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

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 $x_n(t)$ of small minimal periods $p_n=2T/n$, for integer $n\rightarrow \infty$, and with correspondingly large amplitudes. Note how $x_n(t)$ are also marginally stable solutions, respectively, of the two standard, non-delayed, Hamiltonian Duffing oscillators $$x''+ ax+(-1)^nbx+x^3=0.$$ Stability changes for the delayed Duffing oscillator. Simultaneously for all sufficiently large $n\geq n_0$, we obtain local exponential stability for $(-1)^nb<0$, and exponential instability for $(-1)^nb>0$, 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 stabilization and destabilization for large amplitude rapidly periodic solutions of the standard Duffing oscillators. We conclude with numerical illustrations of our results for small and moderate $n$ which also indicate a Neimark-Sacker torus bifurcation at the validity boundary of our theoretical results.