Determination of the boundaries between the domains of stability and instability for the Hill's equation
Stability problem for the Hill’s equation containing two parameters is analyzed with computer algebra system M athematica. The characteristic constant is found as a series expansion in powers of a small parameter e. It has been shown that the domains of instability are located only between the c...
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| Цитувати: | Determination of the boundaries between the domains of stability and instability for the Hill's equation / E.A. Grebenikov, A.N. Prokopenya // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 42-51. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1761612021-02-04T01:29:58Z Determination of the boundaries between the domains of stability and instability for the Hill's equation Grebenikov, E.A. Prokopenya, A.N. Stability problem for the Hill’s equation containing two parameters is analyzed with computer algebra system M athematica. The characteristic constant is found as a series expansion in powers of a small parameter e. It has been shown that the domains of instability are located only between the curves a = a(e) on the a−e plane crossing the e = 0 axis at the points a = (2k−1)²/4, k = 1, 2, 3 . . . . The corresponding curves are found as power series in e with accuracy O(e⁶ ). Проблема стабiльностi для рiвняння Хiлла, яке мiстить два параметри, вивчається за допомогою комп’ютерної системи M athematica. Знайдено характеристичну константу в термiнах ряду вiдносно степенiв малого параметра e. Показано, що областi стабiльностi i нестабiльностi знаходяться тiльки на площинi a − e мiж кривими a = a(e), що перетинають вiсь e = 0 в точках a = (2k − 1)²/4, k = 1, 2, 3 . . . . Знайдено вiдповiднi кривi як ряди за степенями e з точнiстю O(e⁶). 2003 Article Determination of the boundaries between the domains of stability and instability for the Hill's equation / E.A. Grebenikov, A.N. Prokopenya // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 42-51. — Бібліогр.: 12 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176161 517.9 en Нелінійні коливання Інститут математики НАН України |
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Stability problem for the Hill’s equation containing two parameters is analyzed with computer algebra
system M athematica. The characteristic constant is found as a series expansion in powers of a small
parameter e. It has been shown that the domains of instability are located only between the curves a = a(e)
on the a−e plane crossing the e = 0 axis at the points a = (2k−1)²/4, k = 1, 2, 3 . . . . The corresponding
curves are found as power series in e with accuracy O(e⁶ ). |
| format |
Article |
| author |
Grebenikov, E.A. Prokopenya, A.N. |
| spellingShingle |
Grebenikov, E.A. Prokopenya, A.N. Determination of the boundaries between the domains of stability and instability for the Hill's equation Нелінійні коливання |
| author_facet |
Grebenikov, E.A. Prokopenya, A.N. |
| author_sort |
Grebenikov, E.A. |
| title |
Determination of the boundaries between the domains of stability and instability for the Hill's equation |
| title_short |
Determination of the boundaries between the domains of stability and instability for the Hill's equation |
| title_full |
Determination of the boundaries between the domains of stability and instability for the Hill's equation |
| title_fullStr |
Determination of the boundaries between the domains of stability and instability for the Hill's equation |
| title_full_unstemmed |
Determination of the boundaries between the domains of stability and instability for the Hill's equation |
| title_sort |
determination of the boundaries between the domains of stability and instability for the hill's equation |
| publisher |
Інститут математики НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176161 |
| citation_txt |
Determination of the boundaries between the domains of stability and instability for the Hill's equation / E.A. Grebenikov, A.N. Prokopenya // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 42-51. — Бібліогр.: 12 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT grebenikovea determinationoftheboundariesbetweenthedomainsofstabilityandinstabilityforthehillsequation AT prokopenyaan determinationoftheboundariesbetweenthedomainsofstabilityandinstabilityforthehillsequation |
| first_indexed |
2025-07-15T13:49:45Z |
| last_indexed |
2025-07-15T13:49:45Z |
| _version_ |
1837721084406792192 |
| fulltext |
UDC 517.9
DETERMINATION OF THE BOUNDARIES
BETWEEN THE DOMAINS OF STABILITY
AND INSTABILITY FOR THE HILL’S EQUATION
ВИЗНАЧЕННЯ МЕЖ МIЖ ОБЛАСТЯМИ СТАБIЛЬНОСТI
ТА НЕСТАБIЛЬНОСТI ДЛЯ РIВНЯННЯ ХIЛЛА
E. A. Grebenikov
Computing Center Rus. Acad. Sci.
