Columbia International Publishing
Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 149-156
doi:10.7726/jams.2016.1012
Research Article (Invited)
Necessary and Sufficient Condition for Existence of a
Unique Invariant Torus of Singular Impulsive System
Yurii Korol1*
Received: 13 August 2016; Accepted: 1 October 2016;
© The author(s) 2016. Published with open access at www.uscip.us
Abstract
Under the assumption that linear nonhomogeneous singular system of differential equation with impulse
action defined on the direct product of torus and Euclidean space can be reduced to the central canonical
form and the corresponding homogeneous system is exponential dichotomous on the semiaxes, we obtained
necessary and sufficient condition for existence of a unique invariant torus of singular linear system.
Keywords: Exponential Dichotomy; Invariant Torus; Bounded Solution; Central Canonical Form; Impulsive
System
1. Introduction
Some problems in control theory, radio physics, mathematical economics, and linear programming
are known to be modeled by systems of differential equations with a singular matrix multiplying
the derivative
𝑑𝑥
𝐵(𝑡) = 𝐴(𝑡)𝑥 + 𝑓(𝑡).
𝑑𝑡
Such systems are usually referred to as singular systems of ordinary differential equations (see
Boyarincev et al., 1989; Samoilenko et al., 2000). The notion of central canonical form introduced in
Campbell and Petzold(1983) plays an important role in the theory of such systems. In this paper
we consider the problem of existence a unique invariant torus of singular impulsive system. Similar
problem for a system of equations defined on a direct product of an 𝑚-dimentional torus 𝐓 𝑚 and
the Euclidean space were investigated in Samoilenko(1987). Using these results let us find the
necessary and sufficient condition for existence of a unique invariant torus under the assumption
that singular system of differential equation can be reduced to the central canonical form.
Let us consider system of impulse differential equation in central canonical form
______________________________________________________________________________________________________________________________
* Corresponding e-mail: [email protected]
1 Uzhhorod National University, Uzhhorod, Ukraine
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Yurii Korol / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 149-156
𝑑𝜑
𝑀(𝜑) 0
𝐸
0 𝑑𝑥
]
] 𝑥 + 𝑓(𝜑),
= 𝑎(𝜑), [ 𝑛−𝑠
=[
(1)
0
𝐸𝑠
0
𝐼 𝑑𝑡
𝑑𝑡
Δ𝑥1 |𝜑∈Γ = 𝐵(𝜑)𝑥1 + 𝐼(𝜑),
(2)
𝑛
where 𝑥 ∈ ℝ , 𝑥 = 𝑐𝑜𝑙(𝑥1 , 𝑥2 ), where 𝑥1 (𝑡, 𝜑), 𝑥2 (𝑡, 𝜑) are (𝑛 − 𝑠) and 𝑠 − dimentional
vector
functions respectively, 𝜑 ∈ 𝐓 𝑚 , 𝐓 𝑚 is 𝑚 -dimentional torus, 𝑓(𝜑), 𝐼(𝜑)are continuous(piecewise
continuous with first kind discontinuities in the set Γ) 2𝜋-periodic with respect to each component
𝜑𝑣 , 𝑣 = 1, . . . , 𝑚 and bounded for all 𝜑 ∈ 𝐓 𝑚 vector functions, 𝑓2 (𝜑) ∈ ℂ𝑠−1 (𝐓 𝑚 ); 𝑀(𝜑), 𝐵(𝜑) are
continuous 2𝜋-periodic with respect to each component 𝜑𝑣 square matrices, function 𝑎(𝜑) is
bounded and satisfies Lipschitz condition for 𝜑 ∈ 𝐓 𝑚 , 𝐸𝑛−𝑠 , 𝐸𝑠 are identity matrix of order (𝑛 − 𝑠)
and 𝑠 respectively, 𝐼 is quasidiagonal matrix, 𝑑𝑒𝑡(𝐸𝑛−𝑠 + 𝐵(𝜑)) ≠ 0 for all 𝜑 ∈ 𝐓 𝑚 . We assume that
the set Γ is a subset of the torus 𝐓 𝑚 which is a manifold of dimension 𝑚 − 1 defined by the equation
Φ(𝜑) = 0 for some continuous scalar 2𝜋-periodic with respect to each component 𝜑𝑣 , 𝑣 = 1, . . . , 𝑚,
function.
