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In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h , it has been shown that for any M > 1 there exists s M ∈ R such that the equation x ″ + f (x) x ′ + h (t , x) = s has two positive T -periodic solutions u 1 (⋅ ; s) and u 2 (⋅ ; s) satisfying min ⁡ { u 1 (t ; s) : t ∈ [ 0 , T ] } > M and min ⁡ { u 2 (t ; s) : t ∈ [ 0 , T ] } < 1 / M for every s < s M. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T -periodic solutions of indefinite differential equations. • Establish a refinement result of the Ambrosetti-Prodi type in the absence of anti-coercivity condition. • Investigate the dynamic behaviors of positive periodic solutions to a generalized Liénard equation. • Provide a new approach to study the existence of positive periodic solutions for indefinite differential equations. • Find a relationship between the strict lower functions and Leray-Schauder degree. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with polynomial Liénard equations of type , i.e. planar vector fields associated to a scalar second order differential equation , with f and g polynomials of respective degree m and n. It is shown that, besides compactifying the phase plane, or the Liénard plane, one can also compactify and desingularize the space of Liénard equations of type for each separately, by adding both singular perturbation problems and Hamiltonian perturbation problems. [Copyright &y& Elsevier]