Periodic solutions and limit cycles of mixed Lienard-type differential equations

Abstract

<abstract><p>In the attractive research field of nonlinear differential equations, there are a few studies devoted to finding exact and explicit harmonic and isochronous periodic solutions and limit cycles. In this contribution, we present some classes of polynomial mixed Lienard-type differential equations that can generate many equations with exact solutions. These classes of equations constitute counterexamples of the classical existence theorems.</p></abstract>

Isochronous Oscillations of Nonlinear Systems

“…However, the recent literature shows that the classical existence theorems are not sufficient to predict the behavior of nonlinear dynamic systems. Additionally, qualitative results are not sufficient for engineering and industrial applications [23]. By definition [34,35], a nonautonomous dynamic system is distinguished from an autonomous system by the fact that the solution of the associated initial value problem depends not only on the elapsed time t À t 0 but also on the initial time t 0 .…”

“…Other counterexamples of classical existence theorems can be seen in Refs. [20][21][22][23][24][25][26][27]. If some progress has been made with the work of Calogero and coworkers [28], it will be very difficult to say the same thing concerning the dynamic systems represented by nonlinear differential equations having an exact elementary function solution, more precisely an exact explicit isochronous sinusoidal solution before the contribution of Monsia and his group (see Refs.…”

Defining Jacobian elliptic functions via nonpolynomial differential equations

“…known as the hybrid Rayleigh-Van der Pol system that can exhibit stable limit cycle behavior [10][11][12], where γ is a constant. Equation ( 8) belongs to the following general class of equations:…”

“…called the Lienard-Levinson-Smith system [10][11][12][13][14][15][16][17], where ) , ( x x h  is a function of the solution x and its derivative x  . Equation (9) has been the subject of many studies in connection with the second part of the Hilbert 16th problem for counting the maximum number of limit cycles when ) , ( x x h  and ) (x g are polynomial [18].…”

“…( 9) to have at least one periodic solution have been established through the so-called Lienard-Levinson-Smith theorem [10][11][12][13][14][15][16][17]. However, it has been proven that many differential systems of form ( 9) can exhibit explicit sinusoidal solutions [11,12] with algebraic limit cycles of degree 2. It is interesting to note that as 0 → γ , Eq.…”