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SOME THEOREMS ON THE EXISTENCE OF A PERIODIC SOLUTION OF AN AUTONOMOUS SYSTEM OF SIX DIFFERENTIAL EQUATIONS
N. N. VINOGRADOV
Consider a system of 3 differential equations:
\[ \dot{x}=[\varphi(x)-y][z+a(x)]^{q}=F, \]
\[ \dot{y}=f=[y+\psi(x)]^{p}(mx-z)[\rho(x)-y], \tag{1} \]
\[ \dot{z}=[\gamma_{1}(x)-z][\gamma_{2}(x)-z]. \]
We assume that, in some neighborhood of the origin, which is the unique singular point of system (1), the right-hand sides satisfy the conditions of the existence and uniqueness theorem; the functions \(\varphi,\ \psi,\ \rho,\ a,\ \gamma_{1},\ \gamma_{2}\), as \(|x|\) increases, move monotonically away from the axis of abscissas; \(m\) is a positive constant; for \(x\ne 0\), \(x\varphi>0,\ \gamma_{2}>\gamma_{1}>0,\ \rho>\varphi,\ \psi>0,\ \rho>0,\ |\varphi|<ax<\rho,\ a\ge a_{0},\ a,\ a_{0}\) are positive constants. System (1) can be written in the form
\[ \frac{dz}{dx}=M,\qquad \frac{dz}{dy}=Q. \tag{1′} \]
Assuming that differentiation with respect to a parameter under the integral sign is admissible, we have
\[ z=z_{0}+\int_{x_{0}}^{x} M\,dx+d\,\Delta x\,\Delta y+\int_{y_{0}}^{y} Q(x_{0},y,z)\,dy =\int_{x_{0}}^{x} M\,dx+D, \]
where \(d\) denotes the correction for the inequality of the partial derivatives \(M'_{y}\) and \(Q'_{x}\), and \(\Delta x\) and \(\Delta y\) are the greatest increments of the abscissa and ordinate in the domain under consideration. Periodic solutions of systems of 3 equations were found by V. A. Pliss [1], B. V. Shirokorad [2], E. M. Vaisbord [3], and some other authors. In the present paper, first of all, conditions will be obtained for the existence of a periodic solution of system (1); then a system of 6 equations will be considered. The problem is solved by the method of fixed points.
Theorem 1. If:
1) there exists a closed polyhedral contact-free surface \(K_{1}\), whose faces are orthogonal to the plane \(z=0\), and the positive semitrajectories of system (1) intersect it from the outside inward;
2) inside \(K_{1}\) there exists a closed polyhedral contact-free surface \(K_{2}\), whose faces are orthogonal to the plane \(z=0\), and the positive semitrajectories of system (1) intersect it from the inside outward;
3) trajectories beginning between \(K_1\) and \(K_2\) for \(|z|\leqslant c_1=\mathrm{const}\) intersect the plane \(y=0\) in the strip \(|z|\leqslant c_1\), then system (1) has at least one periodic solution. For the proof it suffices to appeal to Browder’s theorem [4].
Remark 1. The theorem remains valid if, after going around the axis \(Oz\), the surface \(K_1\) bends inward, and the surface \(K_2\) outward.
To construct the surfaces \(K_1\) and \(K_2\) it is enough to find polygonal lines along which they intersect the plane \(z=0\) (respectively the outer [5] \(K_3\) and the inner \(K_4\)). A polygonal line issuing from the point \((x,0)\) will be called inner if the trajectory issuing from the same point as \(t\to\infty\) remains outside it. The slope coefficient \(k\) of a segment of the inner polygonal line is equal either to the upper bound of the fraction \(\dfrac{dy}{dx}\), or to its lower bound in the part of the sector under consideration, depending on its type (5) and on the quadrant. In case 3 of the equations all estimates must be uniform with respect to \(z\).
