Perturbation theorems for a multifrequency system with pulses
The problem of preservation of a piecewise continuous invariant toroidal set for a class of multifrequency systems with impulses at nonfixed moments under perturbations of the right-hand side is considered. New theorems set constraints on perturbation terms not in the whole phase space, but only in...
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irk-123456789-1771552021-02-12T01:25:58Z Perturbation theorems for a multifrequency system with pulses Feketa, P.V. Perestyuk, Y.M. The problem of preservation of a piecewise continuous invariant toroidal set for a class of multifrequency systems with impulses at nonfixed moments under perturbations of the right-hand side is considered. New theorems set constraints on perturbation terms not in the whole phase space, but only in a nonwandering set of dynamical system, to guarantee the existence of exponentially stable invariant toroidal set. Розглянуто задачу збереження кусково-неперервної iнварiантної тороїдальної множини для деякого класу багаточастотних систем з iмпульсами в нефiксованi моменти часу та зi збуренням у правiй частинi. Новi теореми, що задають обмеження на члени збурення не на всьому фазовому просторi, а лише на неблукаючiй множинi динамiчної системи, встановлюють iснування експоненцiально стiйкої iнварiантної тороїдальної множини. 2015 Article Perturbation theorems for a multifrequency system with pulses / P.V. Feketa, Y.M. Perestyuk // Нелінійні коливання. — 2015. — Т. 18, № 2. — С. 280-289 — Бібліогр.: 17 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177155 517.9 en Нелінійні коливання Інститут математики НАН України |
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The problem of preservation of a piecewise continuous invariant toroidal set for a class of multifrequency systems with impulses at nonfixed moments under perturbations of the right-hand side is considered. New theorems set constraints on perturbation terms not in the whole phase space, but only in a nonwandering set of dynamical system, to guarantee the existence of exponentially stable invariant toroidal set. |
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Feketa, P.V. Perestyuk, Y.M. |
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Feketa, P.V. Perestyuk, Y.M. Perturbation theorems for a multifrequency system with pulses Нелінійні коливання |
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Feketa, P.V. Perestyuk, Y.M. |
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Feketa, P.V. |
| title |
Perturbation theorems for a multifrequency system with pulses |
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Perturbation theorems for a multifrequency system with pulses |
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Perturbation theorems for a multifrequency system with pulses |
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Perturbation theorems for a multifrequency system with pulses |
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Perturbation theorems for a multifrequency system with pulses |
| title_sort |
perturbation theorems for a multifrequency system with pulses |
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Інститут математики НАН України |
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2015 |
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Perturbation theorems for a multifrequency system with pulses / P.V. Feketa, Y.M. Perestyuk // Нелінійні коливання. — 2015. — Т. 18, № 2. — С. 280-289 — Бібліогр.: 17 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT feketapv perturbationtheoremsforamultifrequencysystemwithpulses AT perestyukym perturbationtheoremsforamultifrequencysystemwithpulses |
| first_indexed |
2025-07-15T15:11:09Z |
| last_indexed |
2025-07-15T15:11:09Z |
| _version_ |
1837726192684236800 |
| fulltext |
UDC 517.9
PERTURBATION THEOREMS
FOR A MULTIFREQUENCY SYSTEM WITH IMPULSES
ТЕОРЕМИ ПРО ЗБУРЕННЯ
ДЛЯ БАГАТОЧАСТОТНИХ IМПУЛЬСНИХ СИСТЕМ
P. Feketa
Univ. Appl. Sci. Erfurt
Altonaer Str. 25, Erfurt, 99085, Germany
e-mail: petro.feketa@fh-erfurt.de
Yu. Perestyuk
Kyiv Nat. Taras Shevchenko Univ.
Volodymyrska Str. 64, Kyiv, 01601, Ukraine
e-mail: yuriy.perestyuk@gmail.com
The problem of preservation of a piecewise continuous invariant toroidal set for a class of multifrequency
systems with impulses at nonfixed moments under perturbations of the right-hand side is considered. New
theorems set constraints on perturbation terms not in the whole phase space, but only in a nonwandering
set of dynamical system, to guarantee the existence of exponentially stable invariant toroidal set.
