Limit cycles for 3-monomial differential equations

Limit cycles of differential equations

with P and Q polynomial and relatively prime, then the situation is substantially simpler, at least with respect to the richness of the asymptotic behaviour of the orbits. The Poincaré-Bendixson theorem implies that a bounded ω-limit set (and hence also a bounded α-limit set) of an orbit has to be either a singularity (also called a zero, an equilibrium, a critical point, or a stationary point)...

Limit Cycles of Planar Quadratic Differential Equations

Since Hilbert posed the problem of systematically counting and locating lhe limit cycle of polynomial systems on the plane in 1900, much ef Tort has been expended in its investigation. A large body of literature chiefly by Chinese and Soviet authors has addressed this question in the context of differential equations whose field is specified by quadratic polynomials, In this paper we consider t...

LIMIT CYCLES FOR m–PIECEWISE DISCONTINUOUS POLYNOMIAL LIÉNARD DIFFERENTIAL EQUATIONS

We provide lower bounds for the maximum number of limit cycles for the m–piecewise discontinuous polynomial differential equations ẋ = y + sgn(gm(x, y))F (x), ẏ = −x, where the zero set of the function sgn(gm(x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane in sectors of angle 2π/m, and sgn(z) denotes the sign funct...

Limit Cycles of the Generalized Polynomial Liénard Differential Equations

We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ+f(x)ẋ+g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m− 1)/2] limit cycles, where [·] denotes the integer part function.

Advanced Course On Limit Cycles of Differential Equations