48 research outputs found

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

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    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

    Existence of limit cycles for some generalisation of the Liénard equations: the relativistic and the prescribed curvature cases.

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    We study the problem of existence of periodic solutions for some generalisations of the relativistic Liénard equation \begin{equation*} \frac{d}{dt}\frac{\dot{x}}{\sqrt{1-\dot{x}^{2}}}+\hat{f}(x,\dot{x})\dot{x}+g(x)=0 , \end{equation*} and the prescribed curvature Liénard equation \begin{equation*} \frac{d}{dt}\frac{\dot{x}}{\sqrt{1+\dot{x}^{2}}}+\hat{f}(x,\dot{x})\dot{x}+g(x)=0 , \end{equation*} where the damping function depends both on the position and the velocity. In the associated phase-plane this corresponds to a term of the form f(x;y)f(x; y) instead of the standard dependence on x alone. By controlling the continuability of the solutions, we are able to prove the existence of at least a limit cycle in the associated phase-plane for both cases, moreover we provide results with a prefixed arbitrary number of limit cycles. Some examples are given to show the applicability of these results

    On the uniqueness of limit cycle for certain Liénard systems without symmetry

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    The problem of the uniqueness of limit cycles for Liénard systems is investigated in connection with the properties of the function F(x). When α and β (α < 0 < β) are the unique nontrivial solutions of the equation F(x) = 0, necessary and sufficient conditions in order that all the possible limit cycles of the system intersect the lines x = α and x = β are given. Therefore, in view of classical results, the limit cycle is unique. Some examples are presented to show the applicability of our results in situations with lack of symmetry

    On the uniqueness of limit cycle for certain Liénard systems without symmetry

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    The problem of the uniqueness of limit cycles for Liénard systems is investigated in connection with the properties of the function F(x)F(x). When α\alpha and β\beta~(α<0<β)(\alpha<0<\beta) are the unique nontrivial solutions of the equation F(x)=0F(x)=0, necessary and sufficient conditions in order that all the possible limit cycles of the system intersect the lines x=αx=\alpha and x=βx=\beta are given. Therefore, in view of classical results, the limit cycle is unique. Some examples are presented to show the applicability of our results in situations with lack of symmetry

    On the uniqueness of limit cycle for certain Liénard systems without symmetry

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    The problem of the uniqueness of limit cycles for Liénard systems is investigated in connection with the properties of the function F(x). When α and β (α < 0 < β) are the unique nontrivial solutions of the equation F(x) = 0, necessary and sufficient conditions in order that all the possible limit cycles of the system intersect the lines x = α and x = β are given. Therefore, in view of classical results, the limit cycle is unique. Some examples are presented to show the applicability of our results in situations with lack of symmetry

    A new approach to study limit cycles on a cylinder

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    We present a new approach to study limit cycles of planar systems of autonomous differential equations with a cylindrical phase space ZZ. It is based on an extension of the Dulac function which we call Dulac-Cherkas function Ψ\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

    Existence of limit cycles for some generalisation of the Liénard equations : the relativistic and the prescribed curvature cases

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    We study the problem of existence of periodic solutions for some generalisations of the relativistic Liénard equation d dt x˙ 1 − x˙ 2 f(x, x˙)x˙ + g(x) = 0 , and the prescribed curvature Liénard equation d dt x˙ 1 + x˙ 2 f(x, x˙)x˙ + g(x) = 0 , where the damping function depends both on the position and the velocity. In the associated phase-plane this corresponds to a term of the form f(x, y) instead of the standard dependence on x alone. By controlling the continuability of the solutions, we are able to prove the existence of at least a limit cycle in the associated phase-plane for both cases, moreover we provide results with a prefixed arbitrary number of limit cycles. Some examples are given to show the applicability of these results

    PI-controlled bioreactor as a generalized Liénard system

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    "It is shown that periodic orbits can emerge in Cholette’s bioreactor model working under the influence of a PI-controller. We find a diffeomorphic coordinate trans-formation that turns this controlled enzymatic reaction system into a general-ized Lie´nard form. Furthermore, we give sufficient conditions for the existence and uniqueness of limit cycles in the new coordinates. We also perform numerical simu-lations illustrating the possibility of the existence of a local center (period annulus). A result with possible practical applications is that the oscillation frequency is a function of the integral control gain parameter.
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