On a theorem of the existence of periodic solutions

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I. Muntean
Institutul de Calcul

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I. Muntean, On a theorem of the existence of periodic solutions (Russian). Matematica, Cluj, 1, 1959, pag. 287-296.

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Mathematica Cluj

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Published by the Romanian Academy Publishing House

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1222-9016

Online ISSN

2601-744X

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1959-Muntean

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ON A THEOREM OF EXISTENCE OF PERIODIC SOLUTIONS

I. MUNTEANUCluj

Mathematical investigation of various electrical and radio engineering circuits, in particular, the consideration of a vacuum tube generator, as well as the study of some mechanical circuits encountered, for example, in questions of dynamics, leads to systems of two differential equations, generally nonlinear, of the following type:

\begin{aligned} \frac{d x}{d t} = P (x, y), \\ \frac{d y}{d t} = Q (x, y) . \end{aligned}

A number of results in this direction, with technical applications, are given in the second edition of the book by ANDRONOV, VITT and KHAYKIN [1].

In this paper, we indicate the conditions for the implementation of a periodic regime for a certain special case of the above-mentioned system of differential equations. Namely, using the working method developed by LEVISON and SMITH [2], FILIPPOV [3] and DRAGILEV [4] and relying on the idea of introducing a nonlinear term [5], in this paper we give a simple generalization of one of Dragilev's theorem, proving that under certain conditions, the system of differential equations

\begin{aligned} \frac{d x}{d t} = h (y) - F (x) \\ (1) & \frac{d y}{d t} = - g (x) \end{aligned}

admits at least one periodic solution.
The work does not address the problem of uniqueness of the periodic solution found.

Let us assume that the functions included in the system (1)

F (u), g (u), h (u)

are defined and continuous on the entire real axis and satisfy certain conditions that ensure the uniqueness and extendability of the solution to any Cauchy problem for system (1). Under these standard assumptions, the following theorem is established:

THEOREM. If
$1^{\circ} . sgng (x) = sgn x$ and the integral $\int_{0}^{x} g (ξ) d ξ$ diverges at $| x | \to \infty$ ;

2^{\circ}

.

sgn F (x) = - sgn x

at a sufficiently small

∣ x

;

3^{\circ}

there is a positive number

M

and positive constants

k

And

k^{'}, k > k^{'}

, such that

\begin{aligned} F (x) ≧ k, For x > M, \\ F (x) ≦ - k^{'}, For x < - M; \end{aligned}

4^{\circ} . sgn h (y) = sgn y

And

| h (y) | \to \infty

For

| y | \to \infty;

then the system of differential equations (1) has at least one periodic solution.

In proving the theorem, we distinguish several stages.
a). From the conditions

1^{\circ}, 2^{\circ}

And

4^{\circ}

it follows that

g (0) = F (0) = h (0) = 0

, therefore, the origin of coordinates is a singular point of the system. This singular point is unique.

In the phase plane (

x, y

) we will construct a ring region

Z

, which does not contain the origin and has the property that any positive semi-trajectory of system (1), emanating from some point of region 2, remains in this region for all subsequent values of the independent variable.
b). We will implement the indicated construction using the energy level curves of system (1)

λ (x, y) = c

, Where

\begin{matrix} λ (x, y) = G (x) + H (y) \\ G (x) = \int_{0}^{x} g (ξ) d ξ, H (y) = \int_{0}^{y} h (η) d η \end{matrix}

From the conditions

1^{\circ}

And

4^{\circ}

the theorem implies that the function

λ (x, y)

positive definite, so the curve implicitly given by the equation

λ (x, y) = c

, Where

c > 0

, is closed and the interior region bounded by this curve contains the origin. In addition, if