Denote by 𝑡𝑖 (𝜑) the solutions of equation Φ(𝜑𝑡 (𝜑)) = 0 that are moments of impulsive action in
system (1),(2). Assume also that exist constant 𝜃 > 0 such that
𝑡𝑖 (𝜑) − 𝑡𝑖−1 (𝜑) ≥ 𝜃,
(3)
for all 𝑖 ∈ ℤ, 𝜑 ∈ 𝐓 𝑚 . System (1),(2) can be split on two independent systems
𝑑𝜑
𝑑𝑥1
= 𝑎(𝜑),
= 𝑀(𝜑)𝑥1 + 𝑓1 (𝜑), Δ𝑥1 |𝜑∈Γ = 𝐵(𝜑)𝑥1 + 𝐼(𝜑),
(4)
𝑑𝑡
𝑑𝑡
𝑑𝜑
𝑑𝑥2
= 𝑎(𝜑), 𝐼
= 𝑥2 + 𝑓2 (𝜑).
(5)
𝑑𝑡
𝑑𝑡
The general solution of the system (4) has the form
𝑡
𝑥1 (𝑡, 𝜑) = 𝑋(𝑡, 𝜑)𝜉 + ∫ ‍𝑋(𝑡, 𝜑)𝑋 −1 (𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
0
+
‍𝑋(𝑡, 𝜑)𝑋 −1 (𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)),
∑
(6)
0<𝑡𝑖 (𝜑)<𝑡
where 𝑋(𝑡, 𝜑) is fundamental matrix of homogeneous impulsive system
𝑑𝑥1
= 𝑀(𝜑𝑡 (𝜑))𝑥1 ,
𝑑𝑡
Δ𝑥1 |𝜑∈Γ = 𝐵(𝜑𝑡𝑖 (𝜑) (𝜑))𝑥1 .
(7)
Solution of the system (5) we can write in the form (see Boyarincev et al., 1989)
𝑠−1
∂𝑘 𝑓2 (𝜑𝑡 (𝜑))
𝑖
𝑖
𝑎 1 (𝜑). . . 𝑎𝑚𝑚 (𝜑).
𝑖𝑚 1
𝑖1
∂𝜑
.
.
.
∂𝜑
𝑚
1
𝑘=0 𝑖1 +...+𝑖𝑚 =𝑘
By combining (6) and (8), we obtain the general solution of (1),(2) in form
𝑥2 (𝑡, 𝜑) = 𝑟(𝜑𝑡 (𝜑)) = − ∑ ‍
∑
‍𝐼 𝑘
(8)
𝑡
𝑥(𝑡, 𝜑) =
𝑋(𝑡, 𝜑)𝜉 + ∫ 𝑋(𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
0
∑
𝑋(𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
,
0<𝑡𝑖 (𝜑)<𝑡
𝑟(𝜑𝑡 (𝜑))
(
−1
where 𝑋(𝑡, 𝑠, 𝜑) = 𝑋(𝑡, 𝜑)𝑋 (𝑠, 𝜑) and 𝜉 is (𝑛 − 𝑠)-dimentional constant vector.
