Symmetry-breaking analysis for the general Helmholtz–Duffing oscillator

摘要: Abstract The symmetry breaking phenomenon for a general Helmholtz–Duffing oscillator as function of symmetric parameter in the nonlinear force is investigated. Different values this convert into either Helmholtz or Duffing oscillator. Due to variation parameter, phase space patterns unperturbed will cause huge difference between left-hand homoclinic orbit and right-hand one. In particular, area orbits strictly monotonously decreasing function, while varies only very small range. There exist distinct local supercritical subcritical saddle-node bifurcations at two different centers. existing regions harmonic solutions created by bifurcation curves lead transition amplitude–frequency plane. exists also critical frequency which has effect that value equal value. And, if amplitude coefficient used control it larger than same bifurcation, then global stability system be destroyed lowest cost. Besides frequency, are not unequal, but their effects system’s different. Among them, resulting from negligible, large accomplished quite