UDC 517.934

ON PERIODIC SOLUTIONS OF A CERTAIN NONAUTONOMOUS SYSTEM OF DIFFERENTIAL EQUATIONS

M. T. TEREKHIN

In the present note we consider questions of the existence of a unique periodic solution of a first-order differential equation and of a nonautonomous system of two first-order differential equations.

In the first section, the first-order differential equation

\[ \frac{dx}{dt}=f(x,t) \]

is studied, and criteria are established for the existence of a unique periodic solution, different from the criteria established in [1, 2]. In the second section, a certain special system of two first-order differential equations is considered, and on the basis of the theorems of the first section a theorem is proved on the uniqueness of the periodic solution of this system.

§ 1. Consider the equation

\[ \frac{dx}{dt}=f(x,t) \tag{1} \]

in the plane \((x,t)\).

Let \(f(x,t)=[x-x(t)]^k f_1(x,t)\) \((k=1,2,\ldots,m)\), and suppose that there exist two continuously differentiable \(\omega\)-periodic curves \(x=x_1(t)\) and \(x=x_2(t)\) \((x_2(t)>x(t)>x_1(t))\), such that in the closed region \(\bar D\) bounded by these curves for \(0\le t\le \omega\), \(f_1(x,t)\) is a continuous function having a continuous partial derivative with respect to \(x\). We shall assume that in the region \(\bar D\), \(f(x,t)=f(x,t+\omega)\), \(x(t)=x(t+\omega)\), \(x(t)\) is a continuous function, \(f_1(x,t)=f_2(x,t)f_3(t)\), where \(f_2(x,t)\ne 0\), \(f_3(t)\ge 0\) (or \(\le 0\)), the equation \(f_3(t)=0\) can have only isolated roots, \(I_i(t)=f[x_i(t),t]-x_i'(t)\ge 0\) (or \(\le 0\)), and the set \(E[I_i(t)=0]\) can consist only of isolated points, \(i=1,2\).

Theorem 1 (Poincaré). Let \(F(x)=\int \alpha(x)\,dx\) be a single-valued twice continuously differentiable function. If equation (1) has two periodic solutions in the region \(\bar D\), then between them there always exist points at which

\[ \frac{d}{dx}[\alpha(x)f(x,t)]=0. \tag{*} \]

The proof of the theorem is given in [3], p. 96.

Let \(f_2(x,t)=f_2^*(x)+f_2^{**}(x,t)\), where \(\int_0^\omega f_2^{**}(x,t)\,dt=0\) for fixed \(x\), \(f'_{2x}(x,t)=\bar f_2^*(x)+\bar f_2^{**}(x,t)\), where \(\int_0^\omega \bar f_2^{**}(x,t)\,dt=0\) for fixed \(x\).

Definition 1. We shall say that equation (1) (the function \(f(x,t)\)) satisfies conditions \(A\) in the domain \(\overline D\), if
\(f_1(x,t)=f_2(x,t)\times f_3(t)\), where \(f_2(x,t)\ne 0\), \(f_3(t)\ge 0\) (or \(\le 0\)), the equation \(f_3(t)=0\) can have only isolated roots, and at least one of the following conditions holds:

1.
\[ |\Delta|< \left|-\frac{k f_2(x,t) f_2^*(x)}{x-x(t)}\right|, \tag{2} \]

where
\[ \Delta= \begin{vmatrix} f_2^*(x) & f_2^{**}(x,t)\\ \bar f_2^*(x) & \bar f_2^{**}(x,t) \end{vmatrix}; \]

2.
\[ \begin{vmatrix} f_2(x,t) & x(t)-x\\ f'_{2x}(x,t) & k \end{vmatrix} \ne 0. \tag{3} \]

Lemma 1. In the set of continuously differentiable functions not identically equal to zero there exists a function \(\alpha(x)\) such that
\[ B(x,t)=[x-x(t)]\,[\alpha'(x)f_2(x,t)+\alpha(x)\cdot f'_{2x}(x,t)] +k\alpha(x)f_2(x,t)\ne 0, \tag{**} \]
if equation (1) satisfies conditions \(A\).

Proof. Let us note that, by virtue of the inequality \(\alpha(x)f_2(x,t)\ne 0\) in the domain \(\overline D\), \(B(x,t)\ne 0\) along the curve \(x=x(t)\). For the same reason, on the set \(E\,[B(x,t)=0]\) the inequality
\[ \alpha'(x)f_2(x,t)+\alpha(x)f'_{2x}(x,t)\ne 0 \]
holds.