Розглянуто задачу збереження кусково-неперервної iнварiантної тороїдальної множини для де-
якого класу багаточастотних систем з iмпульсами в нефiксованi моменти часу та зi збуренням
у правiй частинi. Новi теореми, що задають обмеження на члени збурення не на всьому фазо-
вому просторi, а лише на неблукаючiй множинi динамiчної системи, встановлюють iснування
експоненцiально стiйкої iнварiантної тороїдальної множини.
1. Introduction. There are several mathematical frameworks to model processes that combine
continuous and discontinuous behavior simultaneously. Among them we would like to emphasi-
ze dynamic equation on time scales [1] and hybrid dynamical systems [3, 4, 16]. However
throughout the paper we will use the framework of impulsive differential equations proposed
in [12]. This framework was first adapted to problems of qualitative analysis of multifrequency
oscillations [10] and is most convenient to consider such problems (see [5, 11] and references
therein for details). It also has a wide application to control theory and a variety of stability-
related problems [2].
In recent years a considerable attention is paid to relaxing the conditions for preservation of
invariant toroidal manifolds of multifrequency systems under perturbations of the right-hand
side [8, 9]. In this paper we will develop new theorems for preservation of the invariant toroidal
set of multifrequency systems with impulses at nonfixed moments. The main object of investi-
gation is a systems of differential equation defined in the direct product of an n-dimensional
torus Tm and an m-dimensional Euclidean space Rn
dϕ
dt
= a(ϕ),
c© P. Feketa, Yu. Perestyuk, 2015
280 ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 2
PERTURBATION THEOREMS FOR A MULTIFREQUENCY SYSTEM . . . 281
dx
dt
= A(ϕ)x+ f(ϕ), ϕ ∈ Tm \ Γ, (1)
∆x|ϕ∈Γ = B(ϕ)x+ g(ϕ),
where ϕ ∈ Tm, x ∈ Rn, A,B ∈ C(Tm), f, g ∈ CΓ(Tm), a ∈ CLip(Tm), C(Tm) (CΓ(Tm)) is
a space of continuous (piecewise continuous with first-kind discontinuities in Γ) functions that
are 2π-periodic with respect to each of the variables ϕj , j = 1, . . . ,m, Γ is a smooth compact
submanifold of the torus Tm of codimension 1.
To the best of our knowledge the problem of existence of invariant sets for such systems
was first considered by Perestyuk in [10] for the case a(ϕ) = ω ≡ const. Later these results
were extended by Tkachenko to a more general case [13]. In [14, 15] a concept of exponenti-
al dichotomy was considered and a problem of its preservation under small perturbation was
investigated. A problem of existence of invariant sets and their smoothness properties were
treated in [13 – 15] as well. Sufficient conditions for existence of exponential dichotomy for
such class of systems were developed in [17].
In this paper we consider a narrower class of systems of type (1) in the case where the
system has an exponentially stable invariant toroidal set. We prove theorems on preservation
of an exponentially stable invariant set under perturbations of the right-hand side which are
less restrictive than in [15]. We use the techniques proposed in [8] for multifrequency systems
without impulses and extend the results of [6, 7]. The rest of the paper is organized as follows.
In Section 2 we give a short introduction to the invariant tori problem for systems defined in
the direct product of a torus and an Euclidean space. The main results are stated in Section 3.
An example and a short discussion completes the paper.
2. Systems defined in Tm × Rn. In this section we recall the basic approach to consider a
problem of existence of invariant sets for systems (1) based on [6, 13]. Let ϕt(ϕ) be a solution of
the first equation from (1) that satisfies the initial condition ϕ0(ϕ) = ϕ. A Lipschitz condition
for the function a(ϕ) guarantees the existence and uniqueness of such a solution.
