On the maximum number of limit cycles of a class of generalized Liénard differential systems

Agraïments: This work is partially supported by grant CONACYT-58968.Applying the averaging theory of first, second and third order to one class generalized polynomial Li'enard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit

Maximum number of limit cycles for generalized Liénard polynomial differential systems

summary:We consider limit cycles of a class of polynomial differential systems of the form $$\begin {cases} \dot {x}=y, \\ \dot {y}=-x-\varepsilon (g_{21}( x) y^{2\alpha +1} +f_{21}(x) y^{2\beta })-\varepsilon ^{2}(g_{22}( x) y^{2\alpha +1}+f_{22}( x) y^{2\beta }), \end {cases}$$ where $\beta$ and $\alpha$ are positive integers, $g_{2j}$ and $f_{2j}$ have degree $m$ and $n$, respectively, for each $j=1,2$, and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot {x}=y$, $\dot {y}=-x$ using the averaging theory of first and second order

A characterization of the generalized Liénard polynomial differential systems having invariant algebraic curves

The generalized Liénard polynomial differential systems are the differential systems of the form x' = y, y' = − f(x)y − g(x), where f and g are polynomials. We characterize all the generalized Liénard polynomial differential systems having an invariant algebraic curve. We show that the first four higher coefficients of the polynomial in the variable y, defining the invariant algebraic curve, determine completely the generalized Liénard polynomial differential system. This fact does not hold for arbitrary polynomial differential systems

On the Qualitative Behavior of a Class of Generalized Li\ue9nard Planar Systems

We study the problem of existence/nonexistence of limit cycles for a class of Lienard generalized differential systems in which, differently from the most investigated case, the function F depends not only on x but also on the y-variable. In this framework, some new results are presented, starting from a case study which, actually, already exhibits the most significant properties. In particular, the so-called "superlinear case" presents some new phenomena of escaping orbits which will be discussed in detail

An upper bound for the amplitude of limit cycles of Liénard-type differential systems

In this paper, we investigate the position problem of limit cycles for a class of Liénard-type differential systems. By considering the upper bound of the amplitude of limit cycles on $\{(x,y)\in\mathbb{R}^2: x0\}$ respectively, we provide a criterion concerning an explicit upper bound for the amplitude of the unique limit cycle of the Liénard-type system on the plane. Here the amplitude of a limit cycle on $\{(x,y)\in\mathbb{R}^2: x0\}$) is defined as the minimum (resp. maximum) value of the $x$-coordinate on such a limit cycle. Finally, we give two examples including an application to predator-prey system model to illustrate the obtained theoretical result, and Matlab simulations are presented to show the agreement between our theoretical result with the simulation analysis

A Bendixson-Dulac theorem for some piecewise systems

The Bendixson-Dulac Theorem provides a criterion to find upper bounds for the number of limit cycles in analytic differential systems. We extend this classical result to some classes of piecewise differential systems. We apply it to three different Liénard piecewise differential systems ¨ x+f±(x)˙ x+x = 0. The first is linear, the second is rational and the last corresponds to a particular extension of the cubic van der Pol oscillator. In all cases, the systems present regions in the parameter space with no limit cycles and others having at most one