arXiv subjects

Si Mohamed Sah

Publications and source records attributed to Si Mohamed Sah.

4 recordsLinked to original sources

arXiv2019

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 $$0 \neq (-1)^{n+1}b\,T^2< \tfrac{3}{2}π^2.$$ 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.

math.DS↗

arXiv2019

Unbounded sequences of stable limit cycles in the delayed Duffing equation: an exact analysis

The delayed Duffing equation $\ddot{x}(t)+x(t-T)+x^3(t)=0$ is shown to possess an infinite and unbounded sequence of rapidly oscillating, asymptotically stable periodic solutions, for fixed delays such that $T^2<\tfrac{3}{2}π^2$. In contrast to several previous works which involved approximate solutions, the treatment here is exact.

math.DS↗

arXiv2017

Background force compensation in dynamic atomic force microscopy

Background forces are linear long-range interactions of the cantilever body with its surroundings that must be compensated for in order to reveal tip-surface force, the quantity of interest for determining material properties in atomic force microscopy. We provide a mathematical derivation of a method to compensate for background forces, apply it to experimental data, and discuss how to include background forces in simulation. Our method, based on linear-response theory in the frequency domain, provides a general way of measuring and compensating for any background force and it can be readily applied to different force reconstruction methods in dynamic AFM.

cond-mat.mes-hall↗

arXiv2016

Delay Terms in the Slow Flow

This work concerns the dynamics of nonlinear systems that are subjected to delayed self-feedback. Perturbation methods applied to such systems give rise to slow flows which characteristically contain delayed variables. We consider two approaches to analyzing Hopf bifurcations in such slow flows. In one approach, which we refer to as approach I, we follow many researchers in replacing the delayed variables in the slow flow with non-delayed variables, thereby reducing the DDE slow flow to an ODE. In a second approach, which we refer to as approach II, we keep the delayed variables in the slow flow. By comparing these two approaches we are able to assess the accuracy of making the simplifying assumption which replaces the DDE slow flow by an ODE. We apply this comparison to two examples, Duffing and van der Pol equations with self-feedback.

math.DS↗