On the number of limit cycles for a generalization of Liénard polynomial differential systems

Agraïments: The second author has been supported by the grant AGAUR PIV-DGR-2010 and by FCT through CAMGSD.We study the number of limit cycles of the polynomial differential systems of the form x˙ = y − g1(x), y˙ = −x − g2(x) − f(x)y, where g1(x) = εg11(x)+ε2g12(x)+ε3g13(x), g2(x) =εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2f2(x) + ε3f3(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 Li'enard 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 ˙x = y, ˙y = −x using the averaging theory of third order

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

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 note on existence and uniqueness of limit cycles for Liénard systems

AbstractWe consider the Liénard equation and we give a sufficient condition to ensure existence and uniqueness of limit cycles. We compare our result with some other existing ones and we give some applications

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 new approach to study limit cycles on a cylinder

We present a new approach to study limit cycles of planar systems of autonomous differential equations with a cylindrical phase space $Z$. It is based on an extension of the Dulac function which we call Dulac-Cherkas function $\Psi$. The level set W:=\{\vf,y) \in Z: \Psi(\vf,y)=0\} plays a key role in this approach, its topological structure influences existence, location and number of limit cycles. We present two procedures to construct Dulac-Cherkas functions. For the general case we describe a numerical approach based on the reduction to a linear programming problem and which is implemented by means of the computer algebra system Mathematica. For the class of generalized Li\'enard systems we present an analytical approach associated with solving linear differential equations and algebraic equations

Analytic reducibility of nondegenerate centers: Cherkas systems

In this paper we study the center problem for polynomial differential systems and we prove that any center of an analytic differential system is analytically reducible. x˙=y,y˙=P0(x)+P1(x)y+P2(x)y2, We also study the centers for the Cherkas polynomial differential systems where Pi(x) are polynomials of degree n, P0(0)=0 and P′0(0)<0. Computing the focal values we find the center conditions for such systems for degree 3, and using modular arithmetics for degree 4. Finally we do a conjecture about the center conditions for Cherkas polynomial differential systems of degree n.The first author is partially supported by a MINECO/FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant 2014SGR 568, and two grants FP7-PEOPLE-2012-IRSES numbers 316338 and 318999