0 < c^{'} < c^{''}

, then the curve

λ (x, y) = c^{'}

is located inside the region bounded by the curve

λ (x, y) = c^{''}

c) Now we choose

| x |

And

| y |

small enough, namely

| x |

we choose so that the condition

2^{\circ}

was fulfilled. Then the derivative function

λ (x, y)

by virtue of system (1) is positive definite. Indeed,

\frac{d λ}{d t} = \frac{\partial λ}{\partial x} \cdot \frac{d x}{d t} + \frac{\partial λ}{\partial y} \cdot \frac{d y}{d t} = g (x) [h (y) - F (x)] + h (y) [- g (x)] = - g (x) F (x)

,
hence, on the basis of the conditional

1^{\circ}

, it follows that

\frac{d λ}{d t} > 0

for the specified values

x

. Therefore, all positive semitrajectories that have common points with the curve

λ (x, y) = c

, Where

c

- a sufficiently small positive constant, intersect this curve from the inside to the outside of the region bounded by this curve. Therefore, the origin is a singular point of repulsion of positive semi-trajectories. The curve constructed in this way

λ (x, y) = c

serves as the inner boundary of the ring region

F

d) Now let

a

some number satisfying the condition

a > max [k, sup_{- M ≦ x ≦ M} | F (x) |]

Then we will construct a rectangle with vertices at the points

V_{1} (M, a)

,

V_{2} (M, - a), V_{3} (- M, - a), V_{4} (- M, a)

(Fig. 1). On a straight line

x = M

we take a point

Q (M, y_{Q})

such that

y_{Q} > a

And

h (y) > a at y > y_{Q} .

This choice is possible due to the condition

4^{\circ}

. Then, denoting by

f (Q, t)

trajectory emanating from a point

Q

, we have at

0 < x ≦ M

\begin{aligned} \frac{d x}{d t} = h (y) - F (x) > a - F (x) > 0 \\ (2) & \frac{d y}{d t} = - g (x) < 0 \end{aligned}

From these inequalities it follows that the negative semi-trajectory

f (Q, t)

goes up to the left and crosses the axes perpendicularly

ABOUT U

at some point

P (0, y p)

, because

lim g (x) = g (0) = 0

The same inequalities show that the positive semi-trajectory

{(Q, t)

directed, at a point

Q

, down to the right; while

y (t)

decreases with increasing

t

And

x > M

.
d). Let us now prove that there exists at least one point

\bar{Q} (\bar{x}, \bar{y})

, located on a positive semi-trajectory such that

x > M

And

h (\bar{y}) - F (\bar{x}) = 0

In fact, due to the choice of point

Q

, coordinates (

x, y

) some point, close enough to the point

Q

and located on a positive semi-trajectory

f (Q, t)

, satisfy the inequality

h (y) - F (x) > 0

To prove the above equality, it is sufficient, in view of the continuity of the function

h (y (x)) - F (x)

, prove the existence of at least one Pair

(x_{1}, y_{1})

, With

x_{1} > M

And

y_{1} < y_{Q}

, such that
19 - Mathematica

\dot{h} (y_{1}) - F (x_{1}) < 0 .

To establish this, let us assume the opposite, namely, that for any

x > M

And

y < y_{Q}

we have

h (y) - F (x) > 0 .

Then inequalities (2) remain valid 11 in the region

D = {x > M, y < y_{Q}}

Let it be now

b

some number satisfying the inequality

b > max [k, sup_{y < y_{Q}} h (y)] .

Taking into account the condition

3^{\circ}

, we have

\frac{d y}{d x} = - \frac{g (x)}{h (y) - F (x)} < - \frac{g (x)}{b - k}

Integrating the last inequality from

M

to

x

, we get

y < y_{Q} - \frac{1}{b - k} \int_{M}^{x} g (ξ) d ξ

From this it is clear that for a sufficiently large

x

, due to the condition

1^{\circ}, y (x)

remains negative. Let

x_{1}

And

y_{1}

such values

x

And

y

. Obviously

x_{1} > M

And

y_{1} < 0

But, taking into account

4^{\circ}

, we know and

h (y_{1}) < 0

. Therefore, even more so,

h (y_{1}) - F (x_{1}) < 0 .

