Noise effect on the signal transmission in an underdamped fractional coupled system

Abstract

The signal transmission phenomenon in an underdamped fractional system composed of two harmonically coupled particles, but only the first particle, which is driven by a periodic force, is studied. We obtain the analytical expressions for the steady-state response of the two coupled particles by applying the stochastic averaging method and define the signal transmission factor to characterize signal transmission efficiency. We analyze the impact of noise on signal transmission efficiency and provide the discriminant criterion for the emergence of the signal transmission enhancement phenomenon. We also analyze the resonance behavior of the signal transmission factor and provide the discriminant criterion for the emergence of different types of resonance behavior based on system parameters. We draw the phase diagrams for varying resonance behavior versus different parameters and study the influences of the parameters on resonance behavior. It is found that the coupling strength and fractional-order exert important influences on the signal transmission efficiency and resonance behavior, and the fractional coupled system exhibits complicated dynamic behaviors than the integer coupled system. Lastly, numerical simulations for the signal transmission factor $\eta$, and the output signal-to-noise ratio (SNR) transmission gain $\text {SNR}_{\eta }$ are performed. We can control signal transmission efficiency and resonance behavior within a certain range by understanding the effects of system parameters on the underdamped fractional coupled system, and it has potential applications in modern science.

Appendix for section 4.1

Coefficients $a_i, b_i, c_i, i=1,2$ of Eq. (18) are as follow:

$a_1=[k^2+b^4+(\gamma b^\alpha )^2]+2kb^2\cos (2\theta )+2k\gamma b^\alpha \cos (\alpha \theta )+2\gamma b^{2+\alpha }\cos [(2-\alpha )\theta ],$

$a_2=\varepsilon ^2,$

$b_1=[2k{\widetilde{g}}_0+2b^2{\widetilde{g}}_2+2\gamma b^\alpha {\widetilde{g}}_3+2k{\widetilde{F}}_{11}]+[2(k{\widetilde{g}}_2+b^2{\widetilde{g}}_{0})+2b^2{\widetilde{F}}_{11}]\cos (2\theta ) +[2(k{\widetilde{g}}_{3}+\gamma b^\alpha {\widetilde{g}}_{0})+2\gamma b^\alpha {\widetilde{F}}_{11}]\cos (\alpha \theta )+2(b^2{\widetilde{g}}_{3}+\gamma b^\alpha {\widetilde{g}}_{2})\cos [(2-\alpha )\theta ]+2b^2{\widetilde{F}}_{12}\sin (2\theta )+2\gamma b^\alpha {\widetilde{F}}_{12}\sin (\alpha \theta ),$

$b_2=(-2\varepsilon )({\widetilde{h}}_{0}+{\widetilde{F}}_{31}),$

$c_1=[{\widetilde{g}}_{0}^2+{\widetilde{g}}_{2}^2+{\widetilde{g}}_{3}^2+2{\widetilde{g}}_{0}{\widetilde{F}}_{11}+{\widetilde{F}}_{11}^2+{\widetilde{F}}_{12}^2]+[2{\widetilde{g}}_{0}{\widetilde{g}}_{2}+2{\widetilde{g}}_{2}{\widetilde{F}}_{11}]\cos (2\theta )+[2{\widetilde{g}}_{0}{\widetilde{g}}_{3}+2{\widetilde{g}}_{3}{\widetilde{F}}_{11}]\cos (\alpha \theta )+2{\widetilde{g}}_{2}{\widetilde{g}}_{3}\cos [(2-\alpha )\theta ]+2{\widetilde{g}}_{2}{\widetilde{F}}_{12}\sin (2\theta )+2{\widetilde{g}}_{3}{\widetilde{F}}_{12}\sin (\alpha \theta ),$

$c_2={\widetilde{h}}_{0}^2+2{\widetilde{h}}_{0}{\widetilde{F}}_{31}+{\widetilde{F}}_{31}^2+{\widetilde{F}}_{32}^2,$

$k=\omega _0^2+\varepsilon ,$

$b=\sqrt{\lambda ^2+\Omega ^2},$

$\theta =\arctan (\Omega /\lambda ),$

$g_1=(\varepsilon ^2-k^2)\gamma \Omega ^\alpha ,$

$g_4=b^4(\Omega ^2-k),$

$g_5=(\gamma b^\alpha )^2(\Omega ^2-k),$

$g_6=2\gamma b^{2+\alpha }(\Omega ^2-k),$

$g_7=2k\gamma \Omega ^\alpha b^2,$

$g_8=2k\gamma ^2(\Omega b)^\alpha$

$g_9=\gamma \Omega ^\alpha b^4,$

$g_{10}=\gamma ^3(\Omega b^2)^\alpha ,$

$g_{11}=2(\gamma b)^2(\Omega b)^\alpha ,$

$h_1=-2\varepsilon kb^2,$

$h_2=-2\varepsilon k\gamma b^\alpha ,$

$h_3=-\varepsilon b^4,$

$h_4=-\varepsilon (\gamma b^\alpha )^2,$

$h_5=-2\varepsilon \gamma b^{2+\alpha },$

${\widetilde{g}}_0=(k-\Omega ^2)(\varepsilon ^2-k^2),$

${\widetilde{g}}_2=2kb^2(\Omega ^2-k),$

${\widetilde{g}}_3=2k\gamma b^\alpha (\Omega ^2-k),$

${\widetilde{h}}_0=\varepsilon (\varepsilon ^2-k^2),$

${\widetilde{F}}_{11}=g_1\cos (\pi \alpha /2)+g_4\cos (4\theta )+g_5\cos (2\alpha \theta )+g_6\cos (2\theta +\alpha \theta )-g_7\cos (2\theta +\pi \alpha /2)-g_8\cos (\pi \alpha /2+\alpha \theta )-g_9\cos (\pi \alpha /2+4\theta )-g_{10}\cos (\pi \alpha /2+2\alpha \theta )-g_{11}\cos (2\theta +\alpha \theta +\pi \alpha /2),$

${\widetilde{F}}_{12}=g_1\sin (\pi \alpha /2)+g_4\sin (4\theta )+g_5\sin (2\alpha \theta )+g_6\sin (2\theta +\alpha \theta )-g_7\sin (2\theta +\pi \alpha /2)-g_8\sin (\pi \alpha /2+\alpha \theta )-g_9\sin (\pi \alpha /2+4\theta )-g_{10}\sin (\pi \alpha /2+2\alpha \theta )-g_{11}\sin (2\theta +\alpha \theta +\pi \alpha /2),$

${\widetilde{F}}_{31}=h_1\cos (2\theta )+h_2\cos (\alpha \theta )+h_3\cos (4\theta )+h_4\cos (2\alpha \theta )+h_5\cos (2\theta +\alpha \theta ),$

${\widetilde{F}}_{32}=h_1\sin (2\theta )+h_2\sin (\alpha \theta )+h_3\sin (4\theta )+h_4\sin (2\alpha \theta )+h_5\sin (2\theta +\alpha \theta ).$