Isochronous waveforms of Liénard equations via commutative factorization

Abstract

Isochronous waveform solutions of homogeneous Liénard equations are obtained by a modification of the nonlinear factorization method of Rosu and Cornejo-Pérez. The scheme is based on the assumption that the intermediate function $Φ$ that can be introduced in this factorization method depends on both the dependent and independent variables of the nonlinear equation. The method is applied to three cases, a noted cubic anharmonic oscillator, a Liénard-reduced form of the Sharma-Tasso-Olver evolution equation, and the cubic-quintic Wilson's Liénard equation. All these cases are written in a commutative factored form that allows to obtain the general solutions as solutions of a certain type of Bernoulli differential equation. A theorem is also given asserting the general form of the Liénard equation, i.e., for given polynomial degree n of its coefficients, which can be solved by this method. The conditions under which these equations can be also approached by non-local transformations are established.

Figures (5)

• Figure 1: (a) Two isochronous solutions $x(t)$, of equal period as determined by the two bullets, of the anharmonic cubic oscillator for the following values of the parameters: $|{\rm A}|=1$, $\delta=0$, $\omega=3$, and ${\rm k}=2.5$ (a bounded case) and 3.5 (a singular case). (b) The corresponding phase portraits.
• Figure 2: (a) Two isochronous solutions $x(t)$, of equal period as determined by the two bullets, of the anharmonic cubic oscillator for the following values of the parameters: $|{\rm A}|=1$, $\delta=0$, $\omega=3$, and ${\rm k}=-2.5$ (a bounded case) and -3.5 (a singular case). (b) The corresponding phase portraits.
• Figure 3: (a) Bounded and singular isochronous solutions $U(z)$ for $b=1$ ($\epsilon=0$) when they are identical to the travelling STO solutions $u(z)$. The other parameters are ${\rm A}=1$, $\delta=0$, $v=1$, and $\alpha=0.2, 0.6$, and 1. (b) The corresponding phase portraits.
• Figure 4: (a) Isochronous waveform solutions of the Wilson polynomial Liénard equation obtained from (\ref{['wpl5']}) for ${\rm A}=0$, $\delta=0$ and three values of the parameter $\mu$; (b) their phase portraits.
• Figure 5: (a) Non-isochronous regular and singular Wilson waveforms from (\ref{['wpl5']}), unbroken lines (${\rm A}=1$) and dotted lines (${\rm A}=-1$), respectively, $\delta=0$, and $\mu=$ 0.5 and 1. (b) their phase portraits.