Formal Asymptotics of Parametric Subresonance Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 5, pp. 927-937. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221220

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 34E10

Formal Asymptotics of Parametric Subresonance

P. Astafyeva, O. Kiselev

The article is devoted to a comprehensive study of linear equations of the second order with an almost periodic coefficient. Using an asymptotic approach, the system of equations for parametric subresonant growth of the amplitude of oscillations is obtained. Moreover, the time of a turning point from the growth of the amplitude to the bounded oscillations in the slow variable is found. Also, a comparison between the asymptotic approximation for the turning time and the numerical one is shown.

Keywords: classical analysis and ODEs, subresonant, almost periodic function,small denominator

1. Introduction

Various linear equations of the second order are the most important models of mechanics because of Newtonian equations of a motion. Among such equations we are interested in an equation with a time-dependent coefficient. This work considers two different equations of such kind. The first one is the equation with an almost periodic coefficient:

where q(t) is an almost periodic function and e is a small positive parameter.

This equation has two important properties which define behaviors of solutions. The first one is a coefficient w2. It defines an oscillation of the solution for the simplest case q(t) = 0.

Another coefficient q(t) can or cannot imply the resonant behavior for the solution. The key work for understanding the behavior of the solution of such kind of equation is the Floquet

Received September 15, 2022 Accepted December 05, 2022

Polina Astafyeva astafyeva.polina2@gmail.com

Institute of Mathematics with Computer Centre of UFRC RAS, Ufa State Petroleum Technological University ul. Kosmonavtov 1, Ufa, 450062 Russia

Oleg Kiselev

o.kiselev@innopolis.ru

Innopolis University, Institute of Mathematics with Computer Centre of UFRC RAS ul. Universitetskaya 1, Innopolis, 420500 Russia

u" + (w2 + eq(t)) u = 0

(1.1)

theory [1]. The parametric map for the parameters (u, e) of the equation with a periodic coefficient is called Arnold's tongues [2]. Such a map defines the zones of parametric resonances [3]. Numerical calculations for Arnold tongues in the case of quasi-periodic systems were considered in [4]. Equations with quasi-periodic coefficients were examined in [5-8]. A two-scale method for quasi-periodic Mathieu-Hill equations was discussed in [9]. Applications in mechanics from the point of view of numerical analysis were considered in [10]. Also, some applications of the dynamics of the Mathieu-Hill equation were considered in [11, 12].

In contrast to the works mentioned above, we investigate almost periodic functions with a given behavior with an unlimited set of frequencies. The frequencies have a condensation point corresponding to the parametric resonance. We have not seen such an analysis before.

One of the famous examples of such an equation is the Mathieu equation for a special kind of the coefficient:

Although the Mathieu equation appeared in the theory of Laplace equations with elliptic boundary [13], this equation is now a widely used model in quantum mechanics and the theory of parametric resonance [3, 14-16].

One of more general equations of such kind appeared in the theory of Moon motion [17], where q(t) is a periodic function defined by their Fourier series.

Here we study a more general and more natural case where the q(t) is almost periodic. Such a view on the coefficient looks more physically important since in a real phenomenon the pure periodic coefficient is enveloped by additional disturbances and perturbations. Moreover, the period of the coefficient often does not coincide with the period of frequency of the solution to an equation such as q(t) = 0.

Here we consider an almost periodic coefficient [18, 19]:

where k > 1 and p > 0.

The expression of the form (1.2) most simply and clearly shows the effects that appear for the subresonance case. The amplitude tends to zero, the frequency tends to resonant, and this is a typical case where subresonant solutions appear. The parameter u differs little from 1: u = = 1 + 5. Here 5 is the parameter of the equation and defines the difference from the exact subresonant frequency. The task is to determine the areas of stability of solutions of the equation depending on the parameters 5 and e.

Equation (1.1) is linear and so the question of stability reduces to analyzing the stability of zero solution.

In Section 2 we formulate the aims of this work. Section 3 contains an asymptotic approach to the result. Section 4 shows similarities of the asymptotic and numerical approaches and is followed by conclusions.

2. Statement of the problem

For an equation with an almost periodic coefficient, one can see three different behaviors with respect to the parameter t ^-<x>. Figure 1 shows a bounded solution to Eq. (1.1). Figure 2 shows the subresonance growth in the initial interval.

The first problem to study is to understand the behavior of the solution depending on the coefficients 5 and e for a given form q(t) as an almost periodic function.

q(t) = a cos(2t).

