A third-order extension to the Liénard oscillator and it’s competitive modes analysis - Van Gorder, R (Springer, 2016)
We study the dynamics of nonlinear differential equations of the form x + f (x)x + g(x, x)x + h(x) = 0, which is a third-order extension to the Liénard oscillator equation. This equation holds a number of interesting and physically relevant third-order dynamical systems as special cases. We present a general competitive modes analysis in order to derive some necessary conditions under which the such systems admit chaos. For several of the interesting reductions of the equations, we demonstrate that the approach allows us to determine parameter values and initial conditions which permit chaotic trajectories. We also demonstrate that, while competitive modes can be useful for finding chaotic regimes, the competitiveness conditions themselves are not a sufficient condition for chaos. In this way, we are able to discuss both the benefits and the limitations of the competitive modes approach. By doing this, we demonstrate that there are several reduction of this general third order equation which give chaos, including those of interest in theoretical physics and electrical engineering.