ESTIMATION OF THE ACCURACY OF THE AVERAGING METHOD FOR SYSTEMS WITH MULTIFREQUENCY PERTURBATIONS Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 2, pp. 379-394. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200211

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 34C29, 34C46

Estimation of the Accuracy of the Averaging Method for Systems with Multifrequency Perturbations

E. I. Kugushev, T. V. Popova

We consider systems of ordinary differential equations whose right-hand sides contain time-periodic functions with some frequencies. An averaged system is constructed by introduction of additional variables and by step-by-step averaging over these variables. An upper estimate of the deviation of the solution to the initial system from the solution to the averaged system is given. Examples are given of mechanical systems in which vibrations with several frequencies occur and for the analysis of which the statements obtained are applied.

Keywords: averaging method, multifrequency perturbations, vibration frequency

The motion of mechanical systems in which elements performing forced oscillations are present can be described by ordinary differential equations involving time-periodic functions with some frequencies. Examples of such systems include systems on a vibrating base. Reference [1] considers single-frequency systems in which the vibration of the base is described by a time-periodic function with a large frequency. In this case the equations of motion can be reduced to a form that is standard for the Krylov-Bogolyubov averaging methods [2], and these methods can be used to analyze the motion of the system. The key point of these methods is an estimate of the deviation of the solution to the averaged system from the solution to the initial system. In the multifrequency case the application of averaging methods is complicated by the presence of resonances, i.e., situations where the frequencies are related by rational linear homogeneous relations. The motion of the system in the resonance case can greatly differ from that of an averaged system. However, even in this situation one can apply averaging methods, for example, in the case of separation of oscillations into fast and slow ones or in the case of strong

Received December 24, 2019 Accepted March 15, 2020

This work was supported by the Russian Foundation for Basic Research (project 18-01-00887).

Eugene I. Kugushev kugushev@keldysh.ru Tatiana V. Popova t.shahova@yandex.ru

Lomonosov Moscow State University GSP-1, Leninskie Gory, Moscow, 119991 Russia

incommensurability of frequencies. An analysis of the application of averaging methods in multifrequency systems can be found in [3-8].

In this paper, we consider multifrequency systems in "slow time" on a finite time interval. By introducing additional variables, we sequentially construct systems averaged over these variables and give estimates of deviations of solutions to the averaged systems from the solution to the initial system. In proving the statements we use direct integral estimates. But it should be noted that the application of the standard method of change of variables [9] could somewhat shorten the exposition.

1. Systems with separable right-hand sides

Consider a system of ordinary differential equations

X = g(x) + fi(x, vit) + f:2(x, v2t), x G Rra, vi > 0, v2 > 0. (1.1)

Let the functions g(x), f1(x,T), f2(x,T) be continuously differentiable with respect to the variables x G Rra, t G R, the functions f1(x,T), f2(x,T) be periodic in t with period 1, independent of v1, v2, and let their average on a period in t be zero. Averaging the right-hand side of the system (1.1)

1 1

J J(g(x) + fi(x, Ti) + f2(x, T:))dTidT2 = g(x), 0 0

we write the averaged system

¿= g(z), z g Rra. (1.2)

Statement 1. Suppose that for some a> 0 and any v1, v2 > 0 the solutions x(t, v1 ,v2,x0) and z(t,x0) to systems (1.1) and (1.2) with initial conditions x(0,v1,v2,x0) = x0 and z(0,X0) = = x0, where x0 lies in some bounded region U С Rra, exist on the interval 0 ^ t ^ a and lie in some bounded region V С Rra. Then one can find constants c1,c2 > 0 such that for all x0 G U one has

Ci C2

|x(t, v\, v2, xo) — 2(t, жо)| ^--1--with 0 ^ t ^ a.

v1 v2

Proof. For brevity we will sometimes write the solutions x(t, v1,v2, x0), z(t, x0) as x(t), z(t). We write the initial and the averaged system in the integral form

t t

x(t) = x0 + j (g(x(s)) + fi(x(s),vi s) + ¡2(x(s),v2 s))ds, z(t) = x0 + j g(z(s))ds.

