| Introduction | p. ix |
| Models and Dynamics | p. 1 |
| Models of the Heart Functioning | p. 1 |
| Processes associated with the periodicity of the cardiac electric signal | p. 1 |
| Van der Pol oscillator and relaxation oscillations | p. 4 |
| Generalized Van der Pol model. FitzHugh-Nagumo model | p. 7 |
| Forms of the FitzHugh-Nagumo system | p. 10 |
| FitzHugh-Nagumo system in the context of mathematical modelling in electrophysiology | p. 12 |
| Elements of Finite-Dimensional Dynamics | p. 17 |
| Dynamical systems and the associated systems of autonomous ODEs | p. 17 |
| Invariant sets in the phase space. Attractors | p. 20 |
| Planar dynamical systems | p. 24 |
| Bifurcation | p. 27 |
| Static and dynamic bifurcation | p. 27 |
| Structural stability. Bifurcation. Codimension of bifurcations | p. 31 |
| Topological normal forms for bifurcations | p. 33 |
| Bifurcations of planar dynamical systems | p. 37 |
| Regular and Singular Perturbations | p. 47 |
| Asymptotic expansions and matching principles | p. 47 |
| Perturbation problems | p. 48 |
| Models of asymptotic approximation | p. 51 |
| Static Bifurcation and Linearization of the Fitzhugh-Nagumo Model | p. 53 |
| Geometric Properties of Phase Trajectories | p. 53 |
| Nullclines | p. 53 |
| Phase trajectories | p. 54 |
| Inflection points | p. 55 |
| Equilibria | p. 59 |
| Eigenvalues of the Linearized System. Eigenvectors and Eigen-Directions | p. 63 |
| The linearized system | p. 63 |
| The nature and sign of the eigenvalues. Discussion in the (b, x)-plane | p. 65 |
| The nature and sign of the eigenvalues. Discussion in the (b, a)-plane | p. 67 |
| The variation of the eigenvalues with respect to the parameters | p. 75 |
| Eigenvectors and eigen-directions | p. 76 |
| Static Bifurcation Diagrams: Partial Dynamical Characterization | p. 77 |
| Asymptotic Behaviour of the Static Bifurcation Diagrams as c [right arrow] [infinity] | p. 85 |
| Types of Hyperbolic Equilibria | p. 86 |
| The Center Manifold and the Saddle-Node Bifurcation | p. 89 |
| Case of positive b | p. 89 |
| Case of negative b | p. 94 |
| Dynamic Bifurcation for the Fitzhugh-Nagumo Model | p. 97 |
| Hope Bifurcation | p. 98 |
| Locus of Hopf bifurcation points and values | p. 98 |
| An equivalent form of the F-N system | p. 102 |
| Bogdanov-Takens Bifurcation | p. 104 |
| Bogdanov-Takens bifurcation at Q[subscript 1] | p. 104 |
| Bogdanov-Takens bifurcation at Q[subscript 3] | p. 112 |
| Homoclinic Bifurcation | p. 115 |
| Curves of homoclinic bifurcation values | p. 115 |
| Double homoclinic bifurcation | p. 117 |
| Saddle-node homoclinic bifurcation. Saddle-node separatrix loop bifurcation | p. 120 |
| Breaking Saddle Connection Bifurcation | p. 121 |
| Locally stable and unstable manifolds of saddles | p. 121 |
| Curves of breaking saddle connection bifurcation values | p. 123 |
| Double breaking saddle connection bifurcation | p. 127 |
| Saddle-node-saddle connection bifurcation. Saddle-node-saddle with separatrix connection bifurcation | p. 128 |
| Bautin Bifurcation. Non-Hyperbolic Limit Cycle Bifurcation | p. 131 |
| Normal form for Bautin bifurcation. Liapunov coefficients | p. 131 |
| Numerical results showing the degeneration of the Hopf bifurcation into the Bautin bifurcation | p. 141 |
| Locus of Bautin bifurcation values as c varies | p. 144 |
| The system generating h[subscript ij] for the case of cubic nonlinearities | p. 145 |
| Models of Asymptotic Approximation for the Fitzhugh-Nagumo System as c [right arrow] [infinity] | p. 149 |
| Types of Asymptotic Behaviour of the Solution of the F-N Model | p. 149 |
| First Order Asymptotic Approximations as [epsilon] [right arrow] 0 | p. 152 |
| The outer approximation | p. 152 |
| The inner approximation | p. 153 |
| Inner-outer expansion matching | p. 156 |
| Higher Order Asymptotic Approximations as [epsilon] [right arrow] 0 | p. 158 |
| Models of outer asymptotic approximation as [epsilon] [right arrow] 0 | p. 158 |
| Equations of inner asymptotic approximation as [epsilon] [right arrow] 0 | p. 160 |
| Inner-outer expansion matching | p. 161 |
| Some Particular Cases | p. 163 |
| The case of the Van der Pol model | p. 163 |
| Expansions matching and the running time along the limit cycle | p. 166 |
| Asymptotic Results on Ducks (French Canards) and Related Objects | p. 168 |
| Canard phenomenon | p. 168 |
| Relaxation oscillations | p. 172 |
| The ducks: standard versus nonstandard asymptotic analysis | p. 174 |
| Global Bifurcation Diagram and Phase Dynamics for the Fitzhugh-Nagumo Model | p. 179 |
| Global Bifurcation Diagram for the Fitzhugh-Nagumo Model | p. 179 |
| Parametric portrait of the F-N model | p. 179 |
| Types of dynamics in the F-N model | p. 183 |
| Basins of Attraction | p. 193 |
| Transient Regimes and Non-Periodic Oscillations | p. 196 |
| Types of transient regimes in the absence of limit cycles | p. 196 |
| Types of transient regimes in the presence of limit cycles | p. 199 |
| Limit Cycles and Periodic Oscillations | p. 204 |
| The number of the limit cycles for the F-N model | p. 204 |
| Concave limit cycles versus inflection points | p. 205 |
| The Initiation of Heart Beats | p. 212 |
| Concluding Remarks of Interest to Physiologists | p. 214 |
| Open Mathematical Problems | p. 218 |
| Liapunov Coefficients | p. 221 |
| Brief Description of the Soft Diecbi | p. 223 |
| References | p. 227 |
| Index | p. 233 |
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