The resulting contradiction proves the existence of at least one point

\bar{Q} (\bar{x}, \bar{y})

for which

h (\bar{y}) - F (\bar{x}) = 0

. In this case, the tangent to the trajectory

f (Q, t)

at the point

\bar{Q}

directed vertically downwards, because

{(\frac{d x}{d t})}_{\begin{matrix} x = \bar{x} \\ y = \bar{y} \end{matrix}} = 0

Moreover, from the above reasoning, it follows that when

x > x_{1}

And

y < y_{Q}

we have

h (y) - F (x) < 0 .

However, this inequality is obvious when

x ≧ 0

And

y < 0

.
e). Let us now continue the positive semi-trajectory

f (Q, t)

behind the point

\bar{Q} (\bar{x}, \bar{y})

. Then for all the following values

t

we have
therefore, when

x > 0

,

h (y) - F (x) < 0

\begin{aligned} \frac{d x}{d t} = h (y) - F (x) < 0 \\ \frac{d y}{d t} = - g (x) < 0 \end{aligned}

So, the positive semi-trajectory decreases to the left and reaches

direct

x = M

at some point

Q^{'} (x^{'}, y^{'})

, the coordinates of which satisfy the inequality

h (y^{'}) - F (x^{'}) < 0

.
f). On a straight line

x = - M

we're fighting the point

U (- M, y \cup)

, having an ordinate

y_{u} > a

such that

h (y) > a

at

y > y u

. Let us consider the positive semi-trajectory emanating from this point.

f (U, t)

. At

- M ≦ x < 0

And

y > y u

we have

\frac{d x}{d t} > 0 And \frac{d y}{d t} > 0 .

So, the positive semi-trajectory

{(U, t)

will cross the axes

O y

at some point

V

. It can be assumed that the point

V

coincides with the point constructed above

P

.

Thus, we will construct an arc of the trajectory

U P Q Q^{'}

, where the coordinates (

x^{'}, y^{'}

) points

Q^{'}

satisfy the inequality

h (y^{'}) - F (x^{'}) < 0

.
3). Using the method of paragraphs c) - g), we will construct a similar arc of the trajectory emanating from the point

W

direct

x = - M

We select the ordinate

y_{w}

points

W

so that

h (y) < a

at

y < y_{w} < a

We will thus obtain the arc of the trajectory

T S W W^{'}

, where is the point

T

is on a straight line

x = M

not above the point

Q^{'}

, point

S

- on the axis

O y

, and the points

W

And

W^{'}

on a straight line

x = - M

. In addition, the coordinates of the point

W^{'} (x^{''}, y^{''})

satisfy the inequality

h (y^{''}) - F (x^{''}) > 0

.
and). Relative to the relative position of points

W^{'}

And

U

on a straight line

x = - M

, the following situations are possible:
α) point

W^{'}

is located under the point

U

,

^{β}

) point

W

coincides with the point

U

,

γ)

dot

W^{'}

is located above the point

U

.
Under the conditions of the theorem, we will prove that there exists at least one system of arcs UPQQ', TSWW' of the specified type for which situation (t) does not occur.

To prove this, let us assume the opposite, namely, that situation (ү) is realized for any system of arcs of the type UPQQ', TSWW'. Then we continue the long-term semi-trajectory

f (W^{'}, t)

until its intersection at the point

P^{'}

with an axis

O y

Let's consider the areas

G

bounded by a closed curve

Γ

, consisting of arcs

P Q Q^{'}, T S W W^{'} P^{'}

and line segments

P P^{'}

And

Q^{'} T

. Due to the uniqueness of solutions of system (1), positive semi-trajectories can enter the region d only through points of the segment

Q T

and can exit only through the points of the segment

P P^{'}

, perpendicularly intersecting this segment.
k). As a starting point, we now choose the point

A (0, y_{0})

, located on the axis

O y

above the point

P^{'}

and we will construct a positive semi-trajectory

f (A, t)

, emanating from this point; it will have a behavior similar to the behavior of the arcs of the curve

Γ

. Let us denote by

B

And

C

points of its intersection with the line

x = M

, through

D (O, y_{1})

intersection point with

O y

, through

E

And

F

, points of intersection with a straight line

x = - M

and through

\bar{A} (0, \overset{―}{y_{1}})

the point of its secondary intersection with the positive semi-axis

O y

.

