Integrability of Van der Pol-Duffing oscillator system in three-dimensional vector field

In this work, we focus on studying the integrability of the following three-dimensional Van der Pol-Duffing system (x) over dot =-m(x(3) - mu x - y), (y) over dot = x - y - z, (z) over dot = beta y. More precisely, if m beta not equal 0, then the above system has no analytic and nor Darboux first integrals at the neighborhood of the origin. Also, the stability and instability of the singular points are employed to investigate the C-1 integrability of this type of system.

Automated summary

The integrability of the three-dimensional Van der Pol-Duffing system was studied, revealing that under certain conditions the system has no analytic and Darboux first integrals at the neighborhood of the origin. The stability and instability of the singular points were used to investigate the C-1 integrability of this type of system.