Full Text
Certain Theorems on the Existence of a Periodic Solution and on Determining the Boundary of the Region of Asymptotic Stability of the Trivial Solution of a System of Ordinary Differential Equations
N. N. Vinogradov
§ 1
Consider a system of two equations of the form
\[ \dot{x}=F(x,y),\quad \dot{y}=f(x,y) \tag{1.1} \]
and a system of three equations of the form
\[ \dot{x}=F(x,y), \]
\[ \dot{y}=f(x,y),\quad \dot{z}=\Phi(x,y,z). \tag{1.2} \]
Assume that the origin is the unique singular point of each system and that, in some neighborhood of it, the functions \(F, f, \Phi\) satisfy the usual conditions of existence and uniqueness. The system (1.1) was considered in the general case by V. V. Nemytskii [1], [2]. With the aid of polygonal lines constructed according to certain rules, he obtained conditions for the existence of a periodic solution, but his method requires constructing a large number of links.
The study of periodic motions of systems of two equations is the subject of papers [3—12] and some others. Systems of three and more equations are considered in [13—16]. In 1961 the author [17] considered the systems (1.1) and (1.2), and obtained conditions for the existence of a periodic solution. In doing so the author succeeded in obtaining a rather rough estimate of the location of the closed solution itself. In the present paper a method will be indicated for a more accurate estimate of the location of the closed solution and for an approximate determination of the boundary of the region of asymptotic stability of the unperturbed motion of the system (1.2). To this end we construct two families of polygonal lines—outer polygonal lines and inner polygonal lines.
The rules for constructing the outer polygonal line are given in the author’s aforementioned paper [17]. A polygonal line issuing from the point \(x=x_0\) will be called inner if it is situated between the rest point and the trajectory of the system (1.1) issuing from the point \(x=x_0\). The inner polygonal lines, the trajectories of the system (1.1), and the outer polygonal lines generate topological transformations of intervals of the half-axis \(x>0\) into itself. Denote them respectively by \(T^i, T, T^l\). We can now apply the fixed-point method to the solution of the problems posed. A point fixed with respect to \(T^i\) will be denoted by \(x_0^i\), and a point fixed ...
relative to \(T\), denote by \(x_0\) a point fixed relative to \(T^{l}\), and denote it by \(x_0^{l}\).
The trajectory issuing from the point \((x_0^{i},0)\), as \(t\) increases, moves away from the rest point, whereas the trajectory issuing from the point \((x_0^{l},0)\), as time increases, on the contrary is attracted to it. If \(x_0^{i}<x_0^{l}\), then in the interval \((x_0^{i},x_0^{l})\) the axis of abscissas intersects a periodic solution of system (1.1). If in the interval \((0,x_0^{i})\) \(T^{i}x>x\), i.e., the inner polygonal line bends outward, then the closed inner polygonal line passing through the point \((x_0^{i},0)\) may be taken as an approximate boundary of the domain of asymptotic stability of the trivial solution of the system differing from system (1.1) only in the signs of the left-hand sides (to carry out the transition it is enough to let \(t\) tend to \(-\infty\)). If, however, in the interval \((0,x_0^{l})\) \(T^{i}x<x\), then in this interval the half-axis may intersect a periodic solution of system (1.1) different from the solution passing through the point \((x_0,0)\). Therefore the inner closed polygonal line can be taken as the boundary of the domain of attraction only if it has been proved by another method [18] that such a solution does not exist.
Let us give the rules for constructing the links of the inner polygonal line. Denote by \(A(x_0,y_0)\) the beginning of a link, and by \(k\) the bound of the slope coefficient of the trajectory (upper or lower, depending on the quadrant and the type of sector). In a sector of type \(\dot R<0\), the equation of the link has the form
\[ y-y_0=k(x-x_0). \]
In a sector of type \(\dot R>0\), the link is drawn parallel to the ordinate axis, \(x=x_0\). In a sector of type \(\dot\varphi>0\), the link is parallel to the abscissa axis in the 1st and 3rd quadrants; its equation has the form \(x=x_0\) or \(y-y_0=k(x-x_0)\), if the right boundary of the sector coincides with the half-axis \(y>0\) in the 2nd quadrant, or the left boundary of the sector coincides with the half-axis \(y<0\) in the 4th quadrant. In a sector of type \(\dot\varphi<0\), the link is parallel to the abscissa axis in the 2nd and 4th quadrants; its equation has the form \(x=x_0\) or \(y-y_0=k(x-x_0)\), if the left boundary of the sector coincides with the half-axis \(y>0\) in the 1st quadrant, or if the right boundary of the sector coincides with the half-axis \(y<0\) in the 3rd quadrant.