Vavilova St., 37, Moscow, 117312, Russia
e-mail: greben@ccas.ru
A. N. Prokopenya
Brest Technical Univ.
Moskowskaya St., 267, Brest, 224017, Belarus
e-mail: prokopenya@belpak.brest.by
Stability problem for the Hill’s equation containing two parameters is analyzed with computer algebra
system Mathematica. The characteristic constant is found as a series expansion in powers of a small
parameter e. It has been shown that the domains of instability are located only between the curves a = a(e)
on the a−e plane crossing the e = 0 axis at the points a = (2k−1)2/4, k = 1, 2, 3 . . . . The corresponding
curves are found as power series in e with accuracy O(e6).
Проблема стабiльностi для рiвняння Хiлла, яке мiстить два параметри, вивчається за допо-
могою комп’ютерної системиMathematica. Знайдено характеристичну константу в термiнах
ряду вiдносно степенiв малого параметра e. Показано, що областi стабiльностi i нестабiльно-
стi знаходяться тiльки на площинi a − e мiж кривими a = a(e), що перетинають вiсь e = 0
в точках a = (2k − 1)2/4, k = 1, 2, 3 . . . . Знайдено вiдповiднi кривi як ряди за степенями e з
точнiстю O(e6).
Introduction. The Hill’s equation is the second order linear differential equation of the form
z′′(t) + p(t) z(t) = 0, (1)
where p(t) is a periodic function with a period T (i. e. p(t + T ) = p(t) for all t). It describes
dynamical systems with intrinsic periodicity and has a lot of applications [1,2]. Our interest to
the Hill’s equation arises because an equation of this type occurs in the study of stability of
exact symmetrical solutions of Newton’s gravitational many-body problem [3 – 5]. In this case
equation (1) has the form
z′′(t) +
a+ e cos t
1 + e cos t
z(t) = 0, (2)
where a and e are some nonnegative parameters. Equation (2) is just a homogeneous equati-
on and, hence, all its solutions have the same stability, which is just the stability of its trivial
solution. In this sense we can refer to stability of the Hill’s equation.
c© E. A. Grebenikov, A. N. Prokopenya, 2003
42 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
DETERMINATION OF THE BOUNDARIES BETWEEN THE DOMAINS OF STABILITY . . . 43
If e = 0 equation (2) reduces to the equation for a simple harmonic oscillator of frequency√
a and its general solution is
z = C1 cos(
√
a t) + C2 sin(
√
a t). (3)
This solution is bounded and oscillatory, and we can say that in case where e = 0 the trivial
solution of (2) is stable. For e > 0 equation (2) may be regarded as a model of a mechanical
system which can be described by a simple harmonic oscillator of natural frequency
√
a that is
subjected to an internally imposed forcing of period T = 2π. It is known that in such systems
a parametric resonance may occur when the function z(t) increases unboundedly as t → ∞.
Therefore, for some values of the parameter a, the trivial solution of (2) may be unstable even
for very small e. To find the corresponding values of a and e, we should calculate the characteri-
stic exponents for equation (2). This problem for a general equation (1) is discussed in detail
in [1], where a number of stability criteria have been formulated. But some of these criteria,
for instance, Liapunov’s integral criterion or Zhukovskii criterion give too large a domain of
instability for equation (2). Another ones are quite cumbersome and it is difficult to use them.