Note that if the point

C

is located on the segment

Q^{'} T

, then the positive semi-trajectory

f (A, t)

is included in the region

ϱ

, from which

can only exit through the points of the segment

P P^{'}

; but since the point

A

chose above the point

P^{'}

, then for the corresponding system of arcs, situation (t) does not occur. In this case, our statement is proven.

So, let's assume that the point

C

is located above the point

T

straight my

x = M

.

If, in this case, the point

\bar{A}

will be under the point

A

then our statement is proven again.

So, let's assume that always

\overset{―}{y_{1}} > y_{0}

Let

H

intersection point of the positive_trajectory

f (\overset{―}{A, t) с прямой x = M, I -точка}

secondary intersection

f (\bar{A}, t)

with a straight line

x = M

, and finally,

J (O, y_{2})

- intersection point

f (\bar{A}, t)

with a negative axle shaft

O y

. Because

{\bar{y}}_{1} > y_{0}

, we have

y_{2} < y_{1} < 0

.

To arrive at a contradiction, we will first estimate the variation of the function

λ (x, y)

and trajectories (1).
l). Let's calculate the derivative

\frac{d λ}{d y}

along one trasctorni. We have

\frac{d λ}{d y} = \frac{\partial λ}{\partial y} + \frac{\partial λ}{\partial x} \frac{d x}{d y} = h (y) + g \frac{h (y) - F (x)}{- g (x)} = F (x)

Therefore, when

x = 0

\frac{d λ}{d y} = F (x)

Let us introduce the notation

λ (R) = λ (x_{R}, y_{R})

and we integrate the last equality along the arc

B C

and then along

E F

We will receive

λ (C) - λ (B) = \int_{B}^{C} F (x) d y = - \int_{C}^{B} F (x) d y ≦ - k (y_{B} - y_{C})

And

λ (F) - λ (E) = \int_{E}^{F} F (x) d y < k^{'} (y_{F} - y_{E})

Let us now calculate the derivative

\frac{d λ}{d x}

along one trajectory. We have

\frac{d λ}{d x} = \frac{\partial λ}{\partial y} \cdot \frac{d y}{d x} + \frac{\partial λ}{\partial x} = h (y) \frac{- g (x)}{h (y) - F (x)} + g (x) = - \frac{g (x) F (x)}{h (y) - F (x)}

At -

M ≦ x ≦ M

And

h (y) > a

let's enter the numbers

\begin{matrix} L_{P Q} = \int_{P}^{Q} | \frac{g (x) F (x)}{h (y) - F (x)} | d x, L_{T S} = \int_{T}^{S} | \frac{g (x) F (x)}{h (y) - F (x)} | d x \\ L_{S W} = \int_{S}^{W} | \frac{g (x)}{h (y) - F (x)} | d x, L_{U P} = \int_{U}^{P} | \frac{g (x) F (x)}{h (y) - F (x)} | d x \end{matrix}

where the integrals are taken along the specified arcs of the trajectories. Let us denote

L = max (L_{P Q}, L_{T S}, L_{S W}, L_{U P})

Let's consider the integral

| λ (B) - λ (A) | = | \int_{A}^{B} \frac{- g (x)}{h (y)} \frac{F (x)}{F (x)} d x |

By condition 4, we can choose a point

B

so that

y_{B} > y_{P}

And

\begin{array}{r} inf h (y) > sup_{A} h (y) . \\ y_{B} ≦ y ≦ y_{A} y_{Q ≦ y ≦ y_{P}} \end{array}

