Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem
In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations.
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Families of Twodimensional Vector Fields
Limit Periodic Sets
The OParameter Case
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Abelian integrals algebraic analytic family analytic function analytic unfolding analytic vector field Bautin Ideal bifurcation diagram blow-up blowing-up Bogdanov-Takens Bogdanov-Takens bifurcation bound Chapter codimension coefficients compact computation conjecture consider coordinates cusp cusp singularity defined Definition desingularization diffeomorphism Dulac series eigenvalues elementary graphics equivalent exists family of vector Figure finite codimension finite cyclicity finite number foliated local vector formula germ Gi(t Hamiltonian vector field Hence Hilbert's Hilbert's problem holomorphic instance Lemma limit cycles limit periodic set minimal system monomials Mourtada neighborhood non-zero normal form number of limit obtain parameter space periodic orbit phase portrait phase space Poincare-Bendixson theorem polynomial vector fields problem Proposition prove quadratic vector fields regular limit periodic regular orbits rescaling result return map roots saddle connection saddle point separatrix sequence singular points smooth subset suppose topological transition map transversal segments