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Resonant motions in a substantially non-linear system with one degree of freedom in the critical case
RESONANT MOTIONS IN A SUBSTANTIALLY NON-LINEAR SYSTEM WITH ONE DEGREE OF FREEDOM IN THE CRITICAL CASE* L. D.
AKULENKO Moscow
(Received
1.
The
Consider
problem.
solutions
of
the
set
of
the
1966)
November
29
stationary
periodic
or rotatory
resonant
equations
dE =‘Ef(t,
E, 4); e),
at
;
-
(1)
o(E)+ eF(t, E, +; e),
t 5 (-co, 00) Is the independent real where E > 6 is a small parameter, variable, E is the perturbed “energy”, v the perturbed phase, and f,w and F are smooth functions of their arguments, S- and 2a-periodic with The chief method of invest igatlng these respect to t and 01 respectively. solutions
is
by the
so-called
equation
of
phase
balance
[1 - 51
T Pt(r)=
‘2)
1 f(r, EO*, $0; 0)dt = 0, 0
where
Ea* is
the
real
root
of
the
equation
V
_=;
(3)
(v=2?),
o(E)
in which
T = mS,
and m and n are
(n/m)V(t
-
T.
*
Zh.
to)
uychisl.
+
Mat.
mat.
Fit.
7,
relatively
6,
221
prime
1379 - 1385,
Integers;
1967.
\vc =
222
L.D.
It was shown in [21 that the resonant solution of (1) exists a sense, and has the form
Akulenko
stationary (for for E sufficiently
all
t E small,
[to,
CO)) m/n-
Is unique
in
E = EC,*+ eu(t, e), (I; =
(IL / m)v(t
-
to) + T* + ev(t,
(4)
e),
where u and v are functions smooth with respect to t and E aud periodic in t, provided T* Is the real root of (2) and we have o’(Eo’)aP,/&* +o. The solution vided
(4)
is
asymptotically
Lyapunov-stable
and is unstable if one or both these assertions can be made for t L to.
inequalities
for
t ryto,
are reversed.
pro-
Similar
A very common practical case Is when PlCr)= 0, i.e. the phase balance equation Is satisfied independently of -f. We say in this case that motions of higher degrees (above the first) exist. The oscillations of the first. second and third degrees were examined in [41 for the close t 0 conservative system z+F(s) The possibility of such out in [51. The rotatory lar to (5) were studied The present degree
for
the
note
E==E0”+&i(t)
oscillations solutions in [Sl.
obtains
the
more general
2. The solution. of the first degree,
we seek +...,
e)+
(5)
for more general systems was pointed simiof higher degree of an equation
stationary
system
As in the
+e2&(t)
= ef(t,s;%
(1)
resonant
and examines
solutions their
ordinary case of rotations the solution as the series 1P =
(n/m)v(t--to)
of the second
stability. or oscillations
+T+E+)i(t)
+...,
(6)
of course, that the right -hand sides are analytic in the assuming, neighbourhood of E = EO*, li, = $0, e = 0, Substituting (6) In (1) and equatinfinite ing coefficients of like powers of E, we obtain the following set of equations for the unknown periodic functions Ei, vi Ci& 2):
Resonant
motions
in
a substantially
non-linear
223
system
In the case of motions of the first degree, all the coefficients can be found from this set independently of the form of the functions fi_1, of the second degree, we have to know the form of the Fi-1. For motions terms containing gi_2 and vi-gIn particular, the unknown phase constant T of the zero approximation of y is given by the condition for E2 to be periodic, i.e. by
(7)
where
Al(t)
is
a known function
iT
Ai=--
T
tained
from the
expressions
E,
=
&(r)+
for
El
1fodh
s 0
dt
t s
0
to
and ~1
(fo = tl.=o, E=E,*,+%)>
to t W1=B,(t)i-oi’d,(~)(t-t~)+~o’S
t1
dti
to
(1
fodfz+Fo)
=
B,(z)+
~(4 z),
to
in which B1 is a constant of integration, and ?I( t, T) is a T-periodic function. Hence, if -r* is the real root of (‘I), ye and EI will be completely determined. To find Eg and ~1, we use the periodicits conditions for y2 and- Eg. The first condition gives a connection between the as yet unknown constants of integration Bl and AS, namely,
(8) Substituting that
dkP1 / a+
(8) =
into 0 ik =
the 0,1,2,.
periodicity . .), we get
condition
for
Es,
and recalling
L.D.
294
Akulenko
where
It is relations
easily seen that the next coefficients of the form (6) and (9):
The following simgle real root
Ai+l are given
by
conclusion can be drawn from the above. If T* is a of (‘I), a solution of the required form (4) can be found
In the form of series parameter e, provided a simple real root of represent the required degree, unique in the periodic coefficients their index, the solution
Bi,
(6) in increasing integral powers of the small of course that o’(Eo*) # 6, i.e. that Eg. is also (3). If these series are uniformly convergent, they stationary m/n-resonant solution of the second region where f, o and F are analytic. The bounded Ei, vii (i 22 1) can be found, no matter how high
by the method described, in the explicit is, up to e2 for E and E for ly:
form.
In particular,
Resonant
motions
in
I= + Fo -;
a substantially
non-
225
system
linear
t
1 dt
sfo dt, - $
0
to
Our method can be extended to the nonanalytic case when f and w have second part ial derivatives in the neighbourhood of g = go*, v, = ~0, derivative. As before, we only require E = 0, and F has only the first continuity with respect to t. To summarize: Theorem
1
If 1) PI(T)
= 0;
31 equations
(3)
3) o’(BO*)aPg/;)r*
and (7)
have unique
Eo*,
T*;
= 0,
then the disturbed system (1) with E f, o and F are defined and smooth, a solution of the second degree of the ing E = go*, yl = (nlm)V(t - to) + 9 Note solution
roots
> 0 small has, in the region where unique stationary m/n-resonant form (41, reducing to the generatfor E = 0.
1. The uniqueness is understood in the sense corresponding to each fixed set of constant
As a rule, v* in the
(3) has one root go*, while interval (0, 2~) [21.
Note 2. As mentioned above, under more general assumptions
(7)
that there Is one m, n, Eo* and v*.
has a finite
the solution (4) exists regarding the smoothness
number of roots
and is of f,
Note 3. The case when (7) has multiple roots -r* presents difficulties and Involves an expansion in fractional powers parameters [3, 3, 6, 71.
unique o and F.
considerable of the small
Note 4. If o’@e*) # 0, but Pi = 0, the motions realizable In the system have degrees not less than 3. The corresponding solutions can be obtained by a similar method. Notes 1 - 3 also apply to these SOlUtiOns. 3. Stability
of
second
degree
motions.
The case
of multiple
zero
226
L.D.
Akulcnko
characteristic exponents is well known to present great difficulties in studying the stability of motions described by equations with a small parameter. In some simple cases, however, the characteristic equation can be found and the characteristic exponents written as series in lntein the case of the era1 [51 or fractional [2 - 61 powers of E.’ Clearly, set of two equations cl), the expansion will be in powers of 6 = JE [4, 51 . Recalling Lyapunov’ s familiar theorem, we form a system in variations in the Polncare form. I.e. we confine ourselves to the first approximation system:
If both characteristic exponents of (16) have negative real parts, the perturbed solution will be asymptotically Lyapunov-stable, and will be unstable if one or both have a positive real part [4, 51. The other critical cases require the consideration of the still neglected nonlinear terms. The aim of the present section is to find sufficient conditions for the asymptotic stability of our second-degree solution. The solution
of
(16) will
be sought E = z&e=‘,
as q = vest,
where u and v are T-periodic functions ponent. We have the expansions
Substituting
(12)
in (11).
then (11)
(11;
and a is the characteristic
in (lo),
and equating
ex-
coeffi-
cients of like powers of S, we obtain, in particular. from the system of the zero approximation: ue = 6, vu = bg = const. From the system of the first approximation, we have aI = al = const., ~1 = bl = const., where w’(&*)ai = a,bO, In view of this,
the system of the second au2 -=
bo,
dt
*vz
-= dt
(13) approximation
o’(Eo’)
u2 +
bo -
a&o -
sib,.
in 6 is
Resonant
motions
in
a substantially
non-linear
The ug perlodicity condition gives al = 3, which, al = 0, i.e. III = 0. The ~2 periodicits condition
The equations
of the next approximation
au3
-
dl
=
O’(Eg+)UQ +
3F -
227
system
by (131, is
leads
to
are
vi -
aA
-
a&O,
() ti0
whence
u3
a3 +
=
h
where
The perfodicity condition for the function ~4, obtained equation of the fourth approximation, namely,
from the
is (14)
co&P+ cio2 + cz = 0, in view of which: @pi / 8th = 0 (k = &g;.
. .), where
*T co=--;,
ci=~j[(~)o+(;)o] 00
It follows provfded
d4
c2-$.
