The autonomous Duffing–Holmes oscillator ẋ = y, ẏ = x – x3 + by – kz, ż = w(y – z), depending on the three parameters b, k, and w, has been studied previously by several authors who showed that for certain parameter values, it exhibits chaotic motion, or that numerically it has two periodic orbits coming from a Hopf bifurcation, or that it can have three equilibria for some given values of the parameters. Here we provide new results on the integrability and the global dynamics of the autonomous Duffing–Holmes oscillator using its first integrals and Darboux invariants when these exist for some given values of its parameters.
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