1. First let us prove that \(B(x,t)\ne 0\), if inequality (2) holds. Suppose that for any function \(\alpha(x)\), \(B(x,t)=0\). Let
   \[ \alpha(x)=\exp\left(-\int \frac{\bar f_2^*(x)}{f_2^*(x)}\,dx\right), \]
   then
   \[ \alpha(x)\left[ -\frac{\bar f_2^*(x)}{f_2^*(x)} \left(f_2^*(x)+f_2^{**}(x,t)\right) +\bar f_2^*(x)+\bar f_2^{**}(x,t) \right] = -\frac{k\alpha(x)f_2(x,t)}{x-x(t)} \]
   or
   \[ f_2^*(x)\bar f_2^{**}(x,t)-\bar f_2^*(x)f_2^{**}(x,t) = -\frac{k f_2(x,t)f_2^*(x)}{x-x(t)}. \]
   But this contradicts inequality (2).

2. Let inequality (3) hold. We shall prove that there exists a function \(\alpha(x)\) such that \(B(x,t)\ne 0\). Suppose that for any function \(\alpha(x)\ne 0\), \(B(x,t)=0\), although inequality (3) holds.

The expression \(B(x,t)=0\) can be written as
\[ [x-x(t)]\left[\frac{\alpha'(x)}{\alpha(x)}f_2(x,t)+f'_{2x}(x,t)\right] =-k f_2(x,t) \]
or
\[ [x-x(t)]\left[\frac{\alpha'(x)}{\alpha(x)} +\frac{f'_{2x}(x,t)}{f_2(x,t)}\right] =-k, \]
whence
\[ \frac{\alpha'(x)}{\alpha(x)} = \frac{k}{x(t)-x} -\frac{f'_{2x}(x,t)}{f_2(x,t)}. \]

From inequality (3) it follows that there exists a function \(\beta(x)\) such that, in the domain \(\overline D\),

\[ |\beta(x)|<\inf_{0\le t\le \omega}\left|\frac{k}{x(t)-x}-\frac{f'_{2x}(x,t)}{f_2(x,t)}\right|. \]

Then, taking \(\alpha(x)=\exp\int \beta(x)\,dx\), we obtain \(B(x,t)\ne0\), which contradicts the assumption.

Remark. Condition (2) is always fulfilled if \(f_2(x,t)\equiv f_2^*(x)\), since \(\Delta=0\).

Theorem 2. Equation (1) in the domain \(\overline D\) has a unique periodic solution for odd \(k\), provided it satisfies condition A.

Proof. The existence of at least one \(t\)-periodic solution in the domain \(\overline D\) follows from Schauder’s fixed point theorem.

Suppose that in the domain \(\overline D\) there exist two \(t\)-periodic solutions: \(x=x_1(t)\) and \(x=x_2(t)\). Then for any single-valued continuously differentiable function \(\alpha(x)\), by Theorem 1, in the domain \(\overline D\) there exist points at which equality \((*)\) holds. We shall prove that \(\alpha(x)\) can be chosen so that equality \((*)\) will not hold in the domain \(\overline D\). Consider the function \(\Phi[x_1(t)]-\Phi[x_2(t)]\), where \(\Phi(x)=\int \alpha(x)\,dx\). By virtue of the periodicity of \(x_1(t)\) and \(x_2(t)\), there exist extrema of the function \(\Phi[x_1(t)]-\Phi[x_2(t)]\), i.e., there exists a point \(t=t_1\) at which

\[ \frac{d}{dt}\{\Phi[x_1(t)]-\Phi[x_2(t)]\}=0. \]

The last equality can be written in the form

\[ f_3(t_1)\{\alpha[x_1(t_1)][x_1(t_1)-x(t_1)]^k f_2[x_1(t_1),t_1]- \]
\[ -\alpha[x_2(t_1)][x_2(t_1)-x(t_1)]^k f_2[x_2(t_1),t_1]\}=0. \tag{4} \]

We shall prove that the expression in braces does not vanish in the domain \(\overline D\). Suppose the contrary. Then the function \(\alpha(x)[x-x(t_1)]^k f_2(x,t_1)\) in the domain \(\overline D\) attains an extremum as \(x\) varies from \(x_1(t_1)\) to \(x_2(t_1)\). Consequently,

\[ \frac{d}{dx}\,[\alpha(x)(x-x(t))^k f_2(x,t)]=0 \tag{5} \]

at least at one point of the domain \(D\). Equality (5) can be written as

\[ [x-x(t)]^{k-1}\{k\alpha(x)f_2(x,t)+(x-x(t))[\alpha'(x)f_2(x,t)+\alpha(x)f'_{2x}(x,t)]\}=0. \tag{6} \]

Since equation (1) satisfies conditions A, \(\alpha(x)\) can be chosen so that the expression in braces in equality (5) does not vanish. The function \(\alpha(x)[x-x(t)]^k f_2(x,t)\) will not attain an extremum at points of the curve \(x=x(t)\); otherwise the left-hand side of equality (5) in a neighborhood of the curve \(x=x(t)\) would have different signs, which is impossible, since \(k\) is an odd number. Consequently, equality (5) is impossible.

Thus, the expression in braces in equality (4) does not vanish. But equality (4) is also impossible for those values of \(t\) for which \(f_3(t)\) vanishes; otherwise \(f_3(t)\) would change sign when passing through the root.

Remark. If \(f'_x(x,t)\ne 0\) in the domain \(\overline D\), then the uniqueness of the periodic solution in this domain follows from Theorem 1 with \(a(x)=1\).

§ 2. Consider the system of two differential equations

\[ \begin{aligned} \frac{dx}{dt}&=f(x,t),\\ \frac{dy}{dt}&=\psi(x,y,t), \end{aligned} \tag{7} \]

where the function \(f(x,t)\) is the same as the right-hand side of equation (1). Suppose

\[ \psi(x,y,t)=[y-y(x,t)]^k\,\psi_1(x,y,t),\qquad \psi_1[x,y(x,t),t]\ne 0, \]

the function \(y(x,t)\) is defined, continuous, and single-valued in the domain \(\overline D\); hence there exist numbers \(c\) and \(d\) such that \(c<y(x,t)<d\) if \((x,t)\in \overline D\). We shall assume that

\[ \psi(x,y,t)=\psi(x,y,t+\omega),\qquad y(x,t)=y(x,t+\omega), \]

\[ \psi_1(x,y,t)=\psi_2(x,y,t)\psi_3(t), \]

where \(\psi_2(x,y,t)\ne 0\) in the domain \(\overline D\times[c,d]\), \(\psi_3(t)\ge 0\) (or \(\le 0\)), the equation \(\psi_3(t)=0\) can have only isolated roots, \(k>0\), \(k=2n+1\) (\(n\) an integer).

Definition 2. We shall say that system (7) satisfies conditions B if the function \(f(x,t)\) satisfies conditions A in the domain \(\overline D\), and the functions \(\psi,\psi_2\) in the domain \(\overline D\times[c,d]\) satisfy at least one of the following conditions:

1.

\[ \psi'_y(x,y,t)\ne 0, \tag{8} \]

2.

\[ \left| \begin{matrix} \psi_2(x,y,t) & d-c\\ \psi'_{2y}(x,y,t) & k \end{matrix} \right|\ne 0, \tag{9} \]

3.

\[ \left|\frac{\psi'_{2y}}{\psi_2}\right| < \left|\frac{k}{d-c}-1\right| \tag{10} \]

(in inequalities (9) and (10), \(\operatorname{sign}(d-c)=\operatorname{sign}[y(x,t)-y]\)).

Theorem 3. System (7) in the domain \(\overline D\times[a,b]\) has a unique periodic solution, if in this domain it satisfies conditions B and \(k\) is an odd number.

Proof. Since the right-hand side of the first equation of the system does not depend on \(y\), the theorem will be proved if it is shown that: 1) the equation

\[ \frac{dx}{dt}=f(x,t) \]

in the domain \(\overline D\) has a unique periodic solution \(x=\theta(t)\); 2) the equation

\[ \frac{dy}{dt}=\psi[\theta(t),y,t] \]

also has a unique periodic solution in the domain \(c<y<d\) \((0\le t\le \omega)\).

