A revisit on the characteristics of Yao-Cheng non-linear oscillator

Abstract

We revisited the characteristic features of Yao-Cheng non-linear oscillator by employing both analytical as well as numerical approaches. Our study indicates that the solution of the non-linear oscillator shows oscillatory behaviour. Further, the phase portrait for this oscillator system reflects the closed loop signifying the stability of the system. Our results suggest the absence of damping in this physical system.

In recent years, the study on Yao-Cheng non-linear oscillator has attracted the research attention due to its usefulness for understanding the oscillation behaviour associated with many physical systems. ¹ The equation of motion of the Yao-Cheng non-linear oscillator is of the form ¹

$\ddot{u}+\frac{ku}{1+c{\dot{u}}^2}+bu=0$

(1)

where

$\dot{u}=\frac{du}{dt}and\ddot{u}=\frac{d^{2}u}{d{t}^2}$

(2)

In this work, we revisited the characteristic features of the above oscillator by using both analytical and numerical approaches. We solve the non-linear differential equation (equation (1)) numerically by using two different approaches: (i) mathematica tool and (ii) Runge–Kutta method. Both the approaches with the boundary conditions: $u\left(0\right)=A;u^{\prime }\left(0\right)=0$ yield the same results effectively. Figure 1 shows the solution of the equation (1) as obtained by Runge–Kutta method. Further, the information on the stability of the system is obtained by studying phase portrait. Figure 2 and 3 show the typical phase portrait curves reflect the closed loop signifying the stability of the system.^2–4 These results suggest the absence of damping in the system. We therefore propose an analytical solution for the studied system which is discussed in the followings.

Figure 1.

Time evolution of the numerical solution of equation (1) for $k=1,c=2$ and $b=1$ as obtained by Runge–Kutta Method.

Figure 2.

Numerically obtained phase portrait of equation (1) for $k=1,c=2,A=5\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}10$ and $b=1$ by Runge–Kutta Method.

Figure 3.

Numerically obtained phase portrait of equation (1) for $k=1,c=2,A=5\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}10$ and $b=1$ by mathematical tool.

Considering

$u=A\cos \omega t,$

(3)

We find the expression of frequency ( $\omega$ ) corresponding to equation (1) as

${\omega }^2=\frac{k}{1+\frac{c{A}^2{\omega }^2}{2}}+b$

(4)

Considering the parameters; $k=1,c=2$ and $b=1$ , the analytical expression of $\omega$ could easily be simplified as

${\omega }^2=\frac{\left(A^2-1\right)+\sqrt{1+A^4+6A^2}}{2A^2}$

(5)

We found that the solution of equation (1) does not show any damping (Figure 4). Further, the phase portrait obtained analytically also yield closed loop (Figure 5). The outcome of both the numerical results as stated above matched nicely with the analytical result. All the results show the oscillation behaviour for the Hamiltonian without showing any signature of damping. Further, the fractal modification of Yao-Cheng oscillator also reflects the closed phase portrait. ¹

Figure 4.

Time evolution of the analytical solution of equation (1) for $k=1,c=2$ and $b=1.$

Figure 5.

Analytically obtained phase portrait of equation (1) for $k=1,\hspace{0.17em}c=2,A=5\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}10$ and $b=1.$

In conclusion, we revisited the work on the non-linear oscillator with damping term of Yao and Cheng oscillator by employing analytical as well as numerical approaches. In both the approaches, the solutions of the non-linear oscillator show oscillatory nature without showing any signature of damping. Further, the occurrence of closed phase portrait of this system is justifying the stability of the system. It is therefore incorrect to argue the existence of damping in this physical system.