(9)
)
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Yurii Korol / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 149-156
2. Bounded Solution on Whole Axis
Let us find the conditions of existance bounded on whole axis solutions of system (1),(2) in
assumption that linear homogeneous system (7) is exponential dichotomous in semiaxes 𝑅+ and 𝑅−
with projectors 𝐶+ (𝜑) and 𝐶− (𝜑) (𝐶±2 (𝜑) = 𝐶± (𝜑)) respectively. It means that following
inequalities are true:
‖𝛺 + (𝑡, 𝑠, 𝜑)‖ =∥ 𝑋(𝑡, 𝜑)𝐶+ (𝜑)𝑋 −1 (𝑠, 𝜑) ∥≤ 𝐾1 exp−𝛼1 (𝑡−𝑠) , 𝑡 ≥ 𝑠,
‖𝛺̃ + (𝑡, 𝑠, 𝜑)‖ =∥ 𝑋(𝑡, 𝜑)(𝐼 − 𝐶+ (𝜑))𝑋 −1 (𝑠, 𝜑) ∥≤ 𝐾1 exp−𝛼1 (𝑠−𝑡) , 𝑠 ≥ 𝑡, 𝑡, 𝑠 ∈ ℝ+ ,
(10)
‖𝛺 − (𝑡, 𝑠, 𝜑)‖ =∥ 𝑋(𝑡, 𝜑)𝐶− (𝜑)𝑋 −1 (𝑠, 𝜑) ∥≤ 𝐾2 exp−𝛼2 (𝑡−𝑠) , 𝑡 ≥ 𝑠,
‖𝛺̃ − (𝑡, 𝑠, 𝜑)‖ =∥ 𝑋(𝑡, 𝜑)(𝐼 − 𝐶− (𝜑))𝑋 −1 (𝑠, 𝜑) ∥≤ 𝐾2 exp−𝛼2 (𝑠−𝑡) , 𝑠 ≥ 𝑡, 𝑡, 𝑠 ∈ ℝ− .
The general solution of system(1),(2) that is bounded on semiaxes ℝ+ and ℝ− has the form
𝑥(𝑡, 𝜑, 𝜉) =
𝑡
𝛺
+ (𝑡,
0, 𝜑)𝜉 + ∫ 𝛺+ (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
0<𝑡𝑖 (𝜑)≤𝑡
0
∞
− ∫ ‍𝛺̃+ (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 −
, 𝑡 ≥ 0,
𝛺̃+ (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
∑
𝑡≤𝑡𝑖 (𝜑)<∞
𝑡
𝑟(𝜑𝑡 (𝜑))
[
=
𝛺 + (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) −
∑
]
𝑡
𝛺̃− (𝑡, 0, 𝜑)𝜉 + ∫ 𝛺 − (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
∑
𝛺− (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) −
−∞<𝑡𝑖 (𝜑)≤𝑡
−∞
0
− ∫ ‍𝛺̃− (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 −
, 𝑡 ≤ 0.
𝛺̃ − (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
∑
𝑡≤𝑡𝑖 (𝜑)<0
𝑡
𝑟(𝜑𝑡 (𝜑))
{[
]
(11)
Solution (11) will be bounded on whole axis if and only if impulse action is fixed when 𝑡 = 0:
𝑥(0+, 𝜑, 𝜉) − 𝑥(0−, 𝜑, 𝜉) = 𝐼(𝜑).
(12)
It means that vector constant 𝜉 = 𝜉(𝜑) ∈ ℝ𝑛−𝑠 satisfy algebraic system which is received from (11)
when 𝑡 = 0:
∞
0
𝐷(𝜑)𝜉 = 𝐼(𝜑) + ∫ ‍𝛺
− (0,
𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 + ∫ ‍𝛺̃+ (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
−∞
+
∑
−∞<𝑡𝑖 (𝜑)≤0
‍𝛺
− (0,
𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) +
(13)
0
∑
̃ + (0,
‍𝛺
𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) ,
0≤𝑡𝑖 (𝜑)<∞
where 𝐷(𝜑) = 𝐶+ (𝜑) − (𝐼 − 𝐶− (𝜑)) is (𝑛 − 𝑠) × (𝑛 − 𝑠)-dimentional matrix. This algebraic system
has solution if and only if the condition of orthogonality is true:
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Yurii Korol / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 149-156
∞
0
𝑃𝐷𝛵 (𝜑) { ∫ ‍𝛺
− (0,
𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 + ∫ ‍𝛺̃ + (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 + 𝐼(𝜑) +
−∞
+
0
(14)
‍𝛺− (0, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) +
∑
−∞<𝑡𝑖 (𝜑)≤0
∑
‍𝛺̃ + (0, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))} ≡ 0,
0≤𝑡𝑖 (𝜑)<∞
where 𝑃𝐷𝛵 (𝜑) is (𝑛 − 𝑠) × (𝑛 − 𝑠)-dimentional matrix which is ortoprojector to matrix 𝐷𝛵 (𝜑).