Then, whatever it may be

x \in [O, M]

, coordinates

(x, y_{P Q}), (x, y_{A B})

points taken respectively on the arc of the trajectory

P Q, A B

, satisfy the inequalities

h (y_{P Q}) - F (x) < h (y_{A B}) - F (x)

Not because of this

| λ (A) - λ (B) | < | \int_{P}^{Q} \frac{- g (x) F (x)}{h (y) - F (x)} d x | ≦ L_{P Q} ≦ L

Hence,

| λ (B) - λ (A) | < L

It is similarly shown that

| λ (D) - λ (C) | < L, | λ (E) - λ (D) | < L, | λ (\bar{A}) - λ (F) | < L

It follows that the number

2 L

limits the variation of the absolute value of the function

λ (x, y)

along any trajectory located in the regions

B_{1} = {- M ≦ x ≦ M, h (y) > a}, B_{2} = {- M ≦ x ≦ M, h (y) < - a}

m). Now we will prove that the variation

y

also limited in areas

B_{1}

And

B_{2}

. Indeed, from the relation

λ (x, y) = H (y) + G (x)

it follows that when

\bar{y} \overset{―}{\bar{y}} > 0

And

| h (\bar{y}) | > a, | h (\overset{―}{\bar{y}}) | > a

, we have

H (\bar{y}) - H (\bar{y}) = \bar{λ} - \overset{―}{\bar{λ}} + \int_{\bar{x}}^{\bar{x}} g (\bar{ξ}) d ξ

But by virtue of the mean value theorem, we obtain

H (\bar{y}) - H (\overset{―}{\bar{y}}) = \int_{\bar{y}}^{\overset{―}{\bar{v}}} h (η) d η = (\bar{y} - \overset{―}{\bar{y}}) h (η)

where ηє

[\overset{―}{\bar{y}}, \bar{y}]

Therefore, given the fact that in the regions

B_{1}

And

B_{2}

the inequality holds

| \bar{λ} - \overset{―}{\bar{λ}} | < L

we will receive

| \bar{y} - \overset{―}{\bar{y}} | ≦ \frac{\overset{―}{∣ λ} - \overset{―}{\bar{λ}} | + \int_{\bar{x}}^{\overset{―}{\bar{x}}} | g (ξ) ∣ d ξ}{| h (η) |} < \frac{2 L + 2 M \cdot max | g (x) |}{a}

By this we have proved the limitedness of variation

y

along one trajectory located in the regions

B_{1}

or

B_{2}

, exactly

| \bar{y} - \overset{―}{\bar{y}} | < N

where is the number

N

equals

\frac{1}{a} [2 L + 2 M \cdot max_{- M ≦ x ≦ M} | g (x) |]

n). Let us now estimate the difference

λ (J) - λ (D)

Since we assumed that

\overset{―}{y_{1}} > y_{0}

, we have and

y_{2} < y_{1} < 0

But then, on the one hand

\begin{matrix} (3) & λ (J) - λ (D) > 0 \end{matrix}

for

\begin{matrix} λ (J) - λ (D) = λ (0, y_{2}) - λ (0, y_{1}) = H (y_{2}) - H (y_{1}) = \int_{y_{1}}^{y_{2}} h (y) d y = \\ = \int_{y_{2}}^{y_{1}} [- h (y)] d y > 0 \end{matrix}

and, on the other hand, we will prove that the reverse inequality also holds:

λ (J) - λ (D) < 0

But then we will come to a contradiction and thus it is proved that

y_{2} > y_{1}

, i.e.

{\bar{y}}_{1} ≦ y_{0}

. Therefore, for the arc

A B C D E F \bar{A}

, similar to the system of arcs

U P Q Q^{'}, T S W W^{'}

, situation

γ

) relative to points

\bar{A}

And

A

cannot take place and, therefore, the statement of point i) is completely proven.

