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We explore stability and instability of rapidly oscillating solutions $x(t)$for the hard spring delayed Duffing oscillator $$x''(t)+ax(t)+bx(t-T)+x^3(t)=0.$$ Fix $T>0$. We target periodic solutions $x_n(t)$ ofsmall minimal periods $p_n=2T/n$, for integer $n\rightarrow \infty$, and withcorrespondingly large amplitudes. Note how $x_n(t)$ are also marginally stablesolutions, respectively, of the two standard, non-delayed, Hamiltonian Duffingoscillators $$x''+ ax+(-1)^nbx+x^3=0.$$ Stability changes for the delayedDuffing oscillator. Simultaneously for all sufficiently large $n\geq n_0$, weobtain local exponential stability for $(-1)^nb<0$, and exponential instabilityfor $(-1)^nb>0$, provided that $$0 \neq (-1)^{n+1}b\,T^2< \tfrac{3}{2}\pi^2.$$We interpret our results in terms of noninvasive delayed feedback stabilizationand destabilization for large amplitude rapidly periodic solutions of thestandard Duffing oscillators. We conclude with numerical illustrations of ourresults for small and moderate $n$ which also indicate a Neimark-Sacker torusbifurcation at the validity boundary of our theoretical results.

Coexistence of infinitely many large, stable, rapidly oscillating periodic solutions in time-delayed Duffing oscillators