Then the bounded on ℝ general solution of system (1),(2) has the form (11) where constant
𝜉 = 𝜉(𝜑) ∈ ℝ𝑛−𝑠 can be determine from the system (13) in the following way:
𝜉 = 𝑃𝐷 (𝜑)𝑐 + 𝐷 + (𝜑)Θ
where 𝐷+ (𝜑) is Moore-Penrose pseudoinverse matrix (see Boichuk and Samoilenko, 2004) to
matrix 𝐷(𝜑) and
∞
0
Θ = { ∫ ‍𝛺 − (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 + ∫ ‍𝛺̃+ (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 + 𝐼(𝜑) +
−∞
+
∑
0
‍𝛺− (0, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) +
−∞<𝑡𝑖 (𝜑)≤0
∑
‍𝛺̃ + (0, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))}.
0≤𝑡𝑖 (𝜑)<∞
By substituting constant 𝜉 into (11) we obtain the bounded on whole axis solution of (1),(2) in the
form
𝑥(𝑡, 𝜑, 𝜉) =
𝛺+ (𝑡, 0, 𝜑)𝑃𝐷 (𝜑)𝑐 + 𝛺+ (𝑡, 0, 𝜑)𝐷+ (𝜑)Θ +
𝑡
+ ∫ ‍𝛺+ (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
0
∑
0<𝑡𝑖 (𝜑)≤𝑡
∞
− ∫ ‍𝛺̃+ (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 −
∑
, 𝑡 ≥ 0,
‍𝛺̃+ (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
𝑡≤𝑡𝑖 (𝜑)<∞
𝑡
𝑟(𝜑𝑡 (𝜑))
𝛺̃− (𝑡, 0, 𝜑)𝑃𝐷 (𝜑)𝑐 + 𝛺̃− (𝑡, 0, 𝜑)𝐷+ (𝜑)Θ +
[
=
‍𝛺+ (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) −
]
(15)
𝑡
+ ∫ ‍𝛺− (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
∑
‍𝛺− (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) −
−∞<𝑡𝑖 (𝜑)≤𝑡
−∞
, 𝑡 ≤ 0.
0
− ∫ ‍𝛺̃− (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 −
𝑡
{[
∑
‍𝛺̃ − (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
𝑡≤𝑡𝑖 (𝜑)<0
𝑟(𝜑𝑡 (𝜑))
]
Since 𝑃𝐷𝛵 (𝜑)𝐷(𝜑) = 0 then 𝑃𝐷𝛵 (𝜑)𝐶− (𝜑) = 𝑃𝐷𝛵 (𝜑)(𝐼 − 𝐶+ (𝜑)). In this case necessary and
sufficient condition for existence solution of system (1),(2) has the form
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Yurii Korol / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 149-156
+∞
𝑃𝐷𝛵 (𝜑) { ∫ ‍𝛺 − (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
‍𝛺− (0, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) + 𝐼(𝜑)} ≡ 0. (16)
∑
−∞<𝑡𝑖 (𝜑)<∞
−∞
Taking into account equality
and condition (14) we obtain
[𝐶+ (𝜑) − (𝐼 − 𝐶− (𝜑))]𝐷+ (𝜑) = 𝐼 − 𝑃𝐷𝛵 (𝜑),
𝐶+ (𝜑)𝐷+ (𝜑)Θ − 𝐼Θ = (𝐼 − 𝐶− (𝜑))𝐷+ (𝜑)Θ.
Further, since 𝐷(𝜑)𝑃𝐷 (𝜑) = [𝐶+ (𝜑) − (𝐼 − 𝐶− (𝜑))]𝑃𝐷 (𝜑) = 0, we obtain
𝐶+ (𝜑)𝑃𝐷 (𝜑) = (𝐼 − 𝐶− (𝜑)𝑃𝐷 (𝜑)).
Let us consider the case when the homogeneous system (7) don`t have nontrivial bounded on
whole axis solutions. It means
𝐶+ (𝜑)𝑃𝐷 (𝜑) = (𝐼 − 𝐶− (𝜑))𝑃𝐷 (𝜑) = 0.
Then we can rewrite (15) in the form
𝑥(𝑡, 𝜑, 𝜉) =
𝑋(𝑡, 𝜑)𝐶+ (𝜑)𝐷 + (𝜑)Θ +
𝑡
+ ∫ ‍𝛺+ (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
0
∑
0<𝑡𝑖 (𝜑)≤𝑡
∞
− ∫ ‍𝛺̃+ (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 −
∑
, 𝑡 ≥ 0,
‍𝛺̃+ (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
𝑡≤𝑡𝑖 (𝜑)<∞
𝑡
𝑟(𝜑𝑡 (𝜑))
𝑋(𝑡, 𝜑)(𝐶+ (𝜑)𝐷 + (𝜑) − 𝐼)Θ +
[
=
‍𝛺+ (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) −
]
(17)
𝑡
+ ∫ ‍𝛺− (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
∑
‍𝛺− (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) −
−∞<𝑡𝑖 (𝜑)≤𝑡
−∞
, 𝑡 ≤ 0.
0
− ∫ ‍𝛺̃− (𝑡, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 −
∑
‍𝛺̃ − (𝑡, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
𝑡≤𝑡𝑖 (𝜑)<0
𝑡
𝑟(𝜑𝑡 (𝜑))
{[
]
3. Conditions of Existence of Invariant Torus
Let us show that expression
∞
𝑥(0, 𝜑) = 𝑢(𝜑) = [
∫ ‍𝐺0 (𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
−∞
∑
‍𝐺0 (𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))
−∞<𝑡𝑖 (𝜑)<∞
],
(18)
𝑟(𝜑)
where
𝐺0 (𝑠, 𝜑) = {
𝐶+ (𝜑)𝐷 + (𝜑)𝛺− (0, 𝑠, 𝜑), 𝑠 < 0,
(𝐶+ (𝜑)𝐷+ (𝜑) − 𝐼)𝛺̃+ (0, 𝑠, 𝜑), 𝑠 ≥ 0,
(19)
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(2016) Vol. 3 No. 3 pp. 149-156
which is obtained from (17) when 𝑡 = 0 defines for all 𝜑 ∈ 𝒯𝑚 invariant torus of system (1),(2). As
shown if condition (16) is fullfilled then system (1),(2) has bounded on ℝ solution of the form (17)
at fixed 𝜑 ∈ 𝒯𝑚 . Let us show that the condition (16) on solutions 𝜑𝑡 (𝜑) of corresponding Cauchy
problem defines invariant set. Using notations presented in paper Korol(2016) we have for all
𝑡 ∈ ℝ and 𝜑 ∈ 𝒯𝑚
+∞
𝑃𝐷𝛵 (𝜑𝑡 (𝜑)) { ∫ ‍𝛺− (0, 𝑠, 𝜑𝑡 (𝜑))𝑓1 (𝜑𝑠 (𝜑𝑡 (𝜑)))𝑑𝑠 +
−∞
+
∑
‍𝛺− (0, 𝑡𝑖 (𝜑), 𝜑𝑡 (𝜑))𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑𝑡 (𝜑))) + 𝐼(𝜑𝑡 (𝜑))} =
−∞<𝑡𝑖 (𝜑)<∞
+∞
= 𝑋(𝑡, 𝜑)𝑃𝐷𝛵 (𝜑) { ∫ ‍𝛺 − (0, 𝑠 + 𝑡, 𝜑)𝑓1 (𝜑𝑠+𝑡 (𝜑))𝑑𝑠 +
−∞
+
∑
‍𝛺 − (0, 𝑡𝑖 (𝜑) + 𝑡, 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑)+𝑡 (𝜑)) + 𝐼(𝜑𝑡 (𝜑))} = 0.
−∞<𝑡𝑖 (𝜑)<∞
Due to exponential dichotomy of system (7) on semiaxes and estimation (3) integral and sum are
convergent. We will use the boundness of matrix-projector ∥ 𝑃𝐷𝛵 (𝜑𝑡 (𝜑)) ∥≤ 𝐾0 for all 𝜑 ∈ 𝒯𝑚 .