We assume that the set Γ is a smooth submanifold of a torus of codimension 1 and is defi-
ned be the equation Φ(ϕ) = 0, where Φ is a continuous scalar function. Denote by ti(ϕ) the
solutions of equation Φ(ϕt(ϕ)) = 0 that are the moments of impulsive perturbations in system
(1). Assume that there exists θ > 0 such that
ti(ϕ)− ti−1(ϕ) > θ. (2)
Along with system (1), consider a linear system,
dx
dt
= A(ϕt(ϕ))x+ f(ϕt(ϕ)), ϕ ∈ Tm \ Γ,
(3)
∆x|ϕ∈Γ = B(ϕt(ϕ))x+ g(ϕt(ϕ))
that depends on ϕ ∈ T m as a parameter. We get system (3) by substituting ϕ with ϕt(ϕ) in the
second and the third equations of (1).
Definition 1. By an invariant toroidal manifold of system (1) we call a manifold that is defined
by a function u(ϕ) ∈ CΓ(Tm) such that the function x(t, ϕ) = u(ϕt(ϕ)) is a solution of system
(3) for every ϕ ∈ T m.
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 2
282 P. FEKETA, YU. PERESTYUK
Suppose C(ϕ) ∈ CΓ(Tm) is an arbitrary square matrix and Xt
τ (ϕ) is a fundamental matrix
of the linear system
dx
dt
= A(ϕt(ϕ))x, ϕ ∈ Tm \ Γ,
(4)
∆x|ϕ∈Γ = B(ϕt(ϕ))x
that depends on ϕ as a parameter.
Definition 2. The function
G0(τ, ϕ) =
{
X0
τ (ϕ)C(ϕτ (ϕ)), τ ≤ 0,
−X0
τ (ϕ)(E − C(ϕτ (ϕ))), τ > 0,
is called a Green – Samoilenko function of the invariant tori problem for system (1) if the follo-
wing inequality holds:
+∞∫
−∞
‖G0(τ, ϕ)‖ dτ ≤ K < ∞.
The existence of a Green – Samoilenko function along with (2) guarantees existence of an
invariant toroidal set of system (1) of the form
x = u(ϕ) =
+∞∫
−∞
G0(τ, ϕ)f(ϕτ (ϕ))dτ +
∑
−∞<ti(ϕ)<∞
G0(ti(ϕ) + 0, ϕ)g(ϕti(ϕ)(ϕ)), ϕ ∈ Tm,
for arbitrary f, g ∈ CΓ(Tm).
Throughout this paper we will consider a special case where a fundamental matrix Xt
τ (ϕ) of
system (4) satisfies the estimate∥∥Xt
τ (ϕ)
∥∥ ≤ Ke−γ(t−τ) for t ≥ τ. (5)
From (5) it directly follows that a Green – Samoilenko function exists and has the form
G0(τ, ϕ) =
{
X0
τ (ϕ)C(ϕτ (ϕ)), τ ≤ 0,
0, τ > 0.
(6)
An invariant toroidal set then is called asymptotically stable as it is stable and attracts all
trajectories from a vicinity.
3. Main results. Along with system (1), we consider a perturbed system,
dϕ
dt
= a(ϕ),
dx
dt
= [A(ϕ) +A1(ϕ)]x+ f(ϕ), ϕ ∈ Tm \ Γ, (7)
∆x|ϕ∈Γ = [B(ϕ) +B1(ϕ)]x+ g(ϕ),
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 2
PERTURBATION THEOREMS FOR A MULTIFREQUENCY SYSTEM . . . 283
where perturbation terms A1, B1 ∈ C(Tm). Since the function A1 and B1 are continuous on
compact manifold there exist supϕ∈Tm A1(ϕ) = a1 and supϕ∈Tm B1(ϕ) = b1. In [15] a more
general problem was considered for a system that possesses an invariant toroidal set (without
exponential stability property) and with a perturbation term in the first equation of (7). It was
proven that the perturbation terms should be sufficiently small to guarantee the existence of an
invariant toroidal set of the perturbed system. In this paper we consider the case where system
(1) has an exponentially stable invariant toroidal set and develop theorems with less restrictive
constraints.
Theorem 1. Let the fundamental matrix Xt
τ (ϕ) satisfy estimate (5)∥∥Xt
τ (ϕ)
∥∥ ≤ Ke−γ(t−τ) for t ≥ τ
with some K ≥ 1, γ > 0. If
Ka1 +
1
θ
ln (1 +Kb1) < γ, (8)
then system (7) has an exponentially stable invariant toroidal manifold for arbitrary f, g ∈
∈ CΓ(Tm).