(1.2)

3 2 1

a 0 -1

-2

-3

0 100 200 300 400 500

t

Fig. 1. A typical picture for a nonresonant case. Time indicated on the horizontal axis and the function u(t) on the vertical axis. For numeric solutions we have used the Runge-Kutta method of fourth order with a step equal to 0.01 and the initial data t = 0, u =1, u' = 0. In (1.2) we have got 20 terms in formula (1.1) and k = 2 and p = 5. The picture shows bounded oscillations for the equation w = 1 + S, where S = 0.01, e = 0.1

80 60 40 20 a 0 -20 -40 -60 -80

0 200 400 600 800 1000

t

Fig. 2. The subresonance growth in the initial interval. For numeric solutions we have used the Runge-Kutta method of fourth order with a step equal to 0.01 and the initial data t = 0, u = 1, u' = 0. In (1.2) we have got 20 terms in formula (1.1) and k = 2 and p = 5. The picture shows the growing solution as S = 0.00347, e = 0.1

The second and more intriguing aim is to find a dependence of the amplitude of the oscillating solution which may be considered as a similarity to parametric resonance growth and which was derived, for example, in [3].

3. Asymptotic aproach

Let us construct an asymptotic solution in the form

u ~ u0 + ev,i. (3.1)

Substituting (3.1) into (1.1) and combining the terms at the same degrees e, we obtain the equation for the main term:

d2 2

—Uq + u UQ = 0.

Let us look for its solution in the form of u0 = a(r)cos(wt) + b(r)sin(wt), using the two-scale method [3], where t = eYt is slow time. The choice of a slow scale is fundamental for further simplification of the equation of evolution of slow amplitudes. We use a standard approach in the theory of two-scale asymptotics: fast time is associated with oscillations, and slow time determines the change in amplitude. The parameter y > 0 will be determined below from the condition of uniformity of the asymptotic solution for large values of t. The equation for the first correction is

d2 2

+ Ui^2 + q(t)(bsm(uit) + a.cos(wi)) — 2a1ui sin(wi) + 261wcos(wi) = 0, (3.2)

where a1 = eY-1 a' and b1 = eY-1b', the stroke means a slow-time derivative t. The functions a, b, a1, b1 which depend on slow time are considered as constants when they are differentiated and integrated over fast time.

The task is to find the dependence on t for a and b. A bounded solution for the first correction of the function u1 is constructed below. Denote

f (t) = -q(t)(b sin(wt) + a cos(wt)). In this case, the general solution for the first correction equation can be written as

u1 = Acos(wt) + B sin(wt) — b1t sin(wt) — a1t cos(wt)+

t t cos(wt) f „.-. . ~ sin(wt) /■„,-, , + —— / f(t) sin(uit) dt--— / f(t) cos(uit) dt.

0 0

Let us introduce a replacement in the integrand function cot = t + St = t + kt. k = According to the replacement the equation takes the form

u1 = Acos(wt) + B sin(wt) — b1t sin(wt) — a1t cos(wt)+

tt cos(wt) f „.-. . ~ sin(wt) /■„,-, , ,

+ —-—- / f(t) sin(i + kt) dt---—- / f(t) cos(t + kt) dt. (3.3)

w J w J

00

Note that the formula for f (t) contains a sum over n, we change the summation with integration over t. The first integrand expression in the formula (3.3) can be transformed to the form

. (2nPrer+(4nP-l)t>\ , (2nPrer+(4nP-l)F\ a sin ^ np j 0 cos ^ np j

m sin (t + kt) = —-^---

asm bcos(^i±i) 6c0s

4 nk 4 nk 2 nk

and the integrand expression in the formula (3.3):

, . f 2np kt+(4nP-1)iA

o sin I--- J a cos

f(t) cos(t + kt) =---—-- +

2nP kt+jinP -l)t nP

4nk

4nk

+

6 sin ( M^tï) a cos ( a cos ({-

+

\ np J

+

\ np J

+

np

4nk 4nk 2nk

After sequential integration of all parts by t, using Maxima computer algebra systems [20], let us allocate the maximum order with respect to n for the formula obtained:

cos(wt) 4w

np~k (b isin(2h-:r) - sin riKT + Jj J J + a fcos(2 kt) - cos ^2 kt +

n=l ^ ^ ^ / / \ \

_r

nP.

and

sin(wi) 4w

n=1

y^ np k (a ( Sin(2h:r) - sin | 2 kt + —

— b ( cos(2kt) — cos ( 2kt H—-

In the solution u1 the growing terms in t are combined at cos(wt):

