBN:  

9780792364276

ISBN 10:   0792364279
Pages:   238
Publication Date:   31 August 2000
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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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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