Remark 1. If the boundary of a sector of type \(\dot R>0\) on which the point \((x_0,y_0)\) lies (the beginning of the segment) is the semiaxis \(y>0\) (\(y<0\)), then it is disadvantageous to take the segment in the form of the straight-line segment \(x=x_0\), since in that case the polygonal line will enter an equilibrium point. Therefore, as a segment of the polygonal line 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 (if the other boundary coincides with the semiaxis \(x^2>0\), then the 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 zero on the boundary of the sector. In this case we divide the sector into two parts by a ray issuing from a singular point, estimate \(k\) in that part of the sector on whose boundary the point \((x_0,y_0)\) lies. We choose the segment of the polygonal line 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 a segment of the polygonal line we take the segment \(y=y_1\) or \(x=x_1\), depending on the quadrant.
Remark 3. It may happen that a segment of the polygonal line leaves the sector for which it was constructed. Then in this sector one must estimate the maximal displacement of the trajectory in the direction of the corresponding axis.
We place the beginnings of all polygonal lines on the semiaxis \(x>0\): of the outer ones, in the interval \((R,N)\), of the inner ones, in the interval \((R_6,R_1^i)\), where \(0<R_6<R_1^i<R<N\).
Suppose that \(|z|<c_1=\gamma_1(N)\).
We introduce the following notation: \(x_1\) is the abscissa of the point of intersection of \(K_3\) with the straight line \(y=ax\) in the first quadrant; \(y_1\) is the ordinate of the point of intersection of \(K_3\) with the straight line
\[
x=R_1=\frac{c_1}{m}
\]
in the first quadrant; \(y_2\) is the ordinate of the point of intersection of \(K_3\) with the straight line \(x=-R\) in the second quadrant; \(y_3\) is the ordinate of the point of intersection of \(K_3\) with the curve \(-y=\psi\) in the third quadrant; \(y_4\) is the ordinate of the point of intersection of \(K_3\) with the straight line \(x=R_5\) in the fourth quadrant; \(y_5\) is the ordinate of the point of intersection of \(K_4\) with the semiaxis \(y>0\); \(y_6\) is the exact upper bound of the set of ordinates of the points of intersection of the outer polygonal lines with the semiaxis \(y>0\); \(y_7\) is the ordinate of the point of intersection of the straight line issuing from the point \((R_1,y)\), \(y>\hat\beta(R)\), with slope coefficient equal in modulus to the upper bound of the slope coefficient of a trajectory in the interval \((0,R_1)\) in the first quadrant, with the semiaxis \(y>0\); \(y_8\) is the analogous quantity in the interval \((0,-R_1^i)\) in the second quadrant, and \(y_9\) in the fourth quadrant in the interval \((0,R_5)\), \(R_5<R\); \(\Delta=y_9+\psi(R)\); \(x_2\) is the modulus of the abscissa of the point of intersection
\(K_3\) with the half-axis \(x<0\). \(y_7\) is found from the equation \(y_7=\rho(N)+R_1kc_3\), where \(k=mR_1+c_1,\ \alpha_0^q c_3=[y_7+\psi(R_1)]^p\). Similarly \(y_8,\ y_9\) are found. To determine \(K_3\) (it intersects the axis of abscissas at the point \((x_0^e,0)\)) we obtain the equation
\[ x_0^e=R+\frac{[c_1+a(N)]^q y_4(x_2)}{(mR-c_1)\Delta^p}, \tag{3} \]
where
\[ x_2=R+\frac{[c_1+a(-N)]^q y_2(x_0^e)} {(mR-c_1)\psi^{p-1}(-R)[\rho(-R)-y_2]}. \]
Here it must be that \(R\leq x_0^e<x_1<N\). Assuming that \(\varphi(R_6)>\rho(R_1)\) and, for \(-R_1^i<x<0\), \(\rho<\beta x\), where \(\beta\) is a constant, and putting
\[ L_1=(mR_1^i+c_1)[\rho(R_1^i)+\psi(R_1^i)]^p\alpha_0^{-q}, \]
\[ L_2=\alpha_0^{-q}(mR_1^i+c_1)[\rho(-R_1^i)+\psi(-R_1^i)] [-\varphi(-R_1^i)+\psi(-R_1^i)]^{p-1}, \]
\[ L_3=\alpha_0^{-q}(mR_1^i+c_1)[\rho(R_1^i)+\psi(-R_1^i)] \psi^{p-1}(-R_1^i), \]
\[ \gamma_3>\max\left[1,\left|\frac{\psi}{\varphi}\right|\right], \]
we obtain an equation for determining \(K_4\) (it intersects the axis of abscissas at the point \((x_0^i,0)\)):
\[ \varphi\left[\frac{\varphi(x_0^i)-R_1^iL_1}{\beta}\right] - L_2\frac{\varphi(x_0^i)-R_1^iL_1}{\beta} = L_3\gamma_2 x_0^i. \tag{4} \]
In this case it must be that
\[ R_6\leq x_0^i\leq R_1^i,\qquad -R_1^i\leq \frac{y_5}{\beta}\leq -R_6. \]
It remains to estimate \(z\). From the 3rd equation of system (1) it follows that in the region \(z<0\) all trajectories are bounded in height.