Remark 1. If the boundary of a sector of type \(\dot R>0\) on which the point \((x_0,y_0)\) lies is the half-axis \(y>0\) (\(y<0\)), then as the link we take the segment of the straight line \(y=y_0\) from the point \((x_0,y_0)\) to its intersection with the other boundary of the sector under consideration (if the other boundary coincides with the abscissa half-axis, then our system has no periodic solutions).
Remark 2. In a sector of type \(\dot\varphi>0\) the coefficient \(k\) may turn out to be unbounded or equal to 0 on the boundary of the sector. In this case we divide the sector into two parts by a ray issuing from the rest point, estimate \(k\) in that part of the sector on whose boundary the point \((x_0,y_0)\) lies. The link of the polygonal line is chosen in the form of a segment of the straight line \(y-y_0=k(x-x_0)\) from the point \((x_0,y_0)\) to its intersection with the ray at the point \((x_1,y_1)\). In the second part of the sector, as the link of the polygonal line we take the segment \(y=y_1\) or \(x=x_1\), depending on the quadrant. Here \(k\) is set equal to the upper bound of the slope coefficient of the trajectory. The analogous remark concerning a sector of type \(\dot\varphi<0\) is quite evident.
In practice it is much more difficult to construct the inner polygonal line than the outer one. Below we shall consider several systems for which this can be done effectively. In doing so we shall assume that, in the part of the plane under consideration, the half-branches of the curve \(f=0\) monotonically
move away from the axis of abscissas as \(|x|\) increases, and the boundaries of the sectors are situated symmetrically with respect to the singular point. This auxiliary assumption considerably simplifies the formulas. Denoting the distances of the points \(T_x^i, T_x^l\) from the equilibrium point also by \(T_ix, T_lx\), we find closed (respectively, inner and outer) broken lines from the equations:
\[ T_i x_0^i=x_0^i,\qquad T_l x_0^l=x_0^l . \]
§ 2.
Consider a system of two equations of the form
\[ F=[\varphi(x)-y]^p,\qquad f=c(x)[\psi(x)-y]^{n+p}. \tag{2.1} \]
We assume: 1) \(p\) is a positive odd number; 2) \(n\) is a nonnegative even number; 3) for \(0<|x|<N\), where \(N\) is some constant, \(|\psi-\varphi|>mx\), \(|\varphi|\le |a_1x|<|a_2x|<|\psi(x)|\), \(a_1,a_2,m\) are positive constants; 4) the curves \(F=0\), \(f=0\) are situated in the 1st and 3rd quadrants, \(\varphi(0)=\psi(0)=0\); 5) for \(|x|\le M_1\), \(0<c(x)\le c_1\), while for \(|x|\ge M_2>M_1\), \(M_2<N\), \(c(x)>c_2\), where \(c_1\) and \(c_2\) are positive constants.
We shall prove that, under the assumptions made, the system has a periodic solution and estimate its location. We construct a family of inner broken lines. Taking into account that in the 2nd (4th) quadrant
\[ \frac{\psi-y}{\varphi-y}\le \gamma, \]
where \(\gamma\) is a positive constant, we have
\[ T_i x_0^i=\frac{\varphi\left(x_0^i\right)\,[a_2-c_1\psi^n(N)]}{a_2\gamma^p c_1 2^n\psi^n(N)} . \]
Therefore we arrive at the equation
\[ \varphi\left(x_0^i\right)\left[1-\frac{c_1\psi^n(N)}{a_2}\right] =\gamma^p c_1 2^n\psi^n(N)x_0^i . \tag{2.2} \]
If it has a solution \(x_0^i\) in the interval \((0,M_1)\), then the point \(x_0^i\) will be the required one—through it an inner closed broken line will pass. We now construct a family of outer broken lines intersecting the positive semiaxis of abscissas for \(M_2\le |x|<N\). We have
\[ M_2+\psi\left(\frac{c_2m^nM_2^n x_0^l}{c_2m^nM_2^n-a_1}\right) \frac{1}{c_2m^nM_2^n}=x_0^l . \tag{2.3} \]
If equation (2.3) has a solution \(x_0^l\) in the interval \((M_2,N)\) and