So it remains to prove that

λ (J) - λ (D) < 0

. In fact, taking into account the estimates established in points l) and m), we have

\begin{aligned} (4) & λ (E) - λ (D) < L \\ (5) & λ (F) - λ (E) < k^{'} (y_{F} - y_{E}) < k^{'} ({\bar{y}}_{1} + | y_{2} |) \\ (6) & λ (H) - λ (F) < 2 L \\ (7) & λ (I) - λ (H) < - k (y_{H} - y_{I}) \end{aligned}

Because

y_{H} > \overset{―}{y_{1}} - N

And

y_{I} < y_{2} + N

the last inequality becomes

To the previous inequalities we add the following

\begin{matrix} (8) & λ (J) - λ (I) < L \end{matrix}

Adding up inequalities (4), (5), (6), (7) and (8), we have

λ (J) - λ (D) < 4 L + 2 N k - (k - k^{'}) ({\bar{y}}_{1} + | y_{2} |)

from here, since

{\bar{y}}_{1} > y_{0}

, we get

\begin{matrix} (9) & λ (J) - λ (D) < 4 L + 2 N k - (k - k^{'}) y_{0} \end{matrix}

Now we choose a number

y_{0}

more than a number

\frac{4 L + 2 N k}{k - k^{'}}

In this case the value

L (y_{0})

we can consider the same as in point l). Then from inequality (9) it follows

λ (J) - λ (D) < 0

which contradicts condition (3).
o). The resulting contradiction proves that if we choose a number

y_{0}

more than from the very beginning

\frac{4 L + 2 N k}{k - k^{'}}

then the inequality holds

{\bar{y}}_{1} ≦ y_{0}

.
p). Thus, the positive semi-trajectory

f (A, t)

, emanating from a point

A (O, y_{0})

, comes out after the secondary intersection of the positive semi-axis at the point

\bar{A} (O, {\bar{y}}_{1})

, in the region

D

(Bendixson bag), limited by the arc of the trajectory

A \bar{A}

and a straight line segment

A \bar{A}

, located on the axis

O y

and having ends

y_{0}

And

y_{1}

, moreover

y_{0} > {\bar{y}}_{1}

. (In the case when

y_{0} = y_{1}

, trajectory

f (A, t)

is closed, i.e. it is a periodic solution and the theorem is proven). From the region

D

positive semi-trajectories cannot exit, because through the points of the segment

A \bar{A}

they can only enter the area

D

.

Let us denote by

E

region bounded by a closed curve

λ (x, y) = c, c > 0

, built on point c). Ring area

Z = D - \overset{―}{E}

does not contain any singular point of the system (1) and any positive semi-trajectory emanating from some point of this region remains, by virtue of the above-proven, in the ring region 2 for all subsequent values

t

. Then, by virtue of the well-known Bendixson-Poincaré theorem, in region 2 there exists at least one periodic solution of this system of differential equations.

The theorem is proven.

Damn. 1.

LITERATURE

[1] Andronov, A. A., Witt, A. A., and Khaikii, S. E., “Tsoria oscillations”. Moscow, 1959.
[2] Levinson, N., and Smith, O. K., “A general equation of relaxation oscillation”. Duke Math. J., 9 (1942).
[3] Filippov, A. F., Sufficient conditions for the existence of a stable limit cycle for a second-order equation. Mat. Sb., 30(72), 171–180, (1952).
[4] Dragilev, A. V., “Periodic solutions of the differential equation of nonlinear oscillations”. Prikl. Mat. i Mekh. 16, 85–88 (1952).
15] Munteanu, I., “Solutii mårginite si solutii periodice pentru anumite sisteme de ecutati d frentiale”. S'ud. şi cerc. de mat. (Cluj), VIII, 1-2, 125-131 (1957).
Received 26. November 1959.

I.

1959

On a theorem of the existence of periodic solutions

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ON A THEOREM OF EXISTENCE OF PERIODIC SOLUTIONS

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