New aspects of the orbital normal form of the Hopf singularity: The Rayleigh and the van der Pol forms

In this paper, we present different possibilities for the simplest orbital normal form of the Hopf bifurcation. Beyond the classical normal form we present other forms that have the structure of the Rayleigh and the van der Pol equations. The results obtained are applied to obtain orbital normal forms for Newtonian systems with one degree of freedom and Liénard systems. •Different possibilities for simplest orbital normal forms of Hopf bifurcation.•Normal forms with structure of Rayleigh and van der Pol equations.•1DOF Newtonian and Liénard systems preserving Newtonian structure.•Computation of normal form coefficients (invariants of the vector field).•Effect of the terms of the force in the normal form for Newtonian systems.