So the main aim of the present paper is to find the domains of instability for equation (2)
in the a − e plane and to determine their boundaries. Solving this problem is connected with
cumbersome analytical calculations that can be reasonably done only with a computer software.
In the present paper, all calculations are done with computer algebra system Mathematica that
is a very powerful tool for doing both numerical and symbolic calculations [6].
General stability theory for the Hill’s equation. It is easy to see that equation (2) is equi-
valent to the first order system of the form
dx
dt
= A(t) x, (4)
where x is a vector with components x1 = z and x2 = z′ andA(t) is the 2×2 periodic matrix
A(t) =
(
0 1
−a+ e cos t
1 + e cos t
0
)
.
The principal fundamental matrix for the system (4) is
X(t) =
(
z1 z2
z′1 z′2
)
,
where z1(t) and z2(t) are linearly independent solutions of equation (2) satisfying the following
initial conditions:
z1(0) = 1, z′1(0) = 0,
(5)
z2(0) = 0, z′2(0) = 1.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
44 E.A. GREBENIKOV, A.N. PROKOPENYA
The characteristic multipliers ρ for the system (4) are defined as eigenvalues of the matrix
X(T ) ≡ X(2π) and, hence, are given by the characteristic equation
det(X(2π)− ρI2) = 0, (6)
where I2 is a 2× 2 identity matrix. Using (2) and (5) it is easy to prove that
detX(t) = z1z
′
2 − z′1z2 ≡ 1.
So characteristic equation (6) can be rewritten as
ρ2 − 2Bρ+ 1 = 0,
where the characteristic constant B is
B = (z1(2π) + z′2(2π))/2. (7)
Thus, the characteristic multipliers ρ1,2 are functions of the single parameter B and are given
by
ρ1,2 = B ±
√
B2 − 1. (8)
As soon as the parameter B has been found, the characteristic exponents µ1,2 are easily
defined from the relationship
ρ1,2 = exp (µ1,2 T ) = exp (2π µ1,2). (9)
It is evident that the next relations are true:
ρ1ρ2 = 1, ρ1 + ρ2 = 2B. (10)
As a consequence, we may write the next relationships for the characteristic exponents µ1,2
µ1 + µ2 = 0, cosh(µ1T ) = cosh(2πµ1) = B. (11)
Thus, to determine ρ1,2 and µ1,2 we should find two linear independent solutions z1(t) and
z2(t) of equation (2) satisfying the initial conditions (5). Although these solutions are not found
yet, we can characterize properties of ρ1,2 and µ1,2 in terms of the characteristic constant B.
a) If B > 1, then, according to (8), the characteristic multipliers ρ1,2 are both real and
positive (ρ1 > 1 > ρ2 > 0). Consequently (see (9) – (11)), µ1 =
ln ρ1
2π
is real and positive
too, while µ2 = −µ1 is real and negative. From the theory of linear differential equations
with periodic coefficients [1,7] we can deduce that the general solution of equation (2), in this
case, is
z = C1 exp (µ1t)f1(t) + C2 exp (−µ1t)f2(t), (12)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
DETERMINATION OF THE BOUNDARIES BETWEEN THE DOMAINS OF STABILITY . . . 45
where f1(t) and f2(t) are periodic functions with period T = 2π. The solution (12) is not
periodic and, in general, |z| → ∞ as t → ∞. So, in the case of B > 1, the trivial solution of
equation (2) is unstable.
b) If B < −1 then the characteristic multipliers ρ1,2 are both real and negative (ρ2 < −1 <
< ρ1 < 0) and again there is one characteristic exponent with positive real part
(
µ2 = β −
−iπ/T = β − i/2 and β = Reµ2 =
ln |ρ2|
2π
> 0
)
. Now, the general solution of equation (2) is
z = C1 exp (−βt)g1(t) + C2 exp (βt)g2(t), (13)
where g1(t) and g2(t) are periodic functions with period 2T = 4π. The solution (13) is not
periodic and, in general, |z| → ∞ as t → ∞. So, in the case of B < −1, the trivial solution of
equation (2) is also unstable.