0
obviously
from (14) that the solution
i[(%)o+(;)o]
dt
0
ghile the second of (15) is necessary sufficient. Notice that the vanishing
is asymptotically
oo’$
stable
(15)
for stability, it is by no means case is excluded from the theorem.
228
L.D.
Akulcnko
However, if the first expression is zero, the characteristic exponents in the calculated approximation are pure imaginary, and in order to find the sufficient conditions for stability, we have to evaluate a3 etc. As a result we have: Theorem
2
Relations (15) are the stability of our perturbed
sufficient resonant
conditions for solution of the
asymptotic Lyapunovsecond degree for
t > to. 4. Example. Take the problem of the resonant rotatory motions of a plane physical pendulum with moment of inertia I, the point of suspension of which executes small rectilinear oscillations of high frequency through an angle 6 to the vertical according to the law A cos at + (B/4) forces and the second cos 2 at. A simpler problem, In which friction harmonics of the oscillations are Ignored, Is considered in [51. The motion
of the
system
d2x
Iz+pagsinx=
can be described
approximately
by
--pao2sin(z-*)(Acosot+Bcos20t)-g~ dt
’
where x is the angle of deviation of the pendulum from the vertical, u a the distance from the centre of gravity to the mass of the pendulum, the axis of rotation, and q the constant coefficient of viscous friction We introduce the parameter acting on the axis of rotation. forces, e = g’lnl-%o-‘-=q 1, where 1 = I/us is the reduced length of the pendulum. We also denote A -= 1
The equation
eb,
-
B 1
of mot ion d2x -=-
=
ezbc,
9 = OZ
82).
(b, c, h -
11,
0 =
wt
is
s2sinr-ebsin(r-6)(;ose+eccos28)-e2h~.
d02
Introducing the phase ly and “energy” g in accordance with the usual [81, this equation can be reduced to a system with rotating formulae however, since the phase of the form (I). There is no need for this, equation is easily written directly in the forrll (1): dE -=
de
-
e2
sin Q - eb sin(g - 6) (cos 0 + ec cos 0)-
e%E,
motions
Resonant
The equation
of phase
in a substantially
non-linear
229
system
balance is 2mn
P, (7) = - b
Ze+T-8 m
!
cosOde=O
which is easily seen to define the nain resonant motion in the Rotations with m/n # 1 are obviously motions of higher degree. resonance l/2. Equation (7) then becomes P2c-r) = - abc sin(~ 4nh = 0, which, if h < bc/4. has two real roots in an interval 2r: 4h z, = 6 - arcsin -
43.
‘t2= 6+n+arcsin--,
hc’
system. the 6) of length
Take
bc
where 8Pz / ~3%~( 0 for -r* = T 1 and 8Pz / at* > 0 for “I* = 72. Since, for solution can be obtained by evaluA < bc/4, aP&* # 0, any approximate ating the coefficients of the series (6) or by successive approximations. This perturbed solution is asymptotically stable for h > 0 for T* = ~1, and unstable otherwise. Translated
by D.E.
Brown
REFERENCES 1.
BOGOLYUBOV,N.N. Theory
of
nelineinykh 2.
3.
kolebanil),
Oscillations
Yu.A. Asymptotic (Asimptoti~heskie
Moscow, Fizmatglz,
AKULENKO, L.D. On resonance in non-linear freedom. Zh. uychisl. Mat. mat. Fiz. 6, AKULENKO, L. D. and VOLOSOV, V.L. Vest.
4.
and MITROPOL’SKII,
Non- linear
Mosk.
Gas.
Univ.
Ser.
mekhan.,
in
metody
the
v teorii
1963. systems
with
6,
-
On resonance
mat.,
Methods
1126
one degree 1966.
in a rotatory 1,
KATS, A.M. Forced oscillations of near-conservative systems with one degree of freedom, Prikl. Mat. 32, 1955.
13
of
1130,
-
17,
system, 1667.
non-linear Mekh.,
19,
1,
13
5.
MALKIN, 1-G. Some problems of non-linear oscillation theory (Nekotorye zadachi teorii nellneinykh kolebanii). Moscow, Gostekhlzdat, 1956.
6.
AKULENKO, L.D.
and VOLOSOV, V.M. Resonant
rotations
of higher
-
230
L.D.
degrees, 1961. 7.
8.
Vest.
Mosk.
Gos.
Akulsnko
Univ.
Ser.
mat.,
mekhan.,
2,
10
-
14,
PROSKURYUAKOV. A.P. Periodic solutions of quasilinear aUtOnOmOUS systems with one degree of freedom, as series in integral and fractional Powers of a parameter, in hoc. Internat. Symposium Non-linear Oscillations, 1, 367 - 380, Kiev, AN USSR, 1963. voLOsOV, V.M. Averaging tions, Usp. mat. Nauk
in systems of ordinary 17,
6,
3
-
126,
1962.
differential
equa-
on
AKULENKO Moscow
(Received
1.
The
Consider
problem.
solutions
of
the
set
of
the
1966)
November
29
stationary
periodic
or rotatory
resonant
equations
dE =‘Ef(t,
E, 4); e),
at
;
-
(1)
o(E)+ eF(t, E, +; e),
t 5 (-co, 00) Is the independent real where E > 6 is a small parameter, variable, E is the perturbed “energy”, v the perturbed phase, and f,w and F are smooth functions of their arguments, S- and 2a-periodic with The chief method of invest igatlng these respect to t and 01 respectively. solutions
is
by the
so-called
equation
of
phase
balance
[1 - 51
T Pt(r)=
‘2)
1 f(r, EO*, $0; 0)dt = 0, 0
where
Ea* is
the
real
root
of
the
equation
V
_=;
(3)
(v=2?),
o(E)
in which
T = mS,
and m and n are
(n/m)V(t
-
T.
*
Zh.
to)
uychisl.
+
Mat.
mat.
Fit.
7,
relatively
6,
221
prime
1379 - 1385,
Integers;
1967.
\vc =
222
L.D.
It was shown in [21 that the resonant solution of (1) exists a sense, and has the form
Akulenko
stationary (for for E sufficiently
all
t E small,
[to,
CO)) m/n-
Is unique
in
E = EC,*+ eu(t, e), (I; =
(IL / m)v(t
-
to) + T* + ev(t,
(4)
e),
where u and v are functions smooth with respect to t and E aud periodic in t, provided T* Is the real root of (2) and we have o’(Eo’)aP,/&* +o. The solution vided
(4)
is
asymptotically
Lyapunov-stable
and is unstable if one or both these assertions can be made for t L to.
inequalities
for
t ryto,
are reversed.
pro-
Similar
A very common practical case Is when PlCr)= 0, i.e. the phase balance equation Is satisfied independently of -f. We say in this case that motions of higher degrees (above the first) exist. The oscillations of the first. second and third degrees were examined in [41 for the close t 0 conservative system z+F(s) The possibility of such out in [51. The rotatory lar to (5) were studied The present degree
for
the
note
E==E0”+&i(t)
oscillations solutions in [Sl.
obtains
the
more general
2. The solution. of the first degree,
we seek +...,
e)+
(5)
for more general systems was pointed simiof higher degree of an equation
stationary
system
As in the
+e2&(t)
= ef(t,s;%
(1)
resonant
and examines
solutions their
ordinary case of rotations the solution as the series 1P =
(n/m)v(t--to)
of the second
stability. or oscillations
+T+E+)i(t)
+...,
(6)
of course, that the right -hand sides are analytic in the assuming, neighbourhood of E = EO*, li, = $0, e = 0, Substituting (6) In (1) and equatinfinite ing coefficients of like powers of E, we obtain the following set of equations for the unknown periodic functions Ei, vi Ci& 2):
Resonant
motions
in
a substantially
non-linear
223
system
In the case of motions of the first degree, all the coefficients can be found from this set independently of the form of the functions fi_1, of the second degree, we have to know the form of the Fi-1. For motions terms containing gi_2 and vi-gIn particular, the unknown phase constant T of the zero approximation of y is given by the condition for E2 to be periodic, i.e. by
(7)
where
Al(t)
is
a known function
iT
Ai=--
T
tained
from the
expressions
E,
=
&(r)+
for
El
1fodh
s 0
dt
t s
0
to
and ~1
(fo = tl.=o, E=E,*,+%)>
to t W1=B,(t)i-oi’d,(~)(t-t~)+~o’S
t1
dti
to
(1
fodfz+Fo)
=
B,(z)+
~(4 z),
to
in which B1 is a constant of integration, and ?I( t, T) is a T-periodic function. Hence, if -r* is the real root of (‘I), ye and EI will be completely determined. To find Eg and ~1, we use the periodicits conditions for y2 and- Eg. The first condition gives a connection between the as yet unknown constants of integration Bl and AS, namely,
(8) Substituting that
dkP1 / a+
(8) =
into 0 ik =
the 0,1,2,.
periodicity . .), we get
condition
for
Es,
and recalling
L.D.
294
Akulenko
where
It is relations
easily seen that the next coefficients of the form (6) and (9):
The following simgle real root
Ai+l are given
by
conclusion can be drawn from the above. If T* is a of (‘I), a solution of the required form (4) can be found
In the form of series parameter e, provided a simple real root of represent the required degree, unique in the periodic coefficients their index, the solution
Bi,
(6) in increasing integral powers of the small of course that o’(Eo*) # 6, i.e. that Eg. is also (3). If these series are uniformly convergent, they stationary m/n-resonant solution of the second region where f, o and F are analytic. The bounded Ei, vii (i 22 1) can be found, no matter how high
by the method described, in the explicit is, up to e2 for E and E for ly:
form.
In particular,
Resonant
motions
in
I= + Fo -;
a substantially
non-
225
system
linear
t
1 dt
sfo dt, - $
0
to
Our method can be extended to the nonanalytic case when f and w have second part ial derivatives in the neighbourhood of g = go*, v, = ~0, derivative. As before, we only require E = 0, and F has only the first continuity with respect to t. To summarize: Theorem
1
If 1) PI(T)
= 0;
31 equations
(3)
3) o’(BO*)aPg/;)r*
and (7)
have unique
Eo*,
T*;
= 0,
then the disturbed system (1) with E f, o and F are defined and smooth, a solution of the second degree of the ing E = go*, yl = (nlm)V(t - to) + 9 Note solution
roots
> 0 small has, in the region where unique stationary m/n-resonant form (41, reducing to the generatfor E = 0.
1. The uniqueness is understood in the sense corresponding to each fixed set of constant
As a rule, v* in the
(3) has one root go*, while interval (0, 2~) [21.
Note 2. As mentioned above, under more general assumptions
(7)
that there Is one m, n, Eo* and v*.
has a finite
the solution (4) exists regarding the smoothness
number of roots
and is of f,
Note 3. The case when (7) has multiple roots -r* presents difficulties and Involves an expansion in fractional powers parameters [3, 3, 6, 71.
unique o and F.
considerable of the small
Note 4. If o’@e*) # 0, but Pi = 0, the motions realizable In the system have degrees not less than 3. The corresponding solutions can be obtained by a similar method. Notes 1 - 3 also apply to these SOlUtiOns. 3. Stability
of
second
degree
motions.
The case
of multiple
zero
226
L.D.
Akulcnko
characteristic exponents is well known to present great difficulties in studying the stability of motions described by equations with a small parameter. In some simple cases, however, the characteristic equation can be found and the characteristic exponents written as series in lntein the case of the era1 [51 or fractional [2 - 61 powers of E.’ Clearly, set of two equations cl), the expansion will be in powers of 6 = JE [4, 51 . Recalling Lyapunov’ s familiar theorem, we form a system in variations in the Polncare form. I.e. we confine ourselves to the first approximation system:
If both characteristic exponents of (16) have negative real parts, the perturbed solution will be asymptotically Lyapunov-stable, and will be unstable if one or both have a positive real part [4, 51. The other critical cases require the consideration of the still neglected nonlinear terms. The aim of the present section is to find sufficient conditions for the asymptotic stability of our second-degree solution. The solution
of
(16) will
be sought E = z&e=‘,
as q = vest,
where u and v are T-periodic functions ponent. We have the expansions
Substituting
(12)
in (11).
then (11)
(11;
and a is the characteristic
in (lo),
and equating
ex-
coeffi-
cients of like powers of S, we obtain, in particular. from the system of the zero approximation: ue = 6, vu = bg = const. From the system of the first approximation, we have aI = al = const., ~1 = bl = const., where w’(&*)ai = a,bO, In view of this,
the system of the second au2 -=
bo,
dt
*vz
-= dt
(13) approximation
o’(Eo’)
u2 +
bo -
a&o -
sib,.
in 6 is
Resonant
motions
in
a substantially
non-linear
The ug perlodicity condition gives al = 3, which, al = 0, i.e. III = 0. The ~2 periodicits condition
The equations
of the next approximation
au3
-
dl
=
O’(Eg+)UQ +
3F -
227
system
by (131, is
leads
to
are
vi -
aA
-
a&O,
() ti0
whence
u3
a3 +
=
h
where
The perfodicity condition for the function ~4, obtained equation of the fourth approximation, namely,
from the
is (14)
co&P+ cio2 + cz = 0, in view of which: @pi / 8th = 0 (k = &g;.
. .), where
*T co=--;,
ci=~j[(~)o+(;)o] 00
It follows provfded
d4
c2-$.
0
obviously
from (14) that the solution
i[(%)o+(;)o]
dt
0
ghile the second of (15) is necessary sufficient. Notice that the vanishing
is asymptotically
oo’$
stable
(15)
for stability, it is by no means case is excluded from the theorem.
228
L.D.
Akulcnko
However, if the first expression is zero, the characteristic exponents in the calculated approximation are pure imaginary, and in order to find the sufficient conditions for stability, we have to evaluate a3 etc. As a result we have: Theorem
2
Relations (15) are the stability of our perturbed
sufficient resonant
conditions for solution of the
asymptotic Lyapunovsecond degree for
t > to. 4. Example. Take the problem of the resonant rotatory motions of a plane physical pendulum with moment of inertia I, the point of suspension of which executes small rectilinear oscillations of high frequency through an angle 6 to the vertical according to the law A cos at + (B/4) forces and the second cos 2 at. A simpler problem, In which friction harmonics of the oscillations are Ignored, Is considered in [51. The motion
of the
system
d2x
Iz+pagsinx=
can be described
approximately
by
--pao2sin(z-*)(Acosot+Bcos20t)-g~ dt
’
where x is the angle of deviation of the pendulum from the vertical, u a the distance from the centre of gravity to the mass of the pendulum, the axis of rotation, and q the constant coefficient of viscous friction We introduce the parameter acting on the axis of rotation. forces, e = g’lnl-%o-‘-=q 1, where 1 = I/us is the reduced length of the pendulum. We also denote A -= 1
The equation
eb,
-
B 1
of mot ion d2x -=-
=
ezbc,
9 = OZ
82).
(b, c, h -
11,
0 =
wt
is
s2sinr-ebsin(r-6)(;ose+eccos28)-e2h~.
d02
Introducing the phase ly and “energy” g in accordance with the usual [81, this equation can be reduced to a system with rotating formulae however, since the phase of the form (I). There is no need for this, equation is easily written directly in the forrll (1): dE -=
de
-
e2
sin Q - eb sin(g - 6) (cos 0 + ec cos 0)-
e%E,
motions
Resonant
The equation
of phase
in a substantially
non-linear
229
system
balance is 2mn
P, (7) = - b
Ze+T-8 m
!
cosOde=O
which is easily seen to define the nain resonant motion in the Rotations with m/n # 1 are obviously motions of higher degree. resonance l/2. Equation (7) then becomes P2c-r) = - abc sin(~ 4nh = 0, which, if h < bc/4. has two real roots in an interval 2r: 4h z, = 6 - arcsin -
43.
‘t2= 6+n+arcsin--,
hc’
system. the 6) of length
Take
bc
where 8Pz / ~3%~( 0 for -r* = T 1 and 8Pz / at* > 0 for “I* = 72. Since, for solution can be obtained by evaluA < bc/4, aP&* # 0, any approximate ating the coefficients of the series (6) or by successive approximations. This perturbed solution is asymptotically stable for h > 0 for T* = ~1, and unstable otherwise. Translated
by D.E.
Brown
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3.
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AKULENKO, L.D. On resonance in non-linear freedom. Zh. uychisl. Mat. mat. Fiz. 6, AKULENKO, L. D. and VOLOSOV, V.L. Vest.
4.
and MITROPOL’SKII,
Non- linear
Mosk.
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the
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1963. systems
with
6,
-
On resonance
mat.,
Methods
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one degree 1966.
in a rotatory 1,
KATS, A.M. Forced oscillations of near-conservative systems with one degree of freedom, Prikl. Mat. 32, 1955.
13
of
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17,
system, 1667.
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19,
1,
13
5.
MALKIN, 1-G. Some problems of non-linear oscillation theory (Nekotorye zadachi teorii nellneinykh kolebanii). Moscow, Gostekhlzdat, 1956.
6.
AKULENKO, L.D.
and VOLOSOV, V.M. Resonant
rotations
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Gos.
Akulsnko
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PROSKURYUAKOV. A.P. Periodic solutions of quasilinear aUtOnOmOUS systems with one degree of freedom, as series in integral and fractional Powers of a parameter, in hoc. Internat. Symposium Non-linear Oscillations, 1, 367 - 380, Kiev, AN USSR, 1963. voLOsOV, V.M. Averaging tions, Usp. mat. Nauk
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differential
equa-
on