Proof. The fundamental matrix of the perturbed system can be represented as
Ωt
0(ϕ) = Xt
0(ϕ) +
t∫
0
Xt
s(ϕ)A1(ϕs(ϕ))Ωs
0(ϕ)ds+
∑
0≤ti(ϕ)<t)
Xt
ti(ϕ)(ϕ)B1(ϕti(ϕ)(ϕ))Ω
ti(ϕ)
0 (ϕ).
Taking estimate (5) into account we have
∥∥Ωt
0(ϕ)
∥∥ ≤ Ke−γt +
t∫
0
Ke−γ(t−s)a1 ‖Ωs
0(ϕ)‖ds+
∑
0≤ti(ϕ)<t)
Ke−γ(t−ti(ϕ))b1
∥∥∥Ω
ti(ϕ)
0 (ϕ)
∥∥∥,
eγt
∥∥Ωt
0(ϕ)
∥∥ ≤ K +
t∫
0
Keγsa1 ‖Ωs
0(ϕ)‖ds+
∑
0≤ti(ϕ)<t)
Keγti(ϕ)b1
∥∥∥Ω
ti(ϕ)
0 (ϕ)
∥∥∥.
Utilizing a Gronwall – Bellmann type inequality for piecewise continuous functions [12] (Lem-
ma 2) we get
eγt
∥∥Ωt
0(ϕ)
∥∥ ≤ K(1 +Kb1)i(0,t)eKa1t,
where i(a, b) is the number of impulsive perturbation in the interval (a, b). Finally, from (2),∥∥Ωt
0(ϕ)
∥∥ ≤ Ke−(γ−Ka1− 1
θ
ln (1+Kb1))t for t > 0.
From condition (8) it directly follows that the fundamental matrix of the perturbed system
satisfies an estimate of type (5) with the same constant K and possibly different γ̃ = γ−Ka1−
−1
θ
ln (1 +Kb1). It means that there exists a Green – Samoilenko function of the form (6) and
an exponentially stable invariant toroidal set for arbitrary functions f, g ∈ CΓ(Tm).
Theorem 1 is proved.
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 2
284 P. FEKETA, YU. PERESTYUK
Next we will relax the dwell-time condition (8). In particular we will show that it is sufficient
to set restrictions on perturbations not on the whole surface of the torus Tm, but only in a
nonwandering set of trajectories of the dynamical system
dϕ
dt
= a(ϕ).
Definition 3. A point ϕ is called wandering if there exist its neighbourhood U(ϕ) and a posi-
tive number T > 0 such that
U(ϕ) ∩ ϕt(U(ϕ)) = 0 for t ≥ T. (9)
Let W be a set of all wandering points of a dynamical system and Ω = Tm \ W be a set
of nonwandering points. From compactness of a torus it follows that the set Ω is nonempty and
compact. Since the functionA1 andB1 are continuous on a compact set there exist supϕ∈ΩA1(ϕ) =
= ã1 and supϕ∈ΩB1(ϕ) = b̃1.
Theorem 2. Let the fundamental matrix Xt
τ (ϕ) satisfy estimate (5),∥∥Xt
τ (ϕ)
∥∥ ≤ Ke−γ(t−τ) for t ≥ τ
with some K ≥ 1, γ > 0. If the following dwell-time condition holds
Kã1 +
1
θ
ln (1 +Kb̃1) < γ, (10)
then system (7) has an exponentially stable invariant toroidal set for arbitrary f, g ∈ CΓ(Tm).
Proof. The fundamental matrix of the perturbed system can be represented as
Ωt
0(ϕ) = Xt
0(ϕ) +
t∫
0
Xt
s(ϕ)A1(ϕs(ϕ))Ωs
0(ϕ)ds+
∑
0≤ti(ϕ)<t)
Xt
ti(ϕ)(ϕ)B1(ϕti(ϕ)(ϕ))Ω
ti(ϕ)
0 (ϕ).
Taking estimate (5) into account we have
∥∥Ωt
0(ϕ)
∥∥ ≤ Ke−γt +
∫ t
0
Ke−γ(t−s)‖A1(ϕs(ϕ))‖‖Ωs
0(ϕ)‖ds+
+
∑
0≤ti(ϕ)<t
Ke−γ(t−ti(ϕ))‖B1(ϕti(ϕ)(ϕ))‖‖Ωti(ϕ)
0 (ϕ)‖,
(11)
eγt
∥∥Ωt
0(ϕ)
∥∥ ≤ K +
t∫
0
Keγs‖A1(ϕs(ϕ))‖‖Ωs
0(ϕ)‖ds+
+
∑
0≤ti(ϕ)<t
Keγti(ϕ)‖B1(ϕti(ϕ)(ϕ))‖‖Ωti(ϕ)
0 (ϕ)‖.
Now we will employ an approach used in [8]. Let Uε(Ω) be an ε-neighbourhood of the set Ω.
We will show that for any fixed ε > 0 there exists a finite time T > 0 that does not depend on
ϕ such that ϕt(ϕ) ∈ Uε(Ω) for t > T.
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 2
PERTURBATION THEOREMS FOR A MULTIFREQUENCY SYSTEM . . . 285
Indeed, since Tm is a compact set and Uε(Ω) is an open set, the set Tm\Uε(Ω) is compact and
consists of wandering points. Thus, for every pointϕ ∈ Tm\Uε(Ω), there exists a neighbourhood
U(ϕ) satisfying condition (9) for t ≤ T (ϕ). Since the phase space is compact, we can choose
finitely many neighbourhoods of this kind, U1, U2, . . . , UN , such that⋃
k=1,...,n
Uk = Tm \ Uε(Ω)
and denote the corresponding numbers T (ϕ) by T1, T2, . . . , TN .
Let ϕ ∈ Tm \ Uε(Ω) be an arbitrary point from the neighbourhood Un1 . According to (9),
for a period of time not larger than Tn1 , it leaves this neighbourhood forever. Assume that it
then appears in the neighbourhood Un2 and leaves it for a time that does not exceed Tn2 , etc.
Finally, for a time not greater than
∑N
k=1 Tk, the point necessarily appears in Uε(Ω) because,
according to (9), it cannot return to any of the neighbourhoods Uk, k = 1, . . . , N.
Thus the time of stay of the point ϕ ∈ Tm \ Uε(Ω) is limited to
T =
N∑
k=1
Tk.
Since the matrices A1, B1 ∈ C(Tm), for any fixed εa, εb > 0 there exist a positive constant
ε > 0 and a finite time T that does not depend on ϕ such that, for every ϕ ∈ Tm \ Uε(Ω),
‖A1(ϕt(ϕ))‖ ≤ ã+ εa, ‖B1(ϕt(ϕ))‖ ≤ b̃+ εb for t ≥ T.
Then from (11) we get
eγt‖Ωt
0(ϕ)‖ ≤ K +
T∫
0
Keγs‖A1(ϕs(ϕ))‖‖Ωs
0(ϕ)‖ds+
+
∑
0≤ti(ϕ)<T
Keγti(ϕ)‖B1(ϕti(ϕ)(ϕ))‖‖Ωti(ϕ)
0 (ϕ)‖+
t∫
T
Keγs(ã1 + εa)‖Ωs
0(ϕ)‖ds+
+
∑
T≤ti(ϕ)<t
Keγti(ϕ)(b̃1 + εb)‖Ω
ti(ϕ)
0 (ϕ)‖. (12)
Estimating
K +
T∫
0
Keγs ‖A1(ϕs(ϕ))‖ ‖Ωs
0(ϕ)‖ds+
∑
0≤ti(ϕ)<T
Keγti(ϕ)
∥∥B1(ϕti(ϕ)(ϕ))
∥∥∥∥∥Ω
ti(ϕ)
0 (ϕ)
∥∥∥ ≤ K̃
we have
eγt
∥∥Ωt
0(ϕ)
∥∥ ≤ K̃ +
t∫
T
Keγs(ã1 + εa) ‖Ωs
0(ϕ)‖ds+
∑
T≤ti(ϕ)<t
Keγti(ϕ)(b̃1 + εa)
∥∥∥Ω
ti(ϕ)
0 (ϕ)
∥∥∥ ≤
ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 2
286 P. FEKETA, YU. PERESTYUK
≤ K̃ +
t∫
0
Keγs(ã1 + εa)‖Ωs
0(ϕ)‖ds+
∑
0≤ti(ϕ)<t
Keγti(ϕ)(b̃1 + εa)‖Ωti(ϕ)
0 (ϕ)‖.
