Some classes of linear extentions of dynamical systems on a torus

Marek Ryszard Morawiak (Gliwice)

Some classes of linear extentions of dynamical systems on a torus

Abstract This article presents a method of construction of the Lyapunov function for some classes of linear extensions of dynamical systems on a torus. The article is divided into two parts. The first part contains a theoretical introduction including definitions of Green-Samoilenko function or regularity of the system of differen- tial equations. The second part contains the theorem, which allows to determine the regularity of the system. The second part also comprises some examples of the application of the theorem.

2010 Mathematics Subject Classification: 34D99;34D10.

Key words and phrases: dynamical systems; regular systems; differential equations.

1. Introduction Consider the system of differential equations:

dφ

dt = a(φ), dx

dt = A(φ)x, (1)

where φ ∈ R^m, x ∈ Rⁿ, a(φ) is a continuous, 2π-periodic with respect to each variable φ_j, j = 1, m, vector function. In papers [1, 2] it is assumed, that the function a(φ) is determined on m-dimensional torus T_m and writ- ten a(φ) ∈ C(T_m), where C(T_m) is a space of the continuous, 2π-periodic with respect to each variable function. A(φ) is a n × n-dimensional matrix, whose elements are continuous, 2π-periodic with respect to each variable vec- tor functions. For a function a(φ) we additionally assume, that it satisfies the Lipschitz condition. This assumption implies, that the Cauchy problem:

dφ

dt = a(φ), φ |_t=0= φ₀, (2)

has exactly one solution φ(t; φ₀) for every value of φ₀ = (φ₁₀, φ₂₀, · · · , φ_m0).

In this paper we will use the following notation:

• C¹(T_m) means the subspace of C⁰(T_m), that consists of all continuously differentiable functions,

• hx, yi = ^Pn

i=1

xiyi means the inner product in Rⁿ,

• ||x|| =^phx, xi means the norm of x ∈ Rⁿ,

• C⁰(T_m, a) means the subspace of C(Tm) of functions F (φ), such that superposition F (φ(t; φ₀)) is continuously differentiable with respect to t,

• Ω^t_τ(φ) is a fundamental matrix of a linear system of equations ^dx_dt = A(φ(t; φ0))x, normed at t = τ , i.e. Ω^t_τ(φ) |_t=τ= I_n, where I_n is n × n- dimensional unit matrix.

Index 0 in the Cauchy problem is frequently missed φ(t; φ₀) = φ(t; φ).

Let us also consider the system of differential equations:

dφ

dt = a(φ), dx

dt = A(φ)x + h (φ) , (3)

where h (φ) ∈ C (T_m). Now we recall some important definitions.

Definition 1.1 We say, that the system (3) possesses an invariant torus determined by the equality:

x = u (φ) , (4)

if the function u (φ) ∈ C⁰(T_m, a) and also satisfies the identity:

u ≡ A(φ)u + h (φ) ,˙ ∀φ ∈ T_m. (5)

Definition 1.2 System (1) has the Green-Samoilenko function G₀(τ, φ) of the problem of a bounded invariant manifold, if there exists a continu- ous n × n-dimensional matrix C(φ) such that the function of the form:

G₀(τ, φ) =

( Ω⁰_τ(φ) · C(φ_τ(φ)), τ ¬ 0, Ω⁰_τ(φ) · [C(φ_τ(φ)) − I_n], τ > 0, satisfies the estimation:

||G₀(τ, φ)|| ¬ K · exp{−γ|τ |},

where K and γ are positive constants. The function G₀(τ, φ) is said to be the Green-Samoilenko function of the problem of the bounded invariant manifold of the system (1).

Definition 1.3 System (1) is called regular if and only if it has exactly one Green-Samoilenko function of the problem of the bounded invariant manifold of the system (1).

When the Green-Samoilenko function exists, then the invariant torus (6) ex- ists for every function h (φ) ∈ C (T_m). The torus can be defined by the fol- lowing integral formula:

x = u(φ) =

∞

Z

−∞

G₀(τ, φ)h(φ(τ, φ))dτ. (6)

One of the main and most important problems in studying the systems (1) is the problem of finding the conditions of existence of the Green-Samoilenko function. The effective method to determine the existence of the Green- Samoilenko function is the method of the Lyapunov function. It is known [1], that if there exists a quadratic form:

V = hS(φ)x, xi, S(φ) ≡ S^T(φ) ∈ C¹(T_m), (7) which the derivative related to the system (1) is positive definite:

V =˙

*



m

X

j=1

∂S(φ)

∂φ_j a_j(φ) + S(φ)A(φ) + A^T(φ)S(φ)



x, x +

­ γ||x||², (8) γ = const> 0,

with an assumption that det S(φ) 6= 0, ∀φ ∈ T_m, then the system (1) is regular. In the case when det S(φ) = 0, for the value φ = φ₀, the system (1) does not have the Green-Samoilenko function. The quadratic form (7) is often called the Lyapunov function.

2. Main results During studying the problem of regularity of the system (1), we encouter issues which have not been considered in literature. As it was said earlier in the introduction the key role in studies on the regularity of system (1) plays the Lyapunov function of the quadratic form (7). The research on finding Lyapunov function for the system (1) was presented in many papers, for example [4,3], and it was always a very difficult task to find the right one. Therefore, it is proposed to extract some classes of the system (1) for which we can find the Lyapunov function.

Consider the system of differential equations:

dφ

dt = a(φ), dx

dt = [A₀(φ) + A₁(φ)] x, (9) where A₀(φ), A₁(φ) ∈ C(T_m).

Theorem 2.1 Let A(φ) ∈ C(Tm) is a n × n-dimensional matrix, for which the equation:

m

X

j=1

∂L

∂φ_ja_j(φ) + L^hA₁(φ) −A(φ)ⁱ+^hA^T₁(φ) − A^T(φ)ⁱL = 0 (10) has a solution in a form of the symmetric matrix L = S(φ) ∈ C¹(T_m) which also satisfies the inequality:

DnS(φ)^hA₀(φ) − A(φ)ⁱ+^hA^T₀(φ) − A^T(φ)ⁱS(φ)^ox, x^E­ α||x||², (11) where α = const > 0, then the system (9) is regular. The quadratic form, for which the derivative related to the system (9) will be positive definite, has the form V = hS(φ)x, xi.

Proof Let us consider a quadratic form V = hS(φ)x, xi, where S(φ) ∈ C¹(T_m) is a symmetric matrix such, that det S(φ) 6= 0, ∀φ ∈ T_m. The deriva- tive of this quadratic form related to the system (9) is as follows:

V =˙ ^DS(φ)x, x˙ ^E+ hS(φ) ˙x, xi + hS(φ)x, ˙xi

=^DS(φ)x, x˙ ^E+ hS(φ) [A₀(φ) + A₁(φ)] x, xi + hS(φ)x, [A₀(φ) + A₁(φ)] xi

=^DS(φ)x, x˙ ^E+ hS(φ) [A₀(φ) + A₁(φ)] x, xi +^D[A₀(φ) + A₁(φ)]^T S(φ)x, xi

=^DS(φ)x, x˙ ^E+ hS(φ) [A₀(φ) + A₁(φ)] x, xi +^DhA^T₀(φ) + A^T₁(φ)ⁱS(φ)x, x^E

=^DnS(φ) + S(φ) [A˙ 0(φ) + A₁(φ)] +^hA^T₀(φ) + A^T₁(φ)ⁱS(φ)^ox, x^E

=^DnS(φ) + S(φ) [A˙ 0(φ) + A₁(φ)ⁱ+^hA^T₀(φ) + A^T₁(φ)ⁱS(φ) +S(φ)^hA(φ) − A(φ)ⁱ+^hA^T(φ) − A^T(φ)ⁱS(φ)^ox, x^E

=^DnS(φ) + S(φ)˙ ^hA₁(φ) − A(φ)ⁱ+^hA^T₁(φ) − A^T(φ)ⁱS(φ)^ox, x^E +^DnS(φ)^hA₀(φ) + A(φ)ⁱ+^hA^T₀(φ) + A^T(φ)ⁱS(φ)^ox, x^E, where ˙S(φ) =

m

P

j=1

∂S(φ)

∂φj aj(φ). We can note, that the first component of the sum has a form (10), so it is equal to 0. The second component of the sum is positive, according to our assumption. It means, that for this quadratic form the derivative related to the system (9) is positive definite, which means, that

the system (9) is regular. _

The assumption, that det S(φ) 6= 0, ∀φ ∈ T_m is very important, because in the case when det S(φ) = 0, for the value φ₀ ∈ T_m, then there exists x₀ ∈ Rⁿ such, that S(φ₀)x₀ = 0. It implies that:

DnS(φ)^hA0(φ) +A(φ)ⁱ+^hA^T₀(φ) + A^T(φ)ⁱS(φ)^ox, x^E= 0.

This theorem is a very powerful tool in studying the regularity of the system (1).

Example 2.2 Consider the system of differential equations:

dφ

dt = sin φ, dx

dt = (cos 4φ)x. (12)

By transforming cos 4φ to a form 1 + 8 sin⁴φ − 8 sin²φ, the system (12) can be rewritten as:

dφ

dt = sin φ, dx

dt = (1 + 8 sin⁴φ − 8 sin²φ)x. (13) Let A₀(φ) = 1, A₁(φ) = 8 sin⁴φ − 8 sin²φ. It means, that the equation (10) has the form:

dL

dφ sin φ + LA₁(φ) + LA₁(φ) = 0. (14) From the equation (14) we received:

dL

dφsin φ = −2LA₁(φ) = −16 sin⁴φ + 16 sin²φ)
L,

dL

dφ = −16 sin³φ + 16 sin φ)
L,

dL

L = −16 sin³φ + 16 sin φ)
dφ, L = e^{−163 cos3φ}.

Therefore the inequality (11) can be written as:

LA₀(φ) + LA^T₀(φ) = 2e^{−163 cos3φ}­ 2e⁻¹⁶³ > 0. (15) Based on the inequality (15) we find, that the system (12) is regular.

Example 2.3 Consider the system of differential equations:











dφ1

dt = ω₁+ cos φ₁+ 2 cos φ₂+ 3 cos φ₃

dφ2

dt = ω₂+ 2 cos φ₁+ cos φ₂+ 3 cos φ₃,

dφ3

dt = ω₃+ 3 cos φ₁+ 2 cos φ₂+ cos φ₃,

dx

dt = [µ₀(φ₁, φ2, φ3) + µ₁(φ₁, φ2, φ3)] x,

(16)

where µ₀(φ₁, φ2, φ3) ∈ C⁰(T₃), µ₀(φ₁, φ2, φ3) > 0, ω_i = const > 0, i = 1, 2, 3.

µ1(φ₁, φ2, φ3) = λ₁cos φ₁+ λ₂cos φ₂+ λ₃cos φ₃+ 4 cos²φ1+ 5 cos²φ2+ + 6 cos²φ₃+ 7 cos φ₁cos φ₂+ 8 cos φ₁cos φ₃+ 9 cos φ₂cos φ₃.

(17)

For the system (16) we can ask a question: for which values of the parameters ω_i, λ_i will the system (16) be regular?

Let us write a partial differential equation:

ds

dφ₁(ω₁+cos φ₁+2 cos φ₂+3 cos φ₃)+ ds

dφ₂(ω₂+2 cos φ₁+cos φ₂+3 cos φ₃) + + ds

dφ₃(ω₃+3 cos φ₁+2 cos φ₂+cos φ₃) = µ₁(φ₁, φ₂, φ₃)− µ (φ₁, φ₂, φ₃) , (18) where:

µ (φ₁, φ₂, φ₃) = µ₁cos φ₁+ µ₂cos φ₂+ µ₃cos φ₃+ µ₁₁cos²φ₁+ µ₂₂cos²φ₂+ +µ₃₃cos²φ₃+µ₁₂cos φ₁cos φ₂+µ₁₃cos φ₁cos φ₃+µ₂₃cos φ₂cos φ₃. (19) We chose the coefficients µ_i, µij such that the equation (18) has a solution of the form:

s = s0(φ₁, φ2, φ3) = c₁sin φ₁+ c₂sin φ₂+ c₃sin φ₃. (20) By substituting (20) in the equation (18) we obtain:

c₁ω₁ = λ₁− µ₁, c₂ω₂= λ₂− µ₂, c₃ω₃ = λ₃− µ₃, 2c₁+ 2c₂ = 7 − µ₁₂ 3c₁+ 3c₃= 8 − µ₁₃, 3c₂+ 2c₃ = 9 − µ₂₃,

from which we come to the following conclusion: if the function (14) has a form:

µ (φ₁, φ₂, φ₃) = (λ₁− c₁ω₁) cos φ₁+ (λ₂− c₂ω₂) cos φ₂+ (λ₃− c₃ω₃) cos φ₃+ + (4 − c₁) cos²φ1+ (5 − c₂) cos²φ2+ (6 − c₃) cos²φ3+ (7 − 2c₁− 2c₂) ·

· cos φ₁cos φ₂+ (8 − 3c₁− 3c₃) cos φ₁cos φ₃+ (9 − 3c₂− 2c₃) cos φ₂cos φ₃, then the equation (18) will have a desired solution of a form (20)

By choosing:

c_i = λ_i

ω_i, i = 1, 2, 3, (21)

we receive:

µ (φ₁, φ₂, φ₃) = (4 − c₁) cos²φ₁+ (5 − c₂) cos²φ₂+ (6 − c₃) cos²φ₃+ + (7 − 2c₁− 2c₂) cos φ₁cos φ₂+ (8 − 3c₁− 3c₃) cos φ₁cos φ₃

+ (9 − 3c₂− 2c₃) cos φ₂cos φ₃.

(22)

Then by denoting x_i = cos φ_i, i = 1, 2, 3, the previous equation can be written in form (22):

Φ = (4 − c₁) x²₁+ (5 − c₂) x²₂+ (6 − c₃) x²₃+ (7 − 2c₁− 2c₂) x₁x₂+

+ (8 − 3c₁− 3c₃) x₁x₃+ (9 − 3c₂− 2c₃) x₂x₃. (23) We want to choose the constants c_i such, that the quadratic form (23) will be positive definite. Let us write the symmetric matrix of the quadratic form (23):







(4 − c₁) ¹₂(7 − 2c₁− 2c₂) ¹₂(8 − 3c₁− 3c₃)

1

2(7 − 2c₁− 2c₂) (5 − c₂) ¹₂(9 − 3c₂− 2c₃)

1

2(8 − 3c₁− 3c₃) ¹₂(9 − 3c₂− 2c₃) (6 − c₃)







According to Sylvester’s theorem the quadratic form (23) will be positive definite, if it satisfies the following three conditions:

a) (4 − c₁) > 0,

b) (4 − c₁) (5 − c₂) −^7₂ − c₁− c₂^2> 0,

c) (4 −c₁) (5 − c₂) (6 − c₃)+2^7₂ −c₁−c₂^{ 9}₂ −³₂c2− c₃^{ }4 −³₂c1− ³₂c3

+

− (5−c₂)^4−³₂c₁−³₂c₃^2−(6−c₃)^7₂−c₁−c₂^2−(4−c₁)^9₂ −³₂c₂−c₃^2> 0.

It is easy to check, that the conditions a),b) and c) are satisfied with following values:

(c₁, c2, c3) = (1, 1, −1) , (c₁, c2, c3) = (1, 1, 0) , (c₁, c2, c3) = (1, 1, 1), (c₁, c₂, c₃) = (1, 1, 2) , (c₁, c₂, c₃) = (1, 0, 0) , (c₁, c₂, c₃) = (1, 0, 1), (c₁, c₂, c₃) = (1, 1, 1.5) , etc.

It means, that when we substitute in the quadratic form (23) the value (c₁, c2, c3) = (1, 1, 2), then it will be positive definite. It also means, that the function (19) satisfies an inequality:

µ (φ₁, φ₂, φ₃) = 3 cos²φ₁+4 cos²φ₂+4 cos²φ₃+3 cos φ₁cos φ₂−cos φ₁cos φ₃+ +2 cos φ₂cos φ₃ ­ ^cos²φ1+ cos²φ2+ cos²φ3

,  = const > 0.

(24) Now rewrite the fourth equation of the system (16) to the form:

dx

dt = [(µ₀(φ₁, φ2, φ3) + µ (φ₁, φ2, φ3)) + (µ₁(φ₁, φ2, φ3) − µ (φ₁, φ2, φ3))] x If we take a function:

V = x²exp {−2s₀(φ₁, φ₂, φ₃)} ,

where s₀(φ₁, φ₂, φ₃) = sin φ₁ + sin φ₂ + 2 sin φ₃ and calculate its derivative related to the system (16) we receive:

V = 2x ˙˙ x exp {−2s₀(φ₁, φ₂, φ₃)} − 2x²˙s₀(φ₁, φ₂, φ₃) exp {−2s₀(φ₁, φ₂, φ₃)} =

= 2x²(µ₀(φ₁, φ₂, φ₃) + µ (φ₁, φ₂, φ₃)) exp {−2s₀(φ₁, φ₂, φ₃)} +

+2x²[µ₁(φ₁, φ₂, φ₃)−µ (φ₁, φ₂, φ₃) − ˙s₀(φ₁, φ₂, φ₃)] exp{−2s₀(φ₁, φ₂, φ₃)} =

= 2x²(µ₀(φ₁, φ₂, φ₃) + µ (φ₁, φ₂, φ₃)) exp {−2s₀(φ₁, φ₂, φ₃)} .

This implies, that the system (16) will be regular, if the following inequality is satisfied:

µ₀(φ₁, φ₂, φ₃) +µ (φ₁, φ₂, φ₃) > 0.

Remark 2.4 If the coefficients (λ1, λ₂, λ₃) of the function (16) satisfy one of the equalities:

(λ₁, λ2, λ3) = (ω₁, ω2, ω3) , (λ₁, λ2, λ3) = (ω₁, ω2, 2ω3), (λ₁, λ₂, λ₃) = (ω₁, ω₂, −ω₃), (λ₁, λ₂, λ₃) = (ω₁, ω₂, 0), (λ₁, λ₂, λ₃) = (ω₁, 0, ω₃),

and also the function µ₀(φ₁, φ2, φ3) ∈ C⁰(T₃), satisfies the estimation:

µ0(φ₁, φ2, φ3) + cos²φ1+ cos²φ2+ cos²φ3> 0 then the system (16) will be regular.

References

[1] . A. Mitropol~ski$i, A. M. Samo$ilenko, V. L. Kulik, Issle- dovani dihotomii line$inyh sistem differencial~nyh uravneni$i s pomow~ funkci$i Lpunova, “Naukova Dumka”, Kiev, 1990, Title trans.: Studies in the dichotomy of linear systems of differential equations by means of Lyapunov functions. MR 1085029

[2] A. M. Samo$ilenko, lementy matematiqesko$i teorii mno- goqastotnyh kolebani$i. Invariantnye tory, “Nