Abstract
We consider a family of cubic Liénard oscillators with linear damping. Particular cases of this family of equations are abundant in various applications, including physics and biology. There are several approaches for studying integrability of the considered family of equations such as Lie point symmetries, algebraic integrability, linearizability conditions via various transformations and so on. Here we study integrability of these oscillators from two different points of view, namely, linearizability via nonlocal transformations and the Darboux theory of integrability. With the help of these approaches we find two completely integrable cases of the studied equation. Moreover, we demonstrate that the equations under consideration have a generalized Darboux first integral of a certain form if and only if they are linearizable.
Highlights
In this work we study the following family of Liénard equations yzz + ( a1 y + a0 )yz + b3 y3 + b2 y2 + b1 y + b0 = 0, (1)
We find linearization conditions for (2) without this restriction and show that they lead to a new non-trivial integrable case of (2)
We believe that Theorem 2 provides a new integrable case of (2) since linearization problem for (2) via (4) with Fz 6= 0 has not been considered previously
Summary
In this work we study the following family of Liénard equations yzz + ( a1 y + a0 )yz + b3 y3 + b2 y2 + b1 y + b0 = 0,. The main aim of this work is to study various aspects of the integrability of (1) and their interconnections and find new completely integrable cases of (1). We consider linearizability conditions for (1) via different classes of nonlocal transformations. We demonstrate that the general solution for the linearizable cases can be constructed in the parametric form. We consider linearizability conditions for (2) via nonlocal transformations and provide new integrable cases of (2). In the last section we briefly summarise and discuss our results
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