+∞
‖𝑃𝐷𝛵 (𝜑) { ∫ ‍𝛺 − (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
‍𝛺 − (0, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑)) + 𝐼(𝜑)}‖ ≤
∑
−∞<𝑡𝑖 (φ)<∞
−∞
0
≤ ‖𝑃𝐷𝛵 (𝜑)‖ { ∫ ‍‖𝛺− (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠‖ +
‍‖𝛺− (0, 𝑡𝑖 (𝜑), 𝜑)𝐼(𝜑𝑡𝑖 (𝜑) (𝜑))‖ +
∑
−∞<𝑡𝑖 (𝜑)<0
−∞
+∞
+ ∫ ‍‖𝛺̃+ (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠‖ +
∑
‍‖𝛺̃ + (0, 𝑡𝑖 (𝜑), 𝜑)𝐼(𝜑𝑡𝑖 (𝜑) (𝜑))‖ + ‖𝐼(𝜑)‖} ≤
0<𝑡𝑖 (𝜑)<∞
0
𝐾2 𝐾1
𝐾2
𝐾1
≤ 𝐾0 ( + ) ‖𝑓1 (𝜑)‖ + 𝐾0 (
+
+ 1) ‖𝐼(𝜑)‖.
−𝛼
𝜃
𝛼2 𝛼1
1−𝑒 2
1 − 𝑒 −𝛼1 𝜃
Now we will show that if condition (16) fullfills then expression (18) defines invariant torus of
system (1),(2) for all 𝜑 ∈ 𝒯𝑚 . Let us show that the integral and sum in (18) are convergent. Taking
into account inequality (10) we consider the first (𝑛 − 𝑠) rows of expression (18):
0
‖𝐶+ (𝜑)𝐷 + (𝜑) { ∫ ‍𝛺 − (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
−∞
∞
+[𝐶+ (𝜑)𝐷+ (𝜑) − 𝐼 ] {∫ ‍𝛺̃+ (0, 𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠
0
‍𝛺− (0, 𝑡𝑖 (𝜑), 𝜑)𝐼 (𝜑𝑡𝑖 (𝜑) (𝜑))} +
∑
−∞<𝑡𝑖 (𝜑)<0
+
∑
‍𝛺̃ + (0, 𝑡𝑖 (𝜑), 𝜑)𝐼(𝜑𝑡𝑖 (𝜑) (𝜑))}‖ ≤
0<𝑡𝑖 (𝜑)<∞
𝐾5 𝐾2 𝐾6 𝐾1
𝐾5 𝐾2
𝐾6 𝐾1
] ‖𝑓1 ‖ + (
) ‖𝐼(𝜑)‖,
≤[
+
+
−𝛼
𝜃
2
𝛼2
𝛼1
1−𝑒
1 − 𝑒 −𝛼1 𝜃
where ‖𝐶+ (𝜑)𝐷+ (𝜑)‖ = 𝐾5 , ‖𝐶+ (𝜑)𝐷 + (𝜑) − 𝐼‖ = 𝐾6 and therefore as in Samoilenko at al.,(2000),
from condition 𝑓1 (𝜑) ∈ 𝐶(𝒯𝑚 ) it follows that 𝑢(𝜑𝑡 (𝜑)) ∈ 𝐶(𝒯𝑚 ). Belonging the other 𝑠 rows of
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(2016) Vol. 3 No. 3 pp. 149-156
expression (18) to space 𝐶(𝒯𝑚 ) follows from the fact that 𝑓2 (𝜑) ∈ 𝐶 𝑠−1 (𝒯𝑚 ). From estimates (10)
and well-known properties (see Samoilenko,1987)
𝐶+ (𝜑𝜏 (𝜑)) = 𝑋(𝜏, 𝜑)𝐶+ (𝜑)𝑋 −1 (𝜏, 𝜑), 𝐶− (𝜑𝜏 (𝜑)) = 𝑋(𝜏, 𝜑)𝐶− (𝜑)𝑋 −1 (𝜏, 𝜑),
𝑋(𝑡, 𝜏, 𝜑𝑠 (𝜑)) = 𝑋(𝑡 + 𝑠, 𝜏 + 𝑠, 𝜑), 𝑋(𝑡, 𝜏, 𝜑)𝑋(𝜏, 𝑠, 𝜑) = 𝑋(𝑡, 𝑠, 𝜑),
−1
(𝑋(𝑡, 𝜏, 𝜑))
we obtain
𝜑𝑡 (𝜑𝑠 (𝜑)) = 𝜑𝑡+𝑠 (𝜑),
= 𝑋(𝜏, 𝑡, 𝜑),
∞
𝑢(𝜑𝑡 (𝜑)) = [
∫ ‍𝐺0 (𝑠, 𝜑𝑡 (𝜑))𝑓1 (𝜑𝑠 (𝜑𝑡 (𝜑)))𝑑𝑠 +
∑
−∞<𝑡𝑖 (𝜑)<∞
−∞
∫ ‍𝐺𝑡 (𝑠, 𝜑)𝑓1 (𝜑𝑠 (𝜑))𝑑𝑠 +
−∞
]=
𝑟(𝜑𝑡 (𝜑))
∞
=[
‍𝐺0 (𝑡𝑖 (𝜑), 𝜑𝑡 (𝜑))𝐼(𝜑𝑡𝑖 (𝜑) (𝜑𝑡 (𝜑)))
∑
−∞<𝑡𝑖 (𝜑)<∞
‍𝐺𝑡 (𝑡𝑖 (𝜑), 𝜑)𝐼(𝜑𝑡𝑖 (𝜑) (𝜑))
],
𝑟(𝜑𝑡 (𝜑))
for all 𝑡 ∈ ℝ and 𝜑 ∈ 𝒯𝑚 . It proves that 𝑢(𝜑𝑡 (𝜑)) ∈ 𝐶 1 (𝒯𝑚 ) and set 𝑢(𝜑) defines invariant torus of
system (1),(2). Thus we proved the following result.
Theorem 1. Let the system (7) is exponential dichotomous on semiaxes 𝑅+ and 𝑅− with projectors
𝐶± (𝜑), the condition (3) is executed and 𝑓2 (𝜑) ∈ ℂ𝑠−1 (𝐓 𝑚 ). Then
1. system (1),(2) has invariant torus if and only if 𝑓1 (𝜑) ∈ 𝐶(𝒯𝑚 ) satisfy the condition (16);
2. moreover if the homogeneous system (7) don`t have nontrivial solutions bounded on whole
axis then expression (18) defines for all 𝜑 ∈ 𝒯𝑚 a unique invariant torus of system (1),(2).
4. Conclusions
In this paper we showed that if the linear nonhomogeneous singular system can be reduce to the
central canonical form and the corresponding homogeneous system is exponential dichotomous on
semiaxes then this system has a bounded on whole axis solution. The main result was presented in
theorem 1 where we found the necessary and sufficient condition for existence of a unique
invariant torus of singular impulsive system.
References
Boichuk A.A. (2007) A criterion for the existence of the unique invariant torus of a linear extension of
dynamical systems. Ukrainian Mathematical Journal - 2007 - 59(1). - p.1-11
http://dx.doi.org/10.1007/s11253-007-0001-8
Boichuk A.A., Samoilenko A.M. (2004) Generalized inverse operators and fredholm boundary value problems.
- Utrecht; Boston: VSP, 2004. - 317p.
http://dx.doi.org/10.1515/9783110944679
Boyarincev Yu.E., Danilov V.A., Loginov A.A., Chistyakov V.F. (1989) Numerical methods for solving singular
systems. – Novosibirsk: Nauka. 1989. – 223p.
Campbell S.L., Petzold L.R. (1983) Canonical forms and solvable singular systems of differential equations.
SIAM J. Alg. Discrete Methods. – 1983(4). – p.517-521.
http://dx.doi.org/10.1137/0604051
155
Yurii Korol / Journal of Applied Mathematics and Statistics
(2016) Vol. 3 No. 3 pp. 149-156
Korol Yu. Yu. (2016) Existence of an invariant torus for a degenerate linear extension of a dynamical system
Nonlinear Oscillations. - 2016. - 19(2), p.217-226
Samoilenko A.M. (1987) Elements of mathematical theory of multi-frequency oscillations. Invariant tori. – M:
Nauka, 1987. – 304 p.
Samoilenko A.M., Perestyuk N.A. (1987) Differential equations with impulsive actions. – K.: Vyshcha shkola,
1987. – 288 p.
Samoilenko A.M., Shkil M.I., Yakovets V.P. (2000) Linear systems of differential equations with singularities. –
K.: Vyshcha shkola, 2000. – 294p.
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