Phase portraits of the quadratic polynomial Liénard differential systems (original) (raw)
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The applied geometry of a general Liénard polynomial system
Applied Mathematics Letters, 2012
In this work, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Liénard polynomial system with an arbitrary (but finite) number of singular points.
Limit cycle bifurcations of the classical Lienard polynomial system
2011
In this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the problem on the maximum number of limit cycles for the classical Liénard polynomial system which is related to the solution of Smale's thirteenth problem.
International Journal of Bifurcation and Chaos, 2023
Any singular irreducible cubic curve (or simply, cubic) after an affine transformation can be written as either y 2 = x 3 , or y 2 = x 2 (x + 1), or y 2 = x 2 (x − 1). We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic y 2 = x 2 (x+1). We prove that there are 63 different topological phase portraits for such quadratic polynomial differential systems. We control all the bifurcations among these distinct topological phase portraits. These systems have no limit cycles. Only 3 phase portraits have a center, 19 of these phase portraits have one polycycle, 3 of these phase portraits have 2 polycycles. The maximum number of separartices that have these phase portraits is 26 and the minimum number is 9, the maximum number of canonical regions of these phase portraits is 7 and the minimum is 3.
On the Number of Limit Cycles for a Generalization of Liénard Polynomial Differential Systems
International Journal of Bifurcation and Chaos, 2013
We study the number of limit cycles of the polynomial differential systems of the form [Formula: see text] where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = -x using the averaging theory of third order.