4w

cos

(2kt ) x:

np k I cos

n=1

i

-11 - sin(2kt) ^

np k sin

n=1

t

nP

— ( sin(2«r) X]

np k I cos

n=1

_)-l)+cos(2«r)£

np-k sin

n=1

and sin(wt):

b

— ( sin(2kt) £

np~k sin cos(2kt) (l - cos ( —

n=1

a

4w

sin(2kt) ^ np k | cos

Vn=1

n=1 t

nP

np

- 1 ] - cos(2kt) ^ np k sin

Replace:

n=1

nP~k ( cos ( ^ ) - 1 ) = - nP~k'2 sin

p k 2

n=1

n=1

(-L

2np

t

nP

In [21], the asymptotics were calculated:

E

np k sin

n=1

np

t1 -aC\

n/2

Cs ~ i J ta~2 sin(r) dr,

- b1 t.

V ??p_fc2 sin2 f~ 2t1~aCc, ^ \2nP

n=1 N 7

C rs-/- I

c 2^1-Qp(1 - a) p

+ I J T«"2 sin2 (I) dT, 0

a

t

0

n

t

500

Fig. 3. Curve 1 shows the first 500 terms of the series np~k sin Curve 2 shows an asymptotic

n=1

approximation of this sum

t

500

Fig. 4. Curve 1 shows the first 500 terms of the series J2 np~k2 sin2 (t^). Curve 2 shows an asymptotic

n =1

approximation of this sum

where

k - 1

a =-.

P

Let us combine the growing terms in t at cos(wt) and sin(wt), in order for the coefficients with increasing summands to be zeros:

ntl-a bfrl—a

ei~lta' = —-(2Cc cos(2kt) - Cs sin(2h:r)) + ——(2Cc sin(2h:r) + Cs cos(2kt)), (3.4)

4w 4w

btl-a ntl-a

e^Hb' = —-(Cs sin(2h",r) - 2Cccos(2kt)) + —-(2Cc sin(2h:r) + Cs cos(2kt)). (3.5)

4w 4w

In these formulas, we will reduce both parts by t and rescale time as t = This yields

d ClfYa bfYa

e7"1 —a = --(2Cccos(2kt) - C.sin(2Kr)) + --(2Cc sin(2h-,r) + C.cos(2kt)),

dr 4t a wv c v ' s v '' 4rawK c v ' s v

d bfYa (lfYa

e7"1 —b = --(C. sin(2h-,r) - 2Cccos(2kt)) + --(2Cc sin(2h:r) + C.cos(2kt)).

dT 4t a s v 7 c \ >) 4t c v ) s \ !!

Hence,

7 — 1 = 7a, 7 =-.

1 — a

As a result, the system of equations takes the form

da b

—a = ——(2Cccos(2kt) - Cs sin(2hrr)) + --(2Cr sin(2hrr) + C,cos(2kt)),

dT 4t a c ^ ' s \ > > 4t«wv c v > s v >h

d b a

—b = --(C, sin(2h-,r) - 2Cccos(2kt)) + —— (2Cc sin(2hrr) + C,cos(2kt)).

dT 4t a s v 7 c \ >) 4t c v ) s \ !!

Denote

-

A = v/C| + 4C2, </> = arctan •

Denote = B, then:

Then the system takes the form

d A(bs'm((f) + 2 kt) + a,cos((f) + 2kt))

! (x — " .

dr 4ura

d A(as'm((f) + 2kt) — bcos((f) + 2kt))

dr ~ Aujt0

_ — R +

d B(b sin(^ + 2kt ) + a cos(0 + 2kt ))

~ (% — .

dT Ta

d B(as'm((f) + 2 kt) — bcos((f) + 2 kt))

d,T ~ Ta

The system can also be written in the complex form z = a + ib:

d _ _ Bzei<t,+2iKT

d,T^ ~ Ta

5 = eY k.

Denote z = ZeiKT+^

. „ i±+iKT d \ i±+iKT BZeA+lhT

iZne 2 +mr + — Z e 2 +mr =-

\dT J Ta

and simplify

dr Ta

—Z =--iZn.

dT Ta

Let us write Z as the sum of real and imaginary parts: Z = w + iv. As a result, we obtain the following system of equations:

d Bw

—w =--h VK,

dT Ta

d Bv

—v = —wk--. (3.6)

dT Ta v 7

4. Comparison of the asymptotic approaches and numerical results

Let us consider the properties of the derived system (3.6) for different values of the parameters.