Utilizing a Gronwall – Bellmann type inequality for a piecewise continuous function [12] ([Lem-
ma 2) we obtain
eγt
∥∥Ωt
0(ϕ)
∥∥ ≤ K̃(1 +K(b̃1 + εb))
i(0,t)eK(ã1+εb)t,∥∥Ωt
0(ϕ)
∥∥ ≤ K̃e−(γ−K(ã1+εa)− 1
θ
ln (1+K(b̃1+εb)))t for t > 0, ϕ ∈ Tm \ Uε(Ω).
Now consider the case where ϕ ∈ Uε(Ω). It means that for every initial value ϕ ∈ Uε(Ω)
there exists a constant T1(ϕ) such that
‖A1(ϕt(ϕ))‖ ≤ ã+ εa, ‖B1(ϕt(ϕ))‖ ≤ b̃+ εb for t ∈ [0, T1(ϕ)] ∪ [T1(ϕ) + T,+∞). (13)
Remark 1. If ϕ ∈ Ω it means that the trajectory never leaves the nonwandering set of the
dynamical system and estimates (13) are valid for any t ≥ 0. The same situation can happen
when ϕ ∈ Uε(Ω), but the trajectory ϕt(ϕ) never leaves the neighbourhood Uε(Ω). However
next we will treat the worst case, where the trajectory that starts in Uε(Ω) leaves it in a time
T1(ϕ) that depends on ϕ.
Then from (11), considering sufficiently large t > T1(ϕ) + T and utilizing a Gronwall-
Bellmann type inequality for piecewise continuous functions [12](Lemma 2), we get
eγt
∥∥Ωt
0(ϕ)
∥∥ ≤ Ke
∫ t
0 K‖A1(ϕs(ϕ))‖ds
∏
0<ti(ϕ)<t
(
1 +K
∥∥B1(ϕti(ϕ)(ϕ))
∥∥) ≤
≤ KeK(ã1+εa)te
∫ T1(ϕ)+T
T1(ϕ)
K‖A1(ϕs(ϕ))‖ds ∏
0≤ti(ϕ)<t
(
1 +K(b̃1 + εb)
)
×
×
∏
T1(ϕ)≤ti(ϕ)<T1(ϕ)+T
(
1 +K
∥∥B1(ϕti(ϕ)(ϕ))
∥∥) ≤ K̄eK(ã1+εa)te
1
θ
ln (1+K(b̃1+εb))t,
where the constant K̄ does not depend on ϕ. Indeed,
Ke
∫ T1(ϕ)+T
T1(ϕ)
K‖A1(ϕs(ϕ))‖ds ∏
T1(ϕ)≤ti(ϕ)<T1(ϕ)+T
(
1 +K
∥∥B1(ϕti(ϕ)(ϕ))
∥∥) ≤
≤ Ke
∫ T1(ϕ)+T
T1(ϕ)
Ka1ds
∏
T1(ϕ)≤ti(ϕ)<T1(ϕ)+T
(1 +Kb1) ≤ KeKa1T (1 +Kb1)
1
θ = K̄.
Then the estimate for a fundamental matrix has the form∥∥Ωt
0(ϕ)
∥∥ ≤ K̄e−(γ−K(ã1+εa)− 1
θ
ln (1+K(b̃1+εb)))t for t > 0, ϕ ∈ Uε(Ω).
Finally, denoting K̂ = max{K̃, K̄} we arrive at the estimate∥∥Ωt
0(ϕ)
∥∥ ≤ K̂e−(γ−K(ã1+εa)− 1
θ
ln (1+K(b̃1+εb)))t for t > 0, ϕ ∈ Tm.
