This book is concerned with the bifurcation theory, the study of the changes in the structures of the solution of ordinary differential equations as parameters of the model vary. The theory has developed rapidly over the past two decades. Chapters 1 and 2 of the book introduce two systematic methods of simplifying equations: centre manifold theory and normal form theory, by which the dimension of equations may be reduced and the forms changed so that they are as simple as possible. Chapters 3–5 of the book study in considerable detail the bifurcation of those one- or two-dimensional equations with one, two or several parameters. This book is aimed at mathematicians and graduate students interested in dynamical systems, ordinary differential equations and/or bifurcation theory. The basic knowledge required by this book is advanced calculus, functional analysis and qualitative theory of ordinary differential equations.

We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.

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Endovascular treatment is a safe, effective, and very minimally invasive technique that provides good long-term patency rates for patients with NCS, and under the premise morphologic measurements, 14-mm-diameter, 60-mm-long self-expanding stents should be first considered for Chinese patients with NCS.

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Vulvar dystrophy could be effectively treated with focused ultrasound therapy. This approach appears to be a new promising treatment method, although further studies are still needed. LEVEL OF EVIDENCE II-3.

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The paper deals with Liénard equations of the form ẋ=y, ẏ =P(x)+yQ(x) with P and Q polynomials of degree respectively 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree 4 and especially to the study of the related elliptic integrals. Besides some general results the paper contains a complete treatment of the Saddle Loop case and the Two Saddle Cycle case. It is proven that the related elliptic integrals have at most two zeros, respectively one zero, the multiplicity taken into account. The bifurcation diagram of the zeros is also obtained.