The cylinders \(\gamma_1\) and \(\gamma_2\) divide the half-space \(z>0\) into 3 regions \(\Sigma_1\) (under the cylinders), \(\Sigma_2\) (between them), \(\Sigma_3\) (above both cylinders). By \(\Sigma_{ik}\) we denote the part of the region \(\Sigma_i,\ i=1,2,3\), situated in the \(k\)-th quadrant, \(k=1,2,3,4\). In the regions \(\Sigma_1\) and \(\Sigma_3\) the trajectory rises; in the region \(\Sigma_2\) it descends.
By \(\Delta_k\) denote the amount of descent in the region \(\Sigma_{2k}\) on the interval \((\rho_{1k},\rho_{2k})\), \(\rho_{2k}-\rho_{1k}=\delta_k\), assuming the function \(D\) on it to be nonpositive,
\[ \rho_{1k}<\rho_{2k}<x_0^i\ (k=1,4),\qquad \frac{y_5}{\beta}<\rho_{1k}<\rho_{2k}\ (k=2,3). \]
We construct a polyhedral surface \(K_5\), orthogonal to the plane \(y=0\). The first face of \(K_5\) is situated in the 1st octant and has equation \(z=c_1\). The equations of the remaining faces are composed so that trajectories intersecting the plane \(y=0\) for \(x_0^i\leq x\leq x_0^e,\ 0\leq z\leq c_1\) as \(t\to\infty\) remain between \(K_5\) and the plane \(z=0\).
We introduce the notation: \(z_1\) is the height of the line of intersection of \(K_5\) with the cylinder \(\gamma_2\) in the 1st octant; \(R_2\) is its distance from the plane \(x=0\); \(z_2\) is the height of the line of intersection of the cylinder \(\gamma_1\) with \(K_5\) in the 2nd octant; \(-x=R_3\) is the plane with which \(K_5\) intersects in the regions \(\Sigma_{22}, \Sigma_{23}\), and its equation is obtained from an estimate of the derivative in the region \(\Sigma_3\); \(z_1'\) is the height of the line of intersection of the cylinder \(\gamma_2\) with \(K_5\) in the 4th octant; \(R_4\) is its distance from the plane \(x=0\); \(z_2'=\max[z_2,\gamma_1(-N)]\); \(z_3\) is the height of the line of intersection of \(K_5\)
with the plane \(x=R_3\) in the 2nd octant; \(z_4(z'_4)\) is the height of the line of intersection \(K_5\) with the plane \(x=0\) in the region \(y>0\) (\(y<0\)).
Assume that \(\gamma_2(x_0^i)>z'_1,\ z_3<\gamma_2\left(\dfrac{y_5}{\beta}\right)\); this is sufficient so that the trajectory cannot rotate around the \(Oz\) axis without leaving the region \(\Sigma_3\). In the region \(\Sigma_{21}\) we have
\[ \Delta_1= \frac{[\gamma_2(\rho_{11})-c_1][\gamma_1(\rho_{21})-c_1]\delta_1} {-y_1[c_1+\alpha(N)]^q-\delta_1[\gamma_2(\rho_{11})-c_1]} . \]
Similarly we find \(\Delta_2,\ \Delta_3,\ \Delta_4\).
