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UDC 517.917
INVESTIGATION OF A NONLINEAR SYSTEM
V. A. KOBYSHEV, B. I. SHAKHTARIN
Consider the second-order differential equation
\[ \ddot{x}+f(x)\dot{x}+g(x)=0. \tag{1} \]
It is easy to prove that if the functions \(f(x)\) and \(g(x)\) in equation (1) are such that 1) the conditions for existence and uniqueness of a solution are satisfied, 2) \(xg(x)>0\) \((xg(x)<0)\) for
\[ a<x<b,\quad x\int_{0}^{x} f(x)\,dx>0\quad \left(x\int_{0}^{x} f(x)\,dx<0\right) \]
for \(a<x<b\), where \(-\infty\leq a\leq b\leq+\infty\), then equation (1) has no periodic solutions in the strip \(a<x<b\).
Indeed, let us pass to the variables of the Liénard phase plane, putting \(y=\dot{x}+F(x)\), where \(F(x)=\int_{0}^{x} f(x)\,dx\). Then we obtain the following system, equivalent to equation (1):
\[ \begin{aligned} \dot{x}&=y-F(x),\\ \dot{y}&=-g(x). \end{aligned} \tag{2} \]
The only equilibrium state of system (2) is the point \((0,0)\). A periodic solution of equation (1) corresponds to a cycle of system (2).
Suppose that equation (1) has a periodic solution (i.e., system (2) has a cycle). Construct the Lyapunov function
\[ V(x,y)=\frac{1}{2}y^{2}+\int_{0}^{x} g(x)\,dx, \]
then
\[ \left.\frac{dV}{dt}\right|_{(2)} =y\dot{y}+g(x)\dot{x} =-g(x)F(x)<0\quad (>0) \]
for \(a<x<b\).
The function decreases (increases) along the cycle, which contradicts its single-valuedness. Hence it follows that there are no periodic solutions of equation (1) in the strip \(a<x<b\).
Now consider an equation of the form
\[ \ddot{x}+\left(\alpha+\frac{a}{\alpha}\cos x\right)\dot{x}+\sin x=\beta \tag{3} \]
with real nonnegative parameters \(a,\alpha,\beta\). (Equations of this kind find wide application in various devices of radio engineering and automatic control (see, for example, [2, 3]).) We shall prove, contrary to the assumption made in [3], that equation (3) has no periodic solutions.
Equation (3) is equivalent to the system of equations
\[ \dot{x}=y\equiv P(x,y), \]
\[ \dot{y}=-\left(\alpha+\frac{a}{\alpha}\cos x\right)y-\sin x+\beta\equiv Q(x,y). \tag{4} \]
Investigation of a Nonlinear System
The phase space of the system under consideration is the surface of a cylinder with axis parallel to the \(y\)-axis. The qualitative picture of the system in phase space, as is known [1], is determined by the character and position of the special points, separatrices, and limit cycles. On the phase cylinder two types of limit cycles may occur: of the first and of the second kind [1].
Equation (3) is a special case of (1) for
\[ f(x)=\alpha\left(1+\frac{a}{\alpha^{2}}\cos x\right),\qquad g(x)=\sin x-\beta . \]
The coordinates of the equilibrium states are determined by the equalities
\[ \begin{aligned} x_1&=\pi-\arcsin\beta, & y_1&=0,\\ x_2&=\arcsin\beta, & y_2&=0. \end{aligned} \tag{5} \]
The equilibrium state \((x_1,y_1)\), for any \(a\gtreqless 0\), \(\alpha>0\), and \(\beta<1\), is a saddle, while \((x_2,y_2)\) is a focus or a node. For \(\beta=1\) the node \((x_2,y_2)\) and the saddle \((x_1,y_1)\) coalesce, forming a compound equilibrium state of saddle-node type. For \(\beta>1\) the equilibrium states disappear.
From simple geometrical considerations it is clear that if system (4) has no limit cycles of the first kind in the strip of width \(2\pi\)
\[ -\pi-\arcsin\beta<x<\pi-\arcsin\beta,\qquad -\infty<y<+\infty, \tag{6} \]
then there are none in the whole plane. Therefore, to prove that equation (3) has no periodic solutions, it is sufficient to show that system (4) has no limit cycles of the first kind in the strip (6) (they correspond to periodic solutions of equation (3)).
Move the origin to the point \((x_2,y_2)\), putting
\[ x=\varphi+\arcsin\beta, \]
then equation (3) takes the form
\[ \ddot{\varphi}+f_1(\varphi)\dot{\varphi}+g(\varphi)=0, \tag{7} \]
where
\[ f_1(\varphi)=\alpha\left[1+\frac{a}{\alpha^{2}}\cos(\varphi+\arcsin\beta)\right], \tag{8} \]
\[ g(\varphi)=\sin(\varphi+\arcsin\beta)-\beta. \tag{9} \]
The strip (6) becomes the strip
\[ -\pi-2\arcsin\beta<\varphi<\pi-2\arcsin\beta,\qquad -\infty<\dot{\varphi}=z<+\infty. \tag{10} \]
Equation (7) in the strip (10) satisfies the conditions of the assertion proved above. Indeed, condition 1) is obviously satisfied. Further, we have \(\varphi g(\varphi)>0\) for
\[ -\pi-2\arcsin\beta<\varphi<\pi-2\arcsin\beta. \]
Since
\[ \int_0^\varphi f_1(\varphi)\,d\varphi =\alpha\left[\varphi+\frac{a}{\alpha^2}g(\varphi)\right], \]
it follows that
\[ \varphi\int_0^\varphi f_1(\varphi)\,d\varphi>0 \]
for
\[ -\pi-2\arcsin\beta<\varphi<\pi-2\arcsin\beta. \]
Thus, equation (7) in the strip (10) has no periodic solutions. The latter means that system (4) has no cycles of the first kind in the strip (6), and consequently none in the entire phase plane.
References
- Andronov A. A., Vitt A. A., Khaikin S. E. Theory of Oscillations. Fizmatgiz, 1959.
- Belostina L. N. Izvestiya VUZov. Radiofizika, 2, No. 2, 1959.
- Rey T. J. Automatic phase control: theory and design. Proc. IRE, No. 10, 1960.
Received by the editors
March 23, 1965
Leningrad State University
named after A. A. Zhdanov