Integrable modulation, curl forces and parametric Kapitza equation with trapping and escaping

In this present communication, the integrable modulation problem has been applied to study parametric extension of Kapitza rotating shaft problem, which is a prototypical example curl force as formulated by Berry and Shukla in (JPA 45:305201, 2012) associated with simple saddle potential. The problems yield time-dependent systems. Hamiltonian first integrals linear nonlinear equation (PKE) monkey potentials have given. construction illustrated choosing $$ \omega (t)=a +b\cos t$$ that maps Mathieu-type equations, Mathieu PKE. We dynamics these equations. most interesting finding mixed mode particle trapping escaping via heteroclinic orbits depicted Mathieu–Kapitza equation, are absent case nonparametric cases.