In the region \(\Sigma_{3i}\) introduce the notation:
\(|z'_x|\le k_{1i},\ |z'_y|\le k_{2i},\ i=1,2,3,4\). Estimating the derivatives, we obtain
\[ z'_2=9z'_1+3k_{21}y_6+3k_{11}R_2+3dR_2y_6+k_{12}R_3+k_{22}y_2+dR_3y_2-\Delta_2+d\delta_2y_2, \]
\[ z'_1=9z'_3+3R_3k_{13}+3y_3k_{23}+3dR_3y_3+k_{14}R_4+k_{24}y_4+dR_4y_4. \]
The condition of boundedness of the height of trajectories takes the form
\[ z'_2<c_1,\qquad c'_1=z'_1-\Delta_4+d\delta_4y_4<c_1. \tag{5} \]
Theorem 2. If condition (5) is fulfilled, equation (3) has a solution \(x_0^e\) on the interval \((R,N)\), and equation (4) has a solution \(x_0^i\) on the interval \((R_6,R_1^i)\), then system (1) has a periodic solution.
Remark 2. One may dispense with the monotonicity of the functions \(\varphi,\gamma_2,\alpha\), replacing their values at the endpoints of the intervals by maxima of the moduli on the intervals under consideration. In addition, one may allow \(c_1>\gamma_1(N)\).
Example.
\[ \dot{x}=\left[\frac{x^3}{1000^{0.5[1-\operatorname{sign}x]}}-y\right][z+11+x^2]^{60}, \]
\[ \dot{y}=(3\cdot10^{-20}x-z) \left[ 10x^2\left(1+10^{-1200\left(x+\frac14\right)}\right)-y \right]f_1, \tag{6} \]
\[ f_1=\left[ y+\frac{10x^4}{1+10x^4}\, 10^{200\left(x-\frac12\right)(1+\operatorname{sign}x)^5} \right], \]
\[ \dot{z}=(3\cdot10^{-35}x^2-z)(25\cdot10^{10005}x^2-z)d_0^{\left[2\frac{\gamma_2-z}{1+|\gamma_2-z|}-1\right]}, \]
\[ 3\cdot10^{125}\le d_0(y)<10^{200},\qquad x_0^i\approx0.23,\qquad x_0^e\approx8\cdot c'_1<c_1,\qquad z'_2<c_1. \]
Thus, system (6) possesses a periodic solution.
Consider the system of six equations:
\[ \dot{u}=F(x,y,z)-u,\qquad \dot{x}=(\varphi-y)(z+\alpha)^q, \]
\[ \dot{z}=(\gamma_1-z)(\gamma_2-z),\qquad \dot{y}=(y+\psi)^p(mx-z)(\rho-y), \tag{7} \]
\[ \dot{v}=[\varphi_1(v)-w][u+\alpha_1(v)]^{q_1},\qquad \dot{w}=[w+\psi_1(v)]^{p_1}(m_1v-u)[\rho_1(v)-w]. \]
With respect to its right-hand sides we make the same assumptions as with respect to the right-hand sides of system (1). Since \(x, y, z\) are bounded, \(u\) is also bounded: \(|u| \leqslant u_0\). It is now easy to see that system (7) has a periodic solution.
References
- Pliss V. A. Some problems in the theory of stability of motion as a whole. Publishing House of Leningrad State University, 1958.
- Shirokorad B. V. Autom. i telemekh., 19, No. 10, 1958, pp. 953—967.
- Ważewski E. M. Matem. sb., 56 (98), No. 1, 1952, pp. 43—58.
- Brouwer L. E. J. Beweis desebenen Translationssatzes. Math. Ann., 72, 31—54, 1912.
- Vinogradov N. N. Vesti AN BSSR, ser. phys.-techn., No. 2, 1961, pp. 31—37.
- Lavrent'ev M., Lyusternik L. Foundations of the calculus of variations, vol. 1, part 1, 1935, pp. 142—144.
Received by the editors
October 5, 1964
Mogilev Pedagogical
Institute