\[ c_2m^nM_2^n x_0^l\left(c_2m^nM_2^n-a_1\right)^{-1}\le N, \]
then the point \((x_0^l,0)\) will be the required fixed point of the family of outer broken lines. It is easy to see that the distance between the trajectories issuing from the points \((x_0^i,0)\) and \((x_0^l,0)\) is less than the width of the boundary strip (between the corresponding broken lines). Denoting it by \(R\), we obtain
\[ R< \sqrt{ \left(\frac{c_2m^nM_2^n x_0^l}{c_2m^nM_2^n-a_1}\right)^2 + \left[ \psi\left(\frac{c_2m^nM_2^n x_0^l}{c_2m^nM_2^n-a_1}\right) \right]^2 }. \tag{2.4} \]
Theorem 1. If the right-hand sides of system (2.1) satisfy conditions 1–5, equation (2.2) has a solution \(x_0^i\) in the interval \((0,M_1)\), equation (2.3) has a solution \(x_0^l\) in the interval \((M_2,N)\),
\[ \frac{c_2 m^n M_2^n x_0^l}{c_2 m^n M_2^n-a_1}\leqslant N, \]
then, in the interval \((x_0^i,x_0^l)\), the axis of abscissas intersects a periodic solution of system (2.1). In this case the accuracy of its determination is given by formula (2.4). If in the interval \((0,x_0^i)\) \(T_i x>x\), then the polygonal line passing through the point \((x_0^i,0)\) may be taken as an approximate boundary of the region of attraction of the rest point as \(t\to -\infty\).
Example.
\[ \dot x=F=100x-x^3-y;\qquad \dot y=f=c(x)(300x-3x^3-y)^3. \tag{2.5} \]
Define the function \(c(x)\) as follows: \(M_1=2.298,\ M_2=2.300,\ c_1=6\cdot 10^{-6},\ c_2=10^{12}\). We have \(x_0^i\simeq 2.29,\ x_0^l\simeq 2.3\). All solutions beginning in the interval \((0,2.29)\), as \(t\to -\infty\), tend to 0.
§ 3.
Consider a system of two equations of the form
\[ \dot x=F=[\varphi(x)-y]^{n+p},\qquad \dot y=f=c(x)[\psi(x)-y]^p. \tag{3.1} \]
Suppose that for \(|x|\leqslant N\) the functions \(F\) and \(f\) satisfy the following conditions:
-
The curve \(y=\varphi(x)\) moves monotonically away from the axis of abscissas as \(|x|\) increases.
-
\(|a_1x|\leqslant |\varphi|\leqslant |ax|<|\psi(x)|,\ x\varphi>0,\ x\psi>0,\ |\psi-\varphi|\geqslant mx\) for \(x\ne 0\), where \(a,\ a_1,\ m\) are positive constants.
-
\(n\) is an even nonnegative number, \(p\) an odd positive number.
-
For \(|x|\leqslant M_1\), \(0<c(x)\leqslant c_1x^n\), \(0<M_1<M_2<N\); for \(|x|>M_2\), \(c(x)\geqslant c_2x^n\), \(c_1,\ c_2\) are positive constants.
We construct families of inner and outer polygonal lines. In the 2nd (4th) quadrant
\[ \frac{\psi-y}{\varphi-y}\leqslant \gamma, \]
where \(\gamma\) is a positive constant. Therefore we obtain the following equation for determining the inner closed polygonal line:
\[ \varphi(x_0^i)\left(1-\frac{c_1}{am^n}\right)=x_0^i\gamma^p\frac{c_1}{a_1^n}. \tag{3.2} \]
The outer closed polygonal line is found from the equation
\[ \psi\left(\frac{c_2x_0^l}{c_2-a^{n+1}}\right)+ \frac{c_2M_2^{\,n+1}}{[\varphi(N)+\psi(N)]^n} = \frac{c_2M_2^n x_0^l}{[\varphi(N)+\psi(N)]^n}. \tag{3.3} \]
Theorem 2. If the right-hand sides of system (3.1) satisfy conditions 1–4, equation (3.2) has a solution \(x_0^i\) in the interval \((0,M_1)\), equation (3.3) has a solution \(x_0^l\) in the interval \((M_2,N)\),
\[ \frac{c_2x_0^l}{c_2-a^{n+1}}<N, \]
then between the points \((x_0^i,0)\), \((x_0^l,0)\) the axis of abscissas intersects a periodic solution of the sys-
of system (3.1), moreover
\[ R < \sqrt{\left(\frac{c_2 x_0^l}{c_2 - a^{n+1}}\right)^2 + \left[\psi\left(\frac{c_2 x_0^l}{c_2 - a^{n+1}}\right)\right]^2}. \]
And if in the interval \((0, x_0^i)\) \(T_i x > x\), then the polygonal line passing through the point \((x_0^i, 0)\) may be taken as an approximate boundary of the domain of attraction of the equilibrium point as \(t \to -\infty\).
