The phenomenon of period-doubling bifurcations in the Duffing’s oscillator with negative linear stiffness is investigated with the aid of approximate analytical methods and computer simulation. Making use of a Hill’s type variational equation together with the ideas drawn out from Floquet theory, it is found that a particular type of subharmonic instability is the one that is responsible for the occurrence of period-doublings in this system. This fact is confirmed by the good agreement between the true critical forcing frequency at which bifurcations are first observed, and the one obtained theoretically. Finally, a threshold criterion for the onset of period-doublings is also proposed and compared with computer simulation results.

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