In the case k = 0 exact subresonance appears and the system splits into two equations. Their solutions have the following form:

(erl-a \ „ (-Bt 1-a

w = G1 exp - , v = C2 exp

1 I ' 2---i \ 1

1 — a / \ 1 — a

Here one can see that the solution is exponentially growing because of w(t). If k = 0, there is a small difference from the frequency of the subresonance. We divide both parts of the equations by the parameter k and rewrite kt = d. In this case we get the system of equations

d Bw

+ v,

d Bw

—w = Kl-affa

dd

d Bv

Tev = Kl~a6'

w.

Rewriting BKa 1 = A, we obtain

d X d A

—w = —w + v, —v = --^—v — w. dd da ' dd da

Here A > 0 is a parameter of the equation and k ^ 0, A

This system leads to the second-order differential equation

d2 ( A2 aA \

+ t) w = 0■

Here A is a large parameter. Thus, the asymptotic solution can be obtained by the WKB method [22]:

' e

w ~ C1

exp |J dej

4/ a2 ot\ _ ^

02a $a + l

For a detailed justification of the method, we refer the reader to [23]. The turning point in the WKB method is where the WKB decomposition turns out to be unsuitable. So the turning point to changing the growing character of the solution is located in a neighborhood of the point

d, - A1/a.

Let us compare the position of the tuning point obtained here in an asymptotic way and the results shown in the Fig. 5.

For the numerical example we have following data:

k — 1 1 15,

k = 2, p = 5, a =-= 7 =-= A ~ 1.09264275,

p 5 1 — a 4

where the value of A was obtained with respect to the work [21]:

A 1

B ee - ~ -

4u 4

A = 5,

e = 0.1, 5 = eYK,

r-J —

1X5/4

4 )

0* -

0

Xl/a rp _

nej

900473.

These asymptotic results are consistent with Fig. 5. The relative difference between the turning point obtained by the asymptotic method and that obtained by the numerical method

is

t-e„

i 900 473-900 700 i i 900 700 i

0.0003.

s 0

900200 900400

900600 t

900800 901000

Fig. 5. The graph shows the area of the end of the growth of the solution with given parameters. A typical picture for a nonresonant case. The time indicated on the horizontal axis and the function u(t) on the vertical axis. For numeric solutions we have used the Runge-Kutta method of fourth order with a step equal to 0.01 and the initial data t = 0, u = 1, uU = 0. In (1.2) we have got 20 terms in the formula (1.1) and k = 2 and p = 5. The picture shows the final stage of this growing solution as the same S = 0.00347, e = 0.1 in a very large time interval of about t = 900500

We are interested in the specific properties of the asymptotic solution: the growth rate and the area where this growth ends. This is because the problem is very sensitive to the phase shift and, as a rule, solutions with a higher degree of accuracy for a small parameter are needed to find it. The solutions to problems concerned with finding the amplitude of solving equations with an almost periodic function are new. The amplitude is well defined, but the phase shift calculation has not been performed, since this is a different, more technically complex, problem.

Let us compare the numerical solution of Eq. (1.2) with the resulting asymptotic solution. For k = 0, the asymptotics of the amplitude of an asymptotic solution is

U ~ i exp

Bt

1—a

1 — a

, ^ 1 ( BT1-a

(cos[t) - sin(i)) + - exp —--

2 \ 1 — a

(cos(t) +sin(t)).

The M-amplitude of oscillations is the envelope line: M = v/cosh(2T)

T ~ 0.277831 • 0.001 • t4/5 ■ 0.25.

Let us define the integration step s = 300

600 000

0.00157. The oscillation period is close

to 2n, we take n since this is half of the period, the integration interval of the equation is 150,

J

:

n

which is the number of oscillations. 600 000 is the number of integration points, which is about 1000 integration steps per oscillation. This result is shown in Fig. 6. At not very long times, when eY1 is not too large, our approximation is suitable. And then another effect begins which we do not take into account. So we need to look further at other times. We believe that this scale determines the limitation of the growth of d^.

200 400 600 800

t

Fig. 6. The amplitude M of oscillations according to calculations. The time indicated on the horizontal axis and the function u(t) on the vertical axis. For numeric solutions we have used the Runge-Kutta method of fourth order with a step equal to 0.001 and the initial data t = 0, u = 1, U = 0. In (1.2) we have got 1000 terms in the formula (1.1) and k = 0, k = 2 and p = 5

5. Conclusions

We have obtained a system of equations for the parametric subresonant growth of the amplitude of oscillations. The growth depends on the slow variable t = eyt. We have also found the time of the turning point of the growth of the amplitude to the bounded oscillations in the slow variable t. A comparison between the asymptotic approximation for the turning time and the numerical one has been found.

Conflict of interest

The authors declare that they have no conflicts of interest.

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