00

For

u(t,Vi,V2 ,x0 ) = x(t,Vi,V2 ,x0 ) - f(t,x0 )

we have

t t t u(t,vi,V2,f0) = J(g(x(s)) - g(z(s)))ds + J fi(x(s),vis)ds + J f2(x(s),V2.s)ds. (1.3) 0 0 0

Let us fix an arbitrary t from the interval [0,a] and estimate \u(t,v1 ,v2,f0)\. Since the functions g(x), f1(x,T), f2(x,T) are continuously differentiable with respect to all variables and

the functions fi(x,r), f2(x,T) are periodic in t, it follows that in the region V C Mra for any t, vi, v2 the functions g, f\, f2 are bounded by some constant K:

\g(x)\ < K, \f1(x,u11)| < K, \f2@,v2t)\ < K

and satisfy the Lipschitz condition in x with some constant L:

\g(x) - g(x')\ ^ L\x - x'\, \fi(x,vit) - fi(x',vit)\ ^ L\x - x'\, \ f2(x,V21) - f2(x',V2t)\ ^ L\x - x'\.

Then, according to (1.5), for the first term on the right-hand side of (1.3) we have

(1.4)

(1.5)

t t t J (g(x(s)) - g(z(s)))ds ^ J\g(x(s)) - g(z(s))\ds ^ L J\u(s,vi,v2 ,xo)\ds.

k

Let us estimate the second term on the right-hand side of (1.3). Let tk = —, where k = = 0,1,..., m, and let m be the maximal integer for which tm = ^ ^ t. It follows from (1.4)

that \g + fi + f2\ ^ 3K for any t, vi, v2, and therefore \x(t,vi,v2,x0)\ ^ 3K. Hence, on the interval tk ^ s ^ tk+i we have

\x(s,Vi,V2 ,xo) - x(tk ,vi ,V2,xo )\ ^ 3K (s - tk)

and, taking (1.5) into account,

\fi(x(s),vis) - fi(x(tk),vis)\ ^ L\x(s) - x(tk)\ ^ 3LK(s - tk).

The average of the function f]_(x,T) over t on a period is zero. Therefore,

ifc+i

fi(x(tk ),vis)ds = 0.

tk

Then

tk + 1

fi(x(s),vi s)ds

tk

tk + 1

(fi(x(s),vis) - fi(x(tk),vis))ds

tk

<

tk+1

tk+1

^ / \fi(x(s),vis) - fi(x(tk),vis)\ds ^ 3LK / (s - tk)ds =

3LK(tk+i - tk)2 3LK

2

tk

tk

2v2

In view of the fact that m ^ vit ^ via, we have

fi(x(s),vis)ds

m-i tk+1

< E

k=0 tk

fi(x(s),vis)ds

3mLK 3aLK

^-^

2v2

2vi

m

Since \t — tm\ < jjp we have

fi(x(s),vi s)ds

<

K

vi'

Thus, for the second term on the right-hand side of (1.3) we obtain

fi(x(s),vis)ds

<

fi(x(s),vi s)ds

+

fi(x(s),vis)ds

^ 3a,LK K_ _ di ^ 2vi vi vi'

where di is the constant independent of vi.

In a similar manner we estimate the third term on the right-hand side of (1.3)

f2(x(s),v2s)ds

< —

where d2 is the constant independent of v2. Finally, from (1.3) we have

t

\u(t,ui,u2,x0)\ ^ L [\u(s,ui,u2,x0)\ds + — + —.

J vi v2

Then, using the Gronwall-Bellman lemma, we obtain

\vi v2 vi

di , d2^ d\eLa d2eLa _ £i £2

v2 vi v2

for any t £ [0,a|. This proves the statement.

Remark 1. If the system is a single-frequency system, i.e., f2 = 0 or v2 = vi, then Statement 1 gives the same estimate as the Krylov - Bogolyubov averaging theorem.

Remark 2. Consider the system of equations

x = g(x, t) + fi(x, t, vit) + /2 (x, t, v2t), x £ .

vi > 0,v2 > 0,

(1.6)

where g(x, t), f1(x,t,T), f2(x,t,T) £ 1" are functions that are continuously differentiable with respect to the variables x £ 1", t, t £ 1 and the functions f1(x,t,T), f2(x,t,T) are periodic in t with period 1, independent of v1, v2, and their average on a period over t is zero. This system of equations can be autonomized by replacing in the functions g(x, t), f1(x,t,T) and f2(x,t,T) time t by an additional variable x"+1 and by setting x,"+1 = 1. As a result, the system (1.6) is brought to the form (1.1)

x = g(x) + /1 (x, v1t) + /2 (x, v2t),

where

gg

g'

,x"+1 / \ 1

Thus, Statement 1 holds for systems (1.6) too.

f 2

/2

t

t

t

m

t

"

1

0

Statement 1 is generalized to the case of three and more frequencies. Consider the system of ordinary differential equations

Xc = + fi(x, vit) + ... + fk (x, vk t), x e Rra, v > 0, i = 1,...,k. (1.7)

Here the functions g(x), fi(x,r) are continuously differentiable with respect to the variables x e Rra, т e R, the functions fi(Xc, т) are periodic in т with period 1, independent of v1,..., vk, and their average on a period is zero.

Statement 2. Suppose that for some a > 0 and any vi > 0 (i = 1,...,k) the solutions x(t, v1,... ,vk, fo) and f(t, x0) to systems (1.7), (1.2) with initial conditions x(0, v1,...,vk, x0) = = x0 and x(0,X0) = x0, where x0 lies in some bounded region U С Rra, exist on the interval 0 ^ t ^ a and lie in some bounded region V С Rra. Then one can find constants c1 ,...,ck > 0 such that for all x0 e U one has

c1 ck

|x(t, г/i,..., vk, xo) — жо)| ^--1- • • • H--- with 0 ^ i ^ a.

v1 vk

2. Step-by-step averaging in systems of general form

Consider the system of ordinary differential equations

£= f(x,V1t,V2t), x e Rra, V1 > 0,V2 > 0, (2.1)

where f(x,T1,T2) is a function that is continuously differentiable with respect to all variables, periodic in the variables т1, т2 with period 1, and independent of v1, v2. Averaging the right-hand side of the system (2.1) over т1 and т2, we write the averaged system

1 1

:z = g(z), g(z) = J J 1(£,т1 ,т2)dnйт2. (2.2)

oo

Statement 3. Suppose that for some a> 0 and any v1,v2 > 0 the solutions x(t, v1 ,v2,x0) and z(t,x0) to systems (2.1), (2.2) with initial conditions x(0,v1 ,v2,x0) = x0 and z(0,X0) = x0, where x0 lies in some bounded region U С Rra, exist on the interval 0 ^ t ^ a and lie in some bounded region V С Rra. Then one can find constants c1 ,c2,c3 > 0 such that for all x0 e U the estimate of the deviation of the solution z(t, x0) to the averaged system from the solution x(t, v1,v2,x0) to the initial system has the form

\x(t, г/i, г/2, Xo) — z(t, Жо)| ^ — + - + — with O^t^a. (2.3)

v1 v1 v2

Proof. a) Averaging the function f (x, т^т2) over the variable т1

1

J f (x,тl,т2)dтl = д1(:е,т2), o

we write the system (2.1) as

x = g1 (x, V2t) + f1 (x, V11, V21), (2.4)

where the function f1(x,т1,т2) = f (x,^ ,т2) — g^x,^) has a zero average over т1.

Consider the system

w = gi(w,v2t).

(2.5)

Let w(t, v2,x0) be a solution to the system (2.5) on the interval 0 ^ t ^ a with initial conditions w(0, v2,f0) = x0. We show that

-> -)-)-> Ci C2V2

\x(t,ui,u2,xo) — w(t,u2,xo)\ ^--1--with O^t^a.

Vi Vi

(2.6)

To this end, introducing the additional variable xn+i, which changes in accordance with the equation xn+i = v2, we reduce the system (2.4) to the form

x = Xi (x) + fi(x,vit)

(2.7)

where

;=* I x \ ^ ^ gi(x,xn+iA . fi(X,Ti,xn+l)\

X = LJ ' Xi(X)=l V2 ' h{X'Ti >=( 0 )

and write its averaged system

wX

= Xi(w), w G Rn+i

(2.8)

Let us estimate the difference x(t, vi ,x0) — w(t, x0) of the solutions to the systems (2.7) and (2.8) on the interval 0 ^ t ^ a. We have

\x(t,Vi,xo) - w(t,xo)\ = \x(t,Vi,xo)\ ^

gXi(x(s)) - Xi(wx(s)))ds

+

fi(x(s),vis)ds

Since the function f(fx,Ti ,t2) is continuously differentiable with respect to all variables and is periodic in the variables Ti, t2, it follows that in the region V c Rn for any t1,v1,v2 the functions gi, fi are bounded by some constant K:

\gi(x,xn+i)\ ^ K, \fi(x,Ti,xn+i)\ ^ K and satisfy the Lipschitz condition in X with some constant L:

\gi (x,xn+i) - gi (x',x'n+i )\ ^ L\x - x\, \fi (x,Ti,xn+i) - fi(x',Ti,x'n+i)\ ^ L\x - x\ Following the algorithm for proving Statement 1, we obtain

Xi(x(s)) - Xi(w(s)))ds Ji(x(s),vis)ds

^ L \u(s,vi,x0)\ds, \x(t,vi,x0)\ ^ 2K + v2,

t tm

<

fi(x(s),Vi s)ds

+

fi(x(s),Vi s)ds

<

< aL(2K + v2) K_ _ <h (hv2

2v1

Vi Vi Vi

t

t

t

t

t

where d1, d2 > 0 are constants independent of v1, v2. Then

(h d2v2\ La _c_i £2^2

~iГ" I f - I-

v1 v1 v1 v1

Iù(t, i/i, ж0)| < ( 77 + j e = t^ + with 0 < t < a

and hence inequality (2.6) holds. We note that

C'2 = eLa■ (2.9)

This equality is used below in the proof of Statement 5. b) Averaging the function g\(x,T2) over t2

i i

J gi(x, T2)dr2 = J J f(x, Ti, т2)dridr2 = g(x), 0 0 0

we obtain for the system (2.5) an averaged system that coincides with the system (2.2). In accordance with Statement 1, the estimate of the deviation of the solution f(t, x0) to the system (2.2) from the solution w(t,v2,x0) to the system (2.5) has the form

|w(t, u2,x0) - z(t, ж0)| ^ — with 0 < i < a. (2.10)

v2

Then, using (2.6) and (2.10) with 0 ^ t ^ a, we obtain

\x(t,V1 ,V2,xo) — f(t,xo)\ ^ \x(t,V1,V2, fo) — W (t,V2 ,xo )| + \W (t,V2, fo) — Z(t,xo )| ^

< °2U'2 £l ^ Ui Ui u2'

This completes the proof.

If one changes the order of averaging and averages first over т2 and then over т1 , the estimate of the deviation of the solution to the averaged system from the solution to the initial system will have the form

c* c* V1 c*

\x(t, v\, v2,xq) — z(t, a?o)| ^ — + - + — with 0 < t < a, (2.11)

v1 v2 v2

where c1, c*, c3 > 0 are some constants independent of v1, v2. From the two estimates, (2.3) and (2.11), one should choose the best. For example, for the case of large frequencies, if v1 » v2 » 1, the best estimate is (2.3), from which we obtain

\x(t,vi,v2,xo) — z(t,xo)\ < c[ —+ —) with 0 ^ i ^ a,

v2 v1

where c > 0 is some constant independent of v1, v2.

Statement 3 is generalized to the case of three and more frequencies. Consider the system of ordinary differential equations

x = f(x,V1 t,V2t,...,Vkt), x e Rra, (2.12) _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2020, 16(2), 379-394_"j^

i

where f(X,Ti,T2,... ,Tk) is a function that is continuously differentiable with respect to all variables, periodic in the variables Ti,...,Tk with period 1, and independent of Vi,...,Vk. We assume that the frequencies Vi have been renumbered in decreasing order

v1 > v2 > ... > Vk > 0.

Let us average the right-hand side of the system (2.12) over Ti,...,Tk and write the averaged system

i i

f = g(z), where g(z)= ... f(z,Ti ,...,Tk )dTi ...dTk. (2.13)

Statement 4. Suppose that for some a > 0 and any Vi,...,Vk > 0 the solutions X(t,Vi,...,Vk,fo) and f(t,Xo) to the systems (2.12), (2.13) with initial conditions X(0,v1,...,Vk,fo) = Xo and f(0,fo) = Xo, where Xo lies in some bounded region U C Rra, exist on the interval 0 ^ t ^ a and lie in some bounded region V C Rra. Then one can find constants ci ,...,ck ,ci,..., c*k_ i > 0 such that for all X0 G U with 0 ^ t ^ a one has

+ ^ + + ^ + ^ + ^ + + (2.14)

Vi V2 Vk Vi V2 Vk-i

Proof. Averaging step by step the function f(X,Ti,T2,...,Tk) over the variables t1,t2,..., Tk-i, we obtain the averaged systems

fi = gi(fi ,Vi+i t,...,Vk t), i = 1,...,k - 1, (2.15)

i i

where gi(X,Ti+i,... ,Tk) = J ... J f(X,Ti,...,Tk )dTi ...dTi. Let fi(t,Vi+i,...,Vk ,fo)

oo

(i = 1,... ,k — 1) be a solution to the system (2.15) on the interval 0 ^ t ^ a with initial conditions zi(0,Vi+i,...,Vk,f0) = X0. For 0 ^ t ^ a, in accordance with the estimates (2.6) and (2.10) obtained in the proof of Statement 3, we have

_j . _j . .. ci CIV2

\x{t,Vi, .. .,vk,x0) - Zi{t,V2, .. .,vk,x0)\ ^--1--

V1V1

\Zi-i{t,Vi, .. .,uk,x0) - Zi(t,Ui+1,.. .,uk,x0)\ ^ — + —i = 2,...,k-l

Vi Vi ck

\zk-i(t,uk,x0) -z(t,xo)| ^ —.

Vk

Using these inequalities and the inequality

\X(t,Vi,.. .,Vk,Xo) — f(t,Xo)\ ^ \X(t,Vi,.. .,Vk,Xo) — fi(t,V2,.. .,Vk,fo)| + + \zi(t,V2, . . .,Vk, Xo) — Z2(t,V3, ...,Vk, Xo) \ + ... + \ fk-i(t,Vk ,fo) — f(t,Xo)\,

we obtain the estimate (2.14).

Taking the inequalities v1 > v2 > ... > Vk into account, we can write the expression (2.14) for estimating the deviation of the solution f(t,Xo) to the averaged system from the solution X(t, v1 ,..., Vk,Xo) to the initial system with 0 ^ t ^ a as follows:

\x{t,vi,.. .,vk,x0) - z(t,x0)| < c(— + — + ... + + —). (2.16)

V1 V2 Vk 1 Vk

3. Examples

In this section, we consider a pendulum with a vibrating point of suspension. The problem of the motion of pendulum-type systems with vibrating elements is a classical one [10-12]. Foremost among studies concerned with such systems are [13-19].

3.1. A mathematical pendulum with a vibrating point of suspension in a rotating coordinate system

Consider a mathematical pendulum of length l whose plane undergoes rotational oscillations about a fixed vertical axis Oz and whose point of suspension oscillates along this axis. The change in the coordinate of the point of suspension on the axis Oz and in the angle 6 of rotation of the pendulum's plane about the axis Oz is given by the relations

Zo(t) = ah\(v\t), 6(t) = bh2 V1). (3.1)

Here a, b > 0 are constants, h1(t), h2(T) are continuous periodic functions with zero average value which are different from a constant, and v1, v2 > 0 are oscillation frequencies.

Let y denote the angle of deviation of the pendulum from the descending vertical and let us write the Hamiltonian function of the system. In the fixed coordinate system Oxyz the coordinates of the pendulum are

x = l sin y cos 6, y = l sin y sin 6, z = z0 — l cos y,

therefore, its squared velocity is

v2 = l2y2 + l262 sin2 y + 2lyz0 sin y + z$.

The Lagrangian function of the system (up to calibration) has the form

, ly2 l62 sin2 y

L(tp, f,t) = — + tpz0 sin (f H-----h g cos f.

Introducing the generalized momentum

dL ; . .

p = — = If + z0 sin f, dy

we write the Hamiltonian function of the system

U! (p — zo sin y)2 l62 sin2 y

H{f,'P,t) =--------gcosf, (3.2)

where z0(t), 6(t) are given by (3.1). The Hamiltonian function is represented as a sum of the Hamiltonian function of the system unperturbed by oscillations, and two time-periodic functions with frequencies v\, v2:

p2

H(f,p,t) = H0(f,p) + Hi(f,p,Ti) + H2(f,T2), H0(f,p) = — -gcosf,

ap sin yhi (ti) a2 sin2 yh (ti)

H1(f,p,Ti) =--j-+---, Ti = vit,

■ . lb2 sin2 yh(t2) H2(f, T2) =--:2 2V 27 , T2 = V2t.

Thus, one can use Statement 1 about averaging in a two-frequency system. The right-hand side of the Hamilton equations is given by partial derivatives of the Hamiltonian function in the canonical variables p, p. Therefore, the averaged Hamilton equations are also Hamilton equations with the Hamiltonian (of the initial system) averaged over the variables r\, t2. The average values of the functions Hi(p,p,Ti) and H2(p,T2) over the variables t\ and t2 are, respectively,

■77 / n 1 irr, , , a2 sin2 plif — 1 7 TT lb2 sin2 p h?2

Hi(p) = — / =-^r^, H2(p) = — / H2(tp,t2)d,t2 =--

Si J 21 S2 J 2

0 0

«i

Here Si is the period of the function /¿¿(r) and h2 = — h2(r) dr, i = 1,2. Then the Hamiltonian

si J 0

of the averaged system is

— — — v2 a2h?-l2b2h?

H(p,p) = H0(p,p)+Hi(p)+H2(p) = ^-gcosp + ' 1 2l-^ sinV (3.3)

Let p(t,vi,v2), p(t,vi,v2) be a solution to the Hamilton equations with Hamiltonian (3.2) on the interval 0 ^ t ^ a (a > 0), and let p*(t), p*(t) be a solution with the same initial conditions to the averaged Hamilton equations with Hamiltonian (3.3) on the interval 0 ^ t ^ a. In accordance with Statement 1, the estimate of the deviation of the solution to the averaged system from the solution to the initial system has the form

\ip(t,ui,u2)-ip*(t)\ + \p(t,ui,u2)-p*(t)\^ — + — with O^t^a,

where ci, c2 > 0 are some constants independent of vi, v2. Thus, the solutions of the initial system tend uniformly in t to solutions of the averaged system if the oscillation frequencies vi, v2 tend to infinity.

Let us investigate the stability of the equilibrium points of the averaged system. According to (3.3), the potential energy V* (p) of the averaged system is

V*(p) = -gcosp + ^ 7 sin2 p. (3.4)

21

The equilibrium points of the averaged system are found from the equation

dV * ( k

= sin p \ g + — cos p = 0

dp \ 1 J

where k = a2h2 — l2b2h2. The system always has two equilibrium points: pi =0 (the lower one) and p2 = n (the upper one). When \k\ > gl, the system has two more equilibrium points (lateral equilibria)

gV

<£>3,4 = ± arccos ( —

d2 V *

The lower equilibrium point is stable when

dp2

d2 V*'

upper equilibrium point is stable when

k

= g + — > 0, i.e., when k > —gl. The

p=0 l k

= — g + — > 0, i.e., when k > gl. The lateral

p=n l

dp

equilibrium points are stable when

d2V *

dp2

<=<3,4

= -jsm2p\v=

<£=<£3,4

> 0, i.e., when k < -gl.

When k = —gl, the lower equilibrium point is degenerately stable. When k = gl, the upper equilibrium point is degenerately unstable.

Thus, we obtain:

When k < —gl, the averaged system has four different equilibria: a lower, an upper and two lateral equilibria located below the point of suspension of the pendulum, the lower and upper equilibria being unstable and the lateral ones stable.

When —gl ^ k ^ gl, the averaged system has only two different equilibria: a stable lower and an unstable upper equilibrium.

When k > gl, the averaged system has four different equilibria: a lower, an upper and two lateral equilibria located above the point of suspension of the pendulum, the lower and the upper equilibria being stable, and the lateral ones unstable.

In contrast to the classical problem of a pendulum with a point of suspension vibrating along a vertical (b = 0), in this problem the upper equilibrium point of the averaged system becomes

stable at a large value of the amplitude of velocity of vertical oscillations: a2 > a situation is possible where the lower equilibrium point becomes unstable.

2 gl + l2b2h22

2 ^ 2 . Also,

h?

3.2. A mathematical pendulum with a vibrating point of suspension

Consider a mathematical pendulum of length l which moves in a fixed vertical plane Oxz and whose point of suspension undergoes oscillations such that the velocity components of the point of suspension in the fixed coordinate system Oxz are

x o(t) = bh2(v2t), zo(t) = ahi(uit),

(3.5)

where a, b > 0 are constants, h1 (t), h2(T) are continuous periodic functions that are different from a constant and have zero mean value, and v1, v2 > 0 are oscillation frequencies. The dynamics of such a system was studied, for example, in [20, 21].

Letting y denote the angle of deviation of the pendulum from the descending vertical, we set up the Hamiltonian function of the system. In the fixed coordinate system Oxz the coordinates of the pendulum are

x = x0 + l sin y, z = z0 — l cos y, and hence its squared velocity is

v = l y + 2lyX0 cos y + 2lyz0 sin y + X0 + zz0.

Then the Lagrange function (up to calibration), the generalized momentum and the Hamiltonian function of the system have the form

,2

T , . lif" L{Lp, <p, t) = ——h <fXo cos ip + tpzo sin ip + g cos p,

dL .. .

p = —— = Ip + .To COS p + Zo sin p, dip

(p - x0 cos p - Zo sin p)2 H(p, p,t) =----g cosp,

Ll

(3.6)

where x0(t), z0(t) are given by (3.5). The Hamiltonian function is represented as a sum of the Hamiltonian function of the system unperturbed by oscillations, and the function containing time-periodic multipliers with frequencies vi, v2:

p2

H{p,p,t) = H0(p,p) + F{p,p,n,T2), H0(p,p) = — -gcosp,

. apsinphi(t1) a2 sin2 ph(t1)

F(p,p, n, r2) = --Z-* +-* 1V -

bp cos ph2(T2) b2 cos2 ph2(t2) ab sin2phi (t1) h2(T2) --1-+-21-+-21-' Tl=Vlt> T2 = V2t

We assume that vi = v2; otherwise we obtain a single-frequency system (t1 = t2 = t) whose right-hand side can be averaged over the variable t and for which we can use Remark 1. The average value of the function F(p,p,Ti ,t2) over the variables Ti and t2 is

—. . 1 if', n , , a2sin2p/?,? b2 cos2 p h?2 F(p) =- / F(p,p,Tl,T2)d,TldT2 =-2

SiS2 J J 1 2 2l 2l

00

Si

where Si is the period of the function hi{r) and h2 = — /?|(r)dr, i = 1,2. Consequently, the

Si

0

Hamiltonian of the averaged system is

H(V,P) - H^,P) + F{V) = + + (3.7)

Let p(t, vi,v2), p(t, vi,v2) be the solution of the Hamilton equations with Hamiltonian (3.6) on the interval 0 ^ t ^ a (a > 0), and let p*(t), p*(t) be the solution with the same initial conditions to the averaged Hamiltonian equations with Hamiltonian (3.7) on the interval 0 ^ t ^ a. In accordance with Statement 3, we obtain an estimate of deviation of the solution to the averaged system from the solution to the initial system

\<p(t,v1,v2)-<p*(t)\ + \p(t,v1,v2)-p*(t)\^— + — + — with O^t^a, (3.8)

Vi Vi V2

where ci, c2, c3 > 0 are some constants independent of vi, v2. Thus, the solutions of the initial

system tend uniformly in t to solutions of the averaged system if the oscillation frequencies vi, v2

V2

tend to infinity and the ratio of frequencies — tends to zero.

Vi

In some cases, one can obtain a more accurate estimate. Suppose, for example, that hi(T) = = h2(T) = cos t, i.e., the point of suspension of the pendulum undergoes oscillations according to the law

xo(t) = —sin v2t, zo(t) = —sinz/it, / z/2.

V2 Vi

Since

hi{ti)h2{t2) = cos t\ cos r2 = - (cos(ri + r2) + cos(ri - r2)), the last term in the function F(p,p,Ti ,t2) can be represented as

ab sin 2p cos t\ cos r2 ab sin 2p cos t-s ab sin 2p cos T4

2l 4l 4l

where t3 = (v1 + v2)t, t4 = (v1 — v2)t. Hence, in this case the Hamiltonian function is represented as a sum of the Hamiltonian function of the system unperturbed by oscillations, and four time-periodic functions with frequencies v1, v2, v1 + v2, v1 — v2:

H (<p,p,t) = Ho(<p,p) + Hi(<p,p,Ti) + H2(p,p,T2) + H3(p,T3) + HA(p,T4).

Therefore, according to Statement 2, we obtain an estimate of the deviation of the solution to the averaged system from the solution to the initial system

c* c* c* c* \<p{t,v1,v2)-<p*{t)\ + \p{t,v1,v2)-p*{t)\^-± + -± +-+ -^-r with O^t^a,

V1 V2 V1 + V2 Vi — V2\

where c*, c*2, c*, c* > 0 are some constants independent of v1, v2. Thus, in this case the solutions of the initial system tend uniformly in t to solutions of the averaged system if the oscillation frequencies v1, v2 and the absolute value of their difference tend to infinity. The last estimate

improves the estimate (3.8). For example, for v2 = 2v1 we have

c

\<p(t,vi,v2)-<p*(t)\ + \p(t,v1,v2)-p*(t)\^i— with 0

where c is some constant independent of v1, v2.

Remark 3. According to (3.7), the potential energy V*(y) of the averaged system is

a2l4-l2b2M . o

v m = -9cos^H----- Siir if

and coincides with the function (3.4). Therefore, the conclusions on the stability of the equilibrium points of the averaged system, as obtained in Example 1, hold in this example as well.

4. Average values of functions on a torus for conditionally periodic motion

Statement 5. Let f (x1,x2) be a function that is continuously differentiable and periodic in the variables x1, x2 with period 1. We assume it to be a function on a two-dimensional torus with angular coordinates x1,x2 mod 1. Then for any conditionally periodic motion

x1 = x1 + w1T, x2 = x2 + u2t, 0 ^ t <

on this torus with frequencies u1,u2 the deviation of the time average from the space average for the function f is estimated by the inequality

T

1 1

^lim 7p J f {x°i + W1TJ x2 + w2t) d,T - j J f{xi,x2)dxidx2

0 0

^ C-,

W1

(4.1)

where c = LeL/2 and L is the Lipschitz constant of the function f (x1,x2).

Proof. Given > 0, v2 > 0, a > 0, consider the ordinary differential equation du

dt

= f (xl + vit,x0 + V2t), u e R, 0 < t < a.

(4.2)

The averaged equation has the form

i i

dv

— = g, g= I I f{xi,x2)dx\dx2, v G R, O^t^a. (4.3)

0 0

Let u(t,vl,v2) and v(t) denote the solutions to Eqs. (4.2), (4.3) on the interval 0 ^ t ^ a with initial conditions u(0, v1,v2) = 0 and v(0) = 0:

t

u(t,vi,v) = J f (x°° + Vis,x0 + V2s)ds, v(t) = gt. 0

According to Statement 3, there are constants cl, c2, c3 > 0 such that the estimate of the deviation of the solution v(t) to the averaged equation from the solution u(t, v1,v2) to the initial equation has the form

Ci C2V2 Сз

|u,(t, 1/1,1/2) - v(t) I < — + + — with 0 < t < a. Vl Vi V2

For t = a we have

a

I f (,i + + - 9a

< £L C2U'2 SI

^ Ui Ui b>2

Taking any T > 0 and rescaling time as s = ^ under the integral, we obtain

T

Irin ат n ат\ ,

/ /(zi + y>x-2 + u'2y) ~ga

< £L c'2U'2 £l

^ Ui Ui b>2

Setting г/i = — wi, г/2 = — W2, a = 1, we obtain

T

J f ix°i + u\T,xl + ш2т) dr-g

< Cl C"2ÜJ"2 сз Тш1 ш1 Тш2'

Here c2 = LeL/2 according to (2.9). For T ^ we obtain inequality (4.1).

We note that, using inequality (2.16), one can obtain a similar statement for conditionally periodic motion on an n-dimensional torus.

Statement 6. Let f(x1,...,xn) be a function that is continuously differentiable and periodic in the variables x1,...,xn with period 1. We will assume it to be a function on an n-dimensional torus with angular coordinates x1,...,xn mod 1. Then for any conditionally periodic motion

x1 = x1 + w1T, ...,xn = x°n + WnT, 0 ^ t <

on this torus with frequencies w1,...,wn the deviation of the time average from the space average for the function f is estimated by the inequality

T

lim —

T—>+oo T

j f (x1 + wit, + w2t ) dT -J ...J f (xi,..., xn)dxi

0

W Wn-i

<

where c = LeL/2 and L is the Lipschitz constant of the function f.

i

i

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