c) If |B| < 1, then ρ1,2 are both complex-valued with unit magnitude (i. e. |ρ1,2| = 1) and
can be represented as
ρ1,2 = exp (±2πσi). (14)
Consequently, the characteristic exponents µ1,2 are both imaginary, i. e., µ1,2 = ± iσ, where σ
is a real number. The corresponding solution of equation (2) is
z = C1Re (exp (iσt)f(t)) + C2Im (exp (iσt)f(t)), (15)
where f(t) is a complex-valued periodic function with period T = 2π. Solution (15) is bounded
and oscillatory, and so the trivial solution of (2) is stable.
d) In the case of B = 1 there is a single characteristic multiplier ρ1 = 1 and a single
characteristic exponent µ1 = 0. It can be regarded as the limit µ1 → 0 in case a) or σ → 0
in case c). According to (12), (15), there should exist at least one periodic solution of equation
(2) with period T = 2π. But, in general, there may exist an additional solution growing linearly
with t → ∞.
e) In case of B = −1 there is again a single characteristic multiplier ρ1 = −1 and a single
characteristic exponent µ1 = iπ/T = i/2. It can be regarded as the limit µ1 → i/2 in case b) or
σ → 1/2 in case c). Consequently, there exists a periodic solution of equation (2) with period
2T = 4π. In addition there may exist a solution growing linearly with t → ∞.
Thus, the inequality |B| < 1 is a necessary and sufficient condition for stability of the tri-
vial solution of equation (2). The equations B = ±1 determine boundaries of the domains of
instability in the a− e plane.
Calculation of the characteristic constant with the method of a small parameter. The functi-
on p(t) =
a+ e cos t
1 + e cos t
may be represented as a series expansion in powers of e
p(t) =
∞∑
k=0
pk(t)ek, (16)
where
p0 = a, pk(t) = (a− 1)(− cos t)k, k = 1, 2, 3, . . . .
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
46 E.A. GREBENIKOV, A.N. PROKOPENYA
The series (16) converges for any t in the domain |e| < 1 and pk(t) are continuous functions.
So, according to Poincare – Liapunov theorem [8 – 10], the general solution of equation (2) may
be also represented as the power series
z(t) =
∞∑
k=0
zk(t)ek (17)
that converges in the domain |e| < 1 for any t and zk(t) are continuous functions.
To find differential equations determining the functions zk(t), let us substitute expansions
(16), (17) into equation (2). Then, equating the coefficients of ek, k = 0, 1, 2, . . . , in the left
and right-hand sides of the equation we obtain the following system of differential equations:
z′′0 + az0 = 0, z′′k + azk = (1− a)
k∑
n=1
(− cos t)n zk−n, k = 1, 2, . . . . (18)
Two linearly independent solutions z0 of the first equation in (18) must satisfy the initial condi-
tions (5). The corresponding functions are easily found from (3) and are given by
z0(t) = cos(
√
a t), z0(t) =
1√
a
sin(
√
a t). (19)
Initial conditions for the functions zk, k = 1, 2, . . . , may be written then as
zk(0) = z′k(0) = 0. (20)
Solving the second equation in (18) with initial conditions (20) we find the recurrence relati-
onship for calculating the functions zk, k = 1, 2, . . . ,
zk =
1− a√
a
t∫
0
sin(
√
a (t− τ))
k∑
n=1
(− cos τ)n zk−n dτ. (21)
Realizing the calculations according to (21) with initial functions z0 given in (19) we find the
parameter B as a power series in e
B = cos(2π
√
a)− 3(a− 1)π
√
a
2(4a− 1)
sin(2π
√
a) e2 −
− 3(a− 1)π
√
a
32(4a− 1)3
(12
√
a (1− 5a+ 4a2) π cos(2π
√
a) +
+ (14− 55a+ 140a2) sin(2π
√
a)) e4 +
3(a− 1)π
√
a
128(4a− 9)(4a− 1)5
(−6
√
a (−126 +
+ 1181a− 4739a2 + 10164a3 − 8720a4 + 2240a5) π cos(2π
√
a) + (462 + 1536a6π2 +
+ 47a2(455 + 48π2)− 64a5(385 + 114π2)− a(3457 + 216π2)− 12a3(4725 + 674π2) +
+ 16a4(4795 + 738π2) sin(2π
√
a)) e6, (22)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
DETERMINATION OF THE BOUNDARIES BETWEEN THE DOMAINS OF STABILITY . . . 47
where the error term is O(e7). It should be noticed that series (22) converges in the domain
|e| < 1 for any a. Calculating the parameter B we can easily find the characteristic multipliers
ρ1,2 according to (8).
Domains of instability of the trivial solution. It follows from (22) that for e = 0, when
equation (2) reduces to a simple harmonic oscillator equation, the characteristic constant is
B = cos(2π
√
a). This means that |B| < 1 for all values of a, except for
a =
k2
4
, k = 0, 1, 2, . . . , (23)
and there is stability of equation (2). The points (23) also correspond to stable behavior of
the trivial solution of (2) because there are two linearly independent oscillatory solutions (3)
in every such point. But, in these points, we have B = ±1 and so they can belong to the
boundaries between the domains of stability and instability.
Since B = B(a, e) is an analytic function of a and e, we may expect that, if an instability
occurs for e > 0, it must do so in a vicinity of points (23). And the boundaries of the domains
of instability on the a−e plane are some curves a = a(e) crossing the e = 0 axis at these points.
So, for sufficiently small e we can represent the stability boundaries a = a(e) as a power series
in e,
a =
k2
4
+ a1e+ a2e
2 + . . . , k = 0, 1, 2 . . . . (24)
To find the coefficients an, let us substitute (24) into (22) and expand the characteristic constant
B in powers of e. Then the coefficients an can be determined from the equationsB = 1 andB =
= −1. Equating the coefficients of en, n = 1, 2, 3, . . . , to zero, we obtain a system of algebraic
equations giving two curves crossing the e = 0 axis in every point a = (2k − 1)2/4, k =
= 1, 2, 3, . . . . They are
a =
1
4
∓ 3
8
e+
15
128
e2 ∓ 45
2048
e3 +
885
32768
e4 ∓ 6105
524288
e5 +
220305
16777216
e6,
a =
9
4
− 135
256
e2 ∓ 45
2048
e3 − 34695
262144
e4 ∓ 23895
2097152
e5 − 8975205
134217728
e6, (25)
a =
25
4
− 525
256
e2 − 141225
262144
e4 ∓ 525
2097152
e5 − 37465575
134217728
e6, . . . ,
where the error term is O(e7). It is easy to see from (25) that, as the number k increases, the
curves, in a point where they intersect the axis e = 0, differ in a term that has a higher order
of e. This means that the width of the domain between these curves becomes smaller for larger
numbers k. At the same time two corresponding curves crossing the e = 0 axis in every point
a = k2, k = 0, 1, 2, . . . , coincide giving one curve of the form
a = 0, a = 1, a = 4− 6
5
e2 − 39
125
e4 − 7023
43750
e6,
(26)
a = 9− 108
35
e2 − 139617
171500
e4 − 177754233
420175000
e6, . . . .
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
48 E.A. GREBENIKOV, A.N. PROKOPENYA
So we may conclude that the zones of instability are only between the curves crossing the e = 0
axis in the points a = (2k − 1)2/4, k = 1, 2, 3, . . . ), and the width of these zones decreases as
the number k increases. This conclusion coincides with the results obtained in [11].
Of course, we may suppose that this conclusion is true only with an accuracy up to e6,
and the distinction of curves crossing the e = 0 axis in the points a = k2, k = 0, 1, 2, . . . ,
will appear in higher orders of e. To check this hypothesis we can calculate the characteristic
constant B numerically for some fixed value of e and different a and compare it with the result
obtained in (22). It should be emphasized that with the system Mathematica we can do numeri-
cal calculations with an arbitrary precision. And our calculations, which were made with an
accuracy of 25 digits, confirmed that the domains where |B| > 1 really exist only near the
points a =
(2k − 1)2
4
, k = 1, 2, . . . . We have found that for every fixed e the characteristic
constant B, as function of a, oscillates and becomes less than −1 only in a vicinity of these
points. At the same time the inequality B ≤ 1 is true for any a and the curve B = B(a) only
touches the line B = 1 near the points a = k2, k = 0, 1, 2, . . . . Positions of the corresponding
points coincide with the results obtained in (26). So, if we show that there are two bounded
linearly independent solutions of equation (2) in every point of the a−e plane belonging to the
curves (26) then we’ll prove that these curves belong to the domain of stability of equation (2).
Infinite determinant method for stability analysis. There is another possibility to prove the
conclusion obtained above. According to the general theory [1, 7, 12], the boundaries between
the domains of stability and instability in the a − e plane are some curves a = a(e) which are
characterized by the presence of periodic solutions with the periods 2π and 4π . Hence, we can
attempt to determine these boundaries directly by seeking a solution of equation (2) in the
form
z = c0 +
∞∑
k=1
(
ck cos
(
k
2
t
)
+ dk sin
(
k
2
t
))
. (27)
Although this is a Fourier series for the function z = z(t) of period 4π, it can also be used
to obtain a solution with period 2π by setting to zero the Fourier coefficients corresponding
to k being an odd integer. On substituting (27) into equation (2) and setting the coefficients
of cos
(k
2
t
)
and sin
(k
2
t
)
to zero we obtain the following infinite sequence of equations for
determining the coefficients of the Fourier series (27):
a c0 = 0,
ec0 + (a− 1)c2 −
3
2
ec4 = 0,
(28)
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
−k(k − 2)
2
ec2k−2 + (a− k2)c2k −
k(k + 2)
2
ec2k+2 = 0;
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
DETERMINATION OF THE BOUNDARIES BETWEEN THE DOMAINS OF STABILITY . . . 49(
a− 1
4
+
3
8
e
)
c1 −
5
8
ec3 = 0,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (29)
−(2k − 5)(2k − 1)
8
ec2k−3 +
(
a−
(
k − 1
2
)2
)
c2k−1 −
(2k − 1)(2k + 3)
8
ec2k+1 = 0;
(a− 1) d2 −
3
2
ed4 = 0,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (30)
−k(k − 2)
2
ed2k−2 + (a− k2)d2k −
k(k + 2)
2
ed2k+2 = 0;
(
a− 1
4
− 3
8
e
)
d1 −
5
8
e d3 = 0,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (31)
−(2k − 5)(2k − 1)
8
e d2k−3 +
(
a−
(
k − 1
2
)2
)
d2k−1 −
(2k − 1)(2k + 3)
8
e d2k+1 = 0.
It can be seen that in fact there are four infinite subsequences of linear homogeneous
equations (28) – (31). Two of these (28) and (30) are for the coefficients c0, c2, . . . , c2k and
d2, . . . , d2k, respectively, and represent solution (27) with period 2π. For a solution to exist,
the corresponding determinants of infinite systems (28), (30) must vanish, thus determining the
stability boundaries in the a − e plane. These boundaries obviously reduce to a = k2, k =
= 0, 1, 2, . . . when e → 0. The remaining two subsequences of equations (29) and (31) are
for c1, c3, . . . , c2k+1 and d1, . . . , d2k+1 , and correspond to those stability boundaries which
reduce to a =
(2k − 1)2
4
, k = 1, 2, . . . , when e → 0.
