Stability for systems of 1-D coupled nonlinear oscillators
In this paper, the authors firstly define a 1-D mechanical system, next they discuss the stability of the null solution of different systems, which correspond to the motion of two coupled nonlinear oscillators, both being under the action of some external forces of differential equations. Each of three cases will be given below is studied by two approaches based on classical arguments, by using differential inequalities and the method of Lyapunov's function. The following ODEs are associated with the given physical system, which describe the motion of oscillators: \[\begin{aligned}\ddot{x}+2f_{1}(t)\dot{x}+\beta(t)x-\gamma(t)y+g_{1}(t,x,y)&=0,\\ \ddot{y}+2f_{2}(t)\dot{y}+\delta(t)y-\gamma(t)x+g_{2}(t,x,y)&=0,\end{aligned}\] where \(\beta(t)=\frac{1}{m}(k_{1}(t)+k_{2}(t))\), \(\delta(t)=\frac{1}{m}k_{2}(t)\), \(f_{1}(t)=\frac{1}{m}\hat{f}_{1}(t)\),\(f_{2}(t)=\frac{1}{m}\hat{f}_{2}(t)\), \(\gamma(t)=\frac{1}{m}k_{2}(t)\),\(g_{1}(t,x,y)=-\frac{1}{m}\hat{g}_{1}(t,x,y)\),\(g_{2}(t,x,y)=-\frac{1}{m}\hat{g}_{2}(t,x,y)\). In Section 2, the main elements of stability theory that will be used throughout the paper are introduced. In Section 3, the stability of the null solution of two coupled damped nonlinear oscillators described by the system above is investigated under the given hypotheses. Later, they give two theorems on uniform stability and asymptotical stability of the null solution to the above equations. In addition, an example for the illustration is also given. In Section 4, the following ODEs, which are associated with a given physical system to describe the motion of oscillators, are given \[\begin{aligned}\ddot{x}+\beta(t)x-\gamma(t)y+g_{1}(t,x,y)&=0,\\ \ddot{y}+2f(t)\dot{y}+\delta(t)y-\gamma(t)x+g_{2}(t,x,y)&=0,\end{aligned}\] here \(\beta(t)=\frac{1}{m}(k_{1}(t)+k_{2}(t))\), \(\delta(t)=\frac{1}{m}k_{2}(t)\), \(\forall t\in\mathbb{R}_{+}\), \(\gamma(t)=\frac{1}{m}k_{2}(t)\),\(\forall t\in\mathbb{R}_{+}\) is the coupling factor of the oscillators. The function \(f:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) is the damping coefficient of the second oscillator and \(g_{1},g_{2}:\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) are external forces acting on oscillators. With admitted hypotheses, a preliminary result, regarding the stability of the null solution to the given system is proved via a lemma. Then, it is exemplified as for small initial data, the solution of the last given system, obtained with Matlab and is plotted on some time intervals. The main result of this section is given with a theorem on stability and limit to the null solution to the second system. Then, the stability result is given with another theorem on uniform stability to the null solution to the system. In Section 5, the stability of the null solution to a system of two 1-D coupled undamped nonlinear oscillators is studied. The following system of ODEs associated with the defined physical application \[\begin{aligned}\ddot{x}+\beta(t)x-\gamma(t)y+g_{1}(t,x,y)&=0,\\\ddot{y}+\delta(t)y-\gamma(t)x+g_{2}(t,x,y)&=0,\end{aligned}\] here \(\beta(t)=\frac{1}{m}(k_{1}(t)+k_{2}(t))\), \(\delta(t)=\frac{1}{m}k_{2}(t)\), \(\forall t\in\mathbb{R}_{+}\),\(\gamma:\mathbb{R}_{+}\rightarrow\mathbb{R}\),\(\gamma(t)=\frac{1}{m}k_{2}(t)\),\(\forall t\in\mathbb{R}_{+}\) is the compiling factor of the oscillators, and \(g_{1},g_{2}:\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) are external forces acting on oscillators. Finally, a theorem is proved so that the null solution of the system given above is uniformly stable. Then, an example is given, the solution of the system above is obtained with Matlab and plotted on some time intervals.
scientific article; zbMATH DE number 7347888
scientific article