The existence of a unique periodic solution of the equation

\[ \frac{dx}{dt}=f(x,t) \]

follows from the fact that the function \(f(x,t)\) satisfies conditions A. To establish the existence of a unique periodic solution of the equation

\[ \frac{dy}{dt}=\psi[\theta(t),y,t], \]

it suffices to prove the validity of inequality \((**)\), where instead of \(x(t)\), \(a(x)\), \(f_2(x,t)\) one must put, respectively,

\[ y[\theta(t),t],\qquad a(y),\qquad \psi_2[\theta(t),y,t], \]

and repeat the reasoning carried out in Theorem 1. Inequality \((**)\) under the indicated changes follows from the fact that system (7) satisfies conditions B. Indeed, if \(\psi(x,y,t)\) satisfies inequality (8),

then \(a(y)\) should be taken equal to unity \((a(y)=1)\); if \(\psi_2(x,y,t)\) satisfies inequality (9), then
\(a(y)=\exp \int \beta(y)\,dy\), where \(\beta(y)\) is determined in the same way as \(\beta(x)\) in Lemma 1, with allowance for the above-mentioned changes; if, however, \(\psi_2(x,y,t)\) satisfies inequality (10), then \(a(y)=e^y\).

Consider the system

\[ \frac{dx_1}{dt}=X_1(x_1,x_2),\quad \frac{dx_2}{dt}=X_2(x_1,x_2),\quad \frac{dx_3}{dt}=X_3(x_1,x_2,x_3), \tag{11} \]

where the functions \(X_1(x_1,x_2)\), \(X_2(x_1,x_2)\), and \(X_3(x_1,x_2,x_3)\) are continuous and have continuous partial derivatives in the space of the variables \(x_1,x_2,x_3\), except possibly at the point \(O(0,0,0)\).

Theorem 3 makes it possible, in some cases, to find periodic solutions of system (11). Introduce the following change of variables \(x_1=\rho\cos\varphi\), \(x_2=\rho\sin\varphi\), \(x_3=z\) with Jacobian \(J=\rho\). Then the system assumes the form

\[ \frac{d\rho}{d\varphi} = \rho\, \frac{x_1X_1(x_1,x_2)+x_2X_2(x_1,x_2)} {x_1X_2(x_1,x_2)-x_2X_1(x_1,x_2)}, \]

\[ \frac{dz}{d\varphi} = \rho^2\, \frac{X_3(x_1,x_2,x_3)} {x_1X_2(x_1,x_2)-x_2X_1(x_1,x_2)}, \tag{12} \]

where, in place of \(x_1,x_2,x_3\), one must substitute respectively \(\rho\cos\varphi\), \(\rho\sin\varphi\), \(z\). Hence it follows that system (11) has a unique periodic solution if system (12) satisfies the conditions of Theorem 3.

Example. We shall prove that the system

\[ \frac{dx_1}{dt} = -x_2+ \left(x_1^2+x_2^2-x_1-x_2-3\sqrt{x_1^2+x_2^2}\right)x_1, \]

\[ \frac{dx_2}{dt} = +x_1+ \left(x_1^2+x_2^2-x_1-x_2-3\sqrt{x_1^2+x_2^2}\right)x_2, \tag{13} \]

\[ \frac{dx_3}{dt} = x_3\sqrt{x_1^2+x_2^2}-x_1-2\sqrt{x_1^2+x_2^2} \]

has a unique periodic solution. Introducing the change of variables, we obtain the system

\[ \frac{d\rho}{d\varphi} = \rho^2(\rho-\cos\varphi-\sin\varphi-3), \]

\[ \frac{dz}{d\varphi} = \rho(z-\cos\varphi-2), \]

which satisfies all the conditions of Theorem 2 in the domain \([1.5;1.3]\). Indeed, since \(f_2=\rho^2\), \(\psi=\rho(z-\cos\varphi-2)\) (here \(\rho=x\), \(y=z\), \(t=\varphi\)), the system under consideration satisfies condition B \((\Delta=0,\ \psi_y'\ne0)\) in the domain \([1.5;1.3]\). In any other domain, system (14) has no periodic solutions, since there either \(\rho\) or \(z\) varies monotonically.

References

1. N. N. Luzin, Matem. sb., 39, no. 3, 1932, pp. 6–26.
2. S. A. Samedova, Tr. In-ta fiziki i matematiki AN AzSSR, 6, 1953, pp. 25–29.
3. A. Poincaré, On curves defined by differential equations. GITTL, 1947.

Received by the editors
14 July 1965

Ryazan State
Pedagogical Institute