Oscillations in biology. Qualitative analysis and models (Q2484075)

This book deals with continuous dynamical system theory, from the point of view of qualitative methods related to nonlinear differential equations and their applications. The ``strategy'' of these methods can be defined noting that the solutions of such equations are in general nonclassical, nontabulated, complex transcendental functions of mathematical analysis. Here, these transcendental functions are not defined by series expansions (case of analytical methods), but by the equation singularities such as equilibrium points, periodic solutions, invariant manifolds passing through saddle singularities, boundary of the domain of attraction (or basin) of a stable stationary state, homoclinic or heteroclinic singularities, or more complex singularities of fractal or nonfractal type. The qualitative methods consider the nature of these singularities in the phase space (state space), and their evolutions when the system parameters vary, or in the presence of a continuous structure modification of this system (study of the bifurcation sets in a parameter space, or in a function space). Although presented for students and researchers of biological sciences, it concerns all the scientific fields using dynamical models, giving the necessary mathematical background for studying them. The four first chapters give the basic tools of the qualitative methods, related to the above typical singularities, the stability theory, the structural stability, the notions of flow, normal form, bifurcation, perturbations, singular perturbations. The fifth chapter is devoted to coupled oscillators. The sixth one tackles the study of some spatiotemporal models, and the last one presents the handling of models coming from the physiology field. This book has the interest to be a ``user's guide'' to tackle the very large field of qualitative methods of nonlinear dynamics. Nevertheless it would have been useful to complete it by a short last part giving suggestions for further reading. More particularly it is the case of the Japanese school contribution (Hayashi, Kawakami, Ueda) on the resonance and synchronization problems, giving rise to the higher harmonics, subharmonics and fractional harmonics phenomena, which can be generated by the most part of the book models. Last point it is surprising that a French author, adopting a bad quotation habit, denotes ``Hopf bifurcation'' a bifurcation qualitatively defined by Poincaré (1892), then analytically studied by Andronov (1938), and after by Hopf (1942).