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PERTURBATION THEOREMS FOR A MULTIFREQUENCY SYSTEM . . . 287
If the dwell-time condition (10) holds then the fundamental matrix of the perturbed system
satisfies the estimate of a type (5) but with possibly different constants K and γ since we can
choose εa and εb to be arbitrarily small. It means that there exists a Green – Samoilenko functi-
on of the form (6) and an exponentially stable invariant toroidal set for arbitrary functions
f, g ∈ CΓ(Tm).
Theorem 2 is proved.
4. Example and discussion. Consider an example that shows the advantages of the proposed
theorems.
Example.
dϕ1
dt
= − sin2 ϕ1
2
,
dϕ2
dt
= ω, ϕ ∈ T2,
(14)
dx
dt
= −x+ f(ϕ), ϕ ∈ T2 \ Γ, ∆x|ϕ∈Γ = g(ϕ),
where ω is a constant. System (14) has an invariant toroidal manifold and the fundamental
matrix Xt
τ (ϕ) satisfy the estimate (5) with constants K = γ ≡ 1
‖Xt
τ (ϕ)‖ ≤ e−(t−τ) for t ≥ τ.
Now perturb system (14) to get
dϕ1
dt
= − sin2 ϕ1
2
,
dϕ2
dt
= ω, ϕ ∈ T2,
dx
dt
= (−1 +A sinϕ1)x+ f(ϕ), ϕ ∈ T2 \ Γ, (15)
∆x|ϕ∈Γ = B sinϕ1 · x+ g(ϕ),
where A and B are arbitrary constants. Suppose that the set Γ is such that the estimate (2) for
the moments of impulsive perturbation holds. The question is does the perturbed system (15)
has an exponentially stable invariant set for arbitrary functions f, g ∈ CΓ(T2)?
The following estimates for the perturbation terms hold:
sup
ϕ∈T2
A sinϕ1 = A, sup
ϕ∈T2
B sinϕ1 = B.
The previously known perturbation theorem for a more general class of systems [15] demands
the norms of the perturbations to be not more than some particular value. However by adjusting
constants A and B one could make it bigger than any fixed δ. So we cannot conclude about the
existence of invariant set of the perturbed system (15). However the unperturbed system (14)
possesses an exponentially stable invariant toroidal set. So we can try to use Theorem 1 or
Theorem 2.
A dwell-time conditions (8) has the form
A+
1
θ
ln (1 +Kb) < 1.
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288 P. FEKETA, YU. PERESTYUK
It is obvious that by adjusting the constants A and B one could make it false. It means that
Theorem 1 couldn’t answer in the affirmative to the example’s question.
However a dynamical system on a two-dimensional torus has a very simple structure of the
limit sets and recurrent trajectories. In particular the nonwandering set Ω consists of only one
meridian ϕ1 = 0,
Ω = {ϕ ∈ T2 : ϕ1 = 0, ϕ2 ∈ T1}.
A point that starts on the meridian is spinning with a constant speed, all other trajectories tend
to Ω by spirals. The estimates for the perturbation terms are
sup
ϕ∈Ω
A sinϕ1 = 0, sup
ϕ∈Ω
B sinϕ1 = 0.
Then the dwell-time condition has the form 0 +
1
θ
ln 1 < 1. It is obvious that for every
1
θ
< ∞
there exist sufficiently small constants εa, εb > 0 such that the dwell-time condition εa + εb < 1
holds. From Theorem 2 it follows that system (15) has an exponentially stable invariant toroidal
set for arbitrary f, g ∈ CΓ(T2) if only the time sequence of impulsive moments is such that
estimate (2) holds.
Proved theorems allow to investigate the qualitative behavior of solutions of a class of
impulsive systems that have a simple structure of limit sets and recurrent trajectories. The
constraints of Theorem 2 are less restrictive than those of Theorem 1. A perturbed system
should satisfy the dwell-time condition not for every ϕ ∈ Tm, but only for ϕ ∈ Ω. However it is
worth to note that if the first equation of system (7) is ϕ̇ = ω = const, that is very frequent in
applications, then its nonwandering set Ω coincides with a whole torus and Theorem 2 has no
advantages compared to Theorem 1.
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Received 16.12.14
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