Of course, it’s impossible to calculate the determinant of an infinite matrix. So to find the
stability boundaries a = a(e) we should truncate the infinite subsequences of equations (28) –
(31) after the k-th term, where k is a suitably large number. The corresponding determinant of
the system (28), for example, can be written as
Dk =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a 0 0 0 . . . 0
c a− 1 −3
2
e 0 . . . 0
0 0 a− 4 −4e . . . 0
0 0 −3
2
e a− 9 . . . 0
. . . . . . . . . . . . . . . . . .
0 0 0 0 −k(k − 2)
2
e a− k2
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
. (32)
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50 E.A. GREBENIKOV, A.N. PROKOPENYA
Setting determinant (32) to zero we obtain an algebraic equation giving an approximation
for the stability boundary a = a(e). An exact expression for the boundary is obtained when
k → ∞. Determinant (32) is best evaluated from the following recurrence relation
Dk = (a− k2)Dk−1 −
e2
2
(k − 2)(k − 1)k(k + 1)Dk−2, k = 3, 4, . . . , (33)
which is readily established from (32). To start the iterative process we observe that
D1 = a, D2 = a(a− 1).
A similar procedure can be followed for the other systems (29) – (31). For instance, the determi-
nant of system (30) is just the same as (32) with the first row and column deleted. The recurrence
relation is again (33) for k ≥ 3 , but the starting values are now given by
D1 = a− 1, D2 = (a− 1)(a− 4).
The corresponding recurrence relation for the determinants of systems (29), (31) is
Dk =
(
a− (2k − 1)2
4
)
Dk−1 −
e2
64
(2k − 5)(2k − 3)(2k − 1)(2k + 1) Dk−2 (34)
with the starting values
D1 = a− 1
4
± 3e
8
, D2 =
(
a− 1
4
± 3e
8
)(
a− 9
4
)
+
15e2
64
.
It’s evident from (33), (34) that in the case of e = 0 the determinants of systems (28) – (31)
will equal to zero when a =
1
4
k2, k = 0, 1, 2, . . . , and the stability boundaries cross the e = 0
axis in the a − e plane at these points. For sufficiently small e we can represent the stability
boundaries a = a(e) in a vicinity of the points a =
1
4
k2 as a power series (24). It is easy to see
from (33), (34) that to find the curves a = a(e) in vicinity of the point a =
1
4
k2 with accuracy
up to e2n, it is necessary to calculate the determinant Dk+n.
Substituting (24) into (33), (34) and setting the corresponding determinants Dk to zero
we obtain just the same curves (25), (26) that we have obtained with the method of a small
parameter. As the systems (28), (30) determine the same curves (26) crossing the e = 0 axis on
the a− e plane in the points a = k2, k = 0, 1, 2, . . . , there exist two linearly independent peri-
odic solutions of equation (2) in every point of these curves. One solution is an even function
determined in (27) for dk = 0 and another one is an odd function corresponding to the case
of ck = 0. So we can conclude that zones of instability for equation (2) exist only between the
curves (25).
It should be noticed that in the case of the corresponding Mathieu equation of the form
z′′(t) + (a+ e cos t)z(t) = 0,
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DETERMINATION OF THE BOUNDARIES BETWEEN THE DOMAINS OF STABILITY . . . 51
the domains of instability exist near every point a =
k2
4
, k = 0, 1, 2, . . . .
Finally, we can formulate the next theorem.
Theorem. The Hill’s equation of the form
z′′(t) +
a+ e cos t
1 + e cos t
z(t) = 0,
is unstable only if the parameters a and e belong to the domains located between curves (25) on
the a− e plane crossing the e = 0 axis in the points a =
(2k − 1)2
4
, k = 1, 2, 3 . . . .
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Received 12.07.2002
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