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In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): x ˙ = y − 12 x 2 + 16 y 2 , y ˙ = − x − 16 x y , and (r20): x ˙ = y + 4 x 2 , y ˙ = − x + 16 x y , and the periodic orbits of the quadratic isochronous centers ( S 1 ) : x ˙ = − y + x 2 − y 2 , y ˙ = x + 2 x y , and ( S 2 ) : x ˙ = − y + x 2 , y ˙ = x + x y . The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system ( S 1 ) and ( S 2 ) are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line y = 0 . It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively 4 n − 3 ( n ≥ 4 ) and 4 n + 3 ( n ≥ 3 ) counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers ( S 1 ) and ( S 2 ) are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively. [ABSTRACT FROM AUTHOR]

Classical theories predict that solutions of differential equations will leave any neighborhood of a stable limit cycle, if white noise is added to the system. In reality, many engineering systems modeled by second order differential equations, like the van der Pol oscillator, show incredible robustness against noise perturbations, and the perturbed trajectories remain in the neighborhood of a stable limit cycle for all times of practical interest. In this paper, we propose a new model of noise to bridge this apparent discrepancy between theory and practice. Restricting to perturbations from within this new class of noise, we consider stochastic perturbations of second order differential systems that –in the unperturbed case– admit asymptotically stable limit cycles. We show that the perturbed solutions are globally bounded and remain in a tubular neighborhood of the underlying deterministic periodic orbit. We also define stochastic Poincaré map(s), and further derive partial differential equations for the transition density function. [ABSTRACT FROM AUTHOR]

In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center, which can be obtained using first order averaging method. This result improves the upper bounds given in [7] . [ABSTRACT FROM AUTHOR]

Consider planar real polynomial differential equations of the form x ˙ = L x + X n ( x ) , where x = ( x , y ) ∈ R 2 , L is a 2 × 2 matrix and X n is a homogeneous vector field of degree n > 1 . Most known results about these equations, valid for infinitely many n , deal with the case where the origin is a focus or a node and give either non-existence of limit cycles or upper bounds of one or two limit cycles surrounding the origin. In this paper we improve some of these results and moreover we show that for n ≥ 3 odd there are equations of this form having at least ( n + 1 ) / 2 limit cycles surrounding the origin. Our results include cases where the origin is a focus, a node, a saddle or a nilpotent singularity. We also discuss a mechanism for the bifurcation of limit cycles from infinity. [ABSTRACT FROM AUTHOR]

Abstract: For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401–1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have? In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.ʼs as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem. [Copyright &y& Elsevier]

Abstract: In this paper we study the maximal number of limit cycles in Hopf bifurcations for two types of Liénard systems and obtain an upper bound of the number. In some cases the upper bound is the least, called the Hopf cyclicity. [Copyright &y& Elsevier]

Abstract: Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the sign of the derivative of the associated bifurcation function at a zero. In this paper we show that, for analytic systems, this result is of topological nature. This means that it is enough to impose a change of sign at the zero, without any assumption on the succesive derivatives. [Copyright &y& Elsevier]

Abstract: In this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane. We study a strip of hyperbolas occurring in some quadratic systems. We deal with the cyclicity of the degenerate graphics from the programme, set up in [F. Dumortier, R. Roussarie, C. Rousseau, Hilbert''s 16th problem for quadratic vector fields, J. Differential Equations 110 (1994) 86–133], to solve the finiteness part of Hilbert''s 16th problem for quadratic systems. Techniques from geometric singular perturbation theory are combined with the use of the Bautin ideal. We also rely on the theory of Darboux integrability. [Copyright &y& Elsevier]

Abstract: In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles. [Copyright &y& Elsevier]

Abstract: In this paper we deal with discontinuous vector fields on and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation. The regularization process developed by Sotomayor and Teixeira is crucial for the development of this work. [Copyright &y& Elsevier]