Remark. One may dispense with the monotone behavior of the function \(\varphi(x)\). In this case, in equality (3.3) \(\varphi(N)\) must be replaced by the upper bound of the modulus of the ordinate of the curve \(F = 0\) in the interval \((-N, +N)\).
§ 4.
Consider a system of two equations of the form
\[ \dot{x} = y - \varphi(x), \qquad \dot{y} = -c(y)\psi(x). \tag{4.1} \]
We assume:
- \(\varphi(0)=0\); for \(|x|<x_0\), \(x\varphi<0\), where \(\varphi(x_0)=0\); \(x\varphi>0\) for \(|x|>x_0\);
- \(\psi(0)=0\), \(x\psi>0\) \((x\ne 0)\); for \(x \in (0,N)\), \(\psi(x)\leq c_1\);
- \(m_1<c(y)\leq |y|+m\), where \(m\) and \(m_1\) are positive constants;
- for \(|x|>x_0\), \(\varphi'(x)\geq 0\);
- the functions \(\varphi(x)\) and \(\psi(x)\) are odd.
System (4.1) is equivalent to one equation, more general than the Van der Pol and Rayleigh equations [19].
We shall prove that system (4.1) has a unique periodic solution. By virtue of conditions 1–5, it is sufficient to construct polygonal lines in the right half-plane \((x>0)\). We shall construct the inner polygonal lines for \(|y|>k\), where \(k\) is some constant. Introduce the notation: \(H(x_H,y_H)\) is the point at which the ordinate of the curve \(F=0\) has the minimum \(\varphi_0\); \(C(x_C,y_C)\) is the point of intersection of the curve \(y=\varphi\) with a segment of the outer closed polygonal line; \(x_0^l\) is the abscissa of the point of intersection of the outer closed polygonal line with the axis of abscissas; \(x_0^i\) is the abscissa of the point of intersection of the inner closed polygonal line with the axis of abscissas. We have
\[ -y_0^i(x_0^i) = \varphi\left\{ \frac{k^2}{2c_1(k+2m)} + \frac{ (2y_0^i-k)\varphi\left[\dfrac{k^2}{2c_1(k+2m)}\right] }{ 2c_1\left[ m+\varphi\left(\dfrac{k^2}{2c_1(k+2m)}\right) \right] } \right\}. \tag{4.2} \]
In addition, the condition
\[ x_0^i \leq x_H \tag{4.3} \]
must be satisfied.
For \(y<2\varphi_0\), \(\dfrac{y}{y-\varphi_0}\leq \gamma\), where \(\gamma\) is some constant; therefore we obtain
\[ y_0^l(x_0^l) = 2\varphi_0 + (x_C-x_0)\frac{c_1(m+2\varphi_0)}{2\varphi_0} + x_0\frac{c_1(m+\gamma\varphi_0)}{\varphi_0}. \tag{4.4} \]
The width of the boundary strip is
\[ R < \sqrt{(x_0^l)^2 + (2y_0^l)^2}. \tag{4.5} \]
If equation (4.4) has a solution \(x_0^l\) in the interval \((x_0,N)\), and equation (4.2) has a solution \(x_0^i\) in the interval \((0,x_H)\), then the system of equations (4.1) has a periodic solution.
For the proof of uniqueness, set
\[ V=\int_{0}^{y}\frac{y\,dy}{c(y)}+\int_{0}^{x}\psi(x)\,dx. \tag{4.6} \]
We have
\[ dV=-\varphi(x)\psi(x)\,dt. \tag{4.7} \]
Further, uniqueness is proved in the same way as in the work of Levinson and Smith [20]. We arrive at the following conclusion.
Theorem 3. If the right-hand sides of system (4.1) satisfy conditions 1–5, equation (4.2) has a solution \(x_0^i\) in the interval \((0,x_H)\), and equation (4.4) has a solution \(x_0^l\) in the interval \((x_0,N)\), then system (4.1) possesses a unique periodic solution, and the inner closed polygonal line may be taken as the boundary of the region of attraction of the rest point as \(t\to-\infty\).
Remark. It follows from the proof of uniqueness that the restriction introduced by us in constructing the inner polygonal lines \((|y|>k)\) is not necessary for the existence of a periodic solution.
Example.
\[ \dot{x}=y-10(x^3-x)\left[1+2^{19\left(x^2-x_H^2\right)}\right],\qquad \dot{y}=-c(y)\psi(x), \tag{4.8} \]
\[ m=10^{-6},\quad m_1=10^{-12},\quad c_1=8. \]
We have \(x_0^i\simeq 0.4475,\quad x_0^l\simeq 1.317\).
Consequently, in the interval \((0.4475,1.317)\) the axis of abscissas is intersected by the unique periodic solution of system (4.8). The width of the boundary strip is \(R<50\).
§ 5.
Consider a system of three equations
\[ -\dot{x}=F(x,y),\qquad -\dot{y}=f(x,y),\qquad \dot{z}=\Phi(x,y,z). \tag{5.1} \]
Suppose that for \(x^2+y^2<\infty\) there exist planes \(z=c_1,\ z=c_2,\ c_1<0<c_2\), which the positive semitrajectories of system (5.1) intersect only in the direction of the plane \(z=0\); for \(z>c_2\) and \(z<c_1\) the positive semitrajectories do not move away from the plane \(z=0\), and, as one approaches the axis \(Oz\), these planes may be chosen arbitrarily close to the plane \(z=0\). Then the following assertion is valid.
Theorem 4. If the functions \(F\) and \(f\) satisfy the conditions of theorem 1(2) and \(T^i x>x\) in the interval \((0,x_0^i)\), then the polyhedral surface passing through the polygonal line issuing from the point \((x_0^i,0)\) orthogonally to the plane \(z=0\) may be taken as an approximate boundary of the region of attraction of the point \((0,0,0)\).
References
- Nemitskii V. V. Matem. sb., 16, No. 3, 1945, pp. 306–337.
- Nemitskii V. V. Uchenye zap. MGU, matem. vyp. 100, 1, 1946, pp. 34–52.
- Voilokov M. I. Nauchnye doklady vysshei shkoly, fiz.-matem. nauki, No. 1, 1959, pp. 18–23.
- Tabueva V. A. Trudy Uralskogo politekhnicheskogo in-ta, sb. 74, matem., 1958, pp. 72–79.
- Samedova S. A. Proceedings of the Institute of Physics and Mathematics, Academy of Sciences of the Azerbaijan SSR, 6, 1953, pp. 25–39.
- Otrokov N. F. Matematicheskii sbornik, 41(83), No. 4, 1957, pp. 417–430.
- Zakharov V. P. Scientific Notes of the Chuvash State Pedagogical Institute, issue 3, 1956, pp. 299–303.
- Peretyagin B. M. Doklady of the Academy of Sciences of the USSR, 114, No. 1, 1957, pp. 29–32.
- Barbashin E. A., Vdovina E. V. Izvestiya VUZov, Mathematics, No. 3 (16), 1960, pp. 43–47.
- Kushkov N. N. Vesti of the Academy of Sciences of the BSSR, Series of Physical-Technical Sciences, No. 3, 1961, pp. 21–24.
- Landis E. M., Petrovskii I. G. Doklady of the Academy of Sciences of the USSR, 113, No. 4, 1957, pp. 209–250.
- Markosyan S. S. Doklady of the Academy of Sciences of the Armenian SSR, 30, No. 1, 1960, pp. 13–18.
- Friedrichs K. O. On Nonlinear Vibrations of Third Order. Studies in Non-Linear Vibrations Theory, 1946, 65–103.
- Pliss V. A. Some Problems in the Theory of Stability of Motion in the Large, 1958.
- Vyborn E. M. Scientific Reports of Higher Education Institutions, Physical and Mathematical Sciences, No. 3, 1959, pp. 10–13.
- Serebryakova V. S., Barbashin E. A. Izvestiya VUZov, Mathematics, No. 2, 1961, pp. 137–146.
- Vinogradov N. N. Vesti of the Academy of Sciences of the BSSR, Series of Physical-Technical Sciences, No. 2, 1961, pp. 31–37.
- Tkachev V. F. and Tkachev Vl. F. Matematicheskii sbornik, No. 52, issue 3, 1960, pp. 811–822.
- Teodorchik K. F. Self-Oscillatory Systems. Gostekhizdat, 1952.
- Levinson N. and Smith O. K. A general equation of relaxation oscillations. Duke Mathematical Journal, 9, 1942.
Received by the editors
November 1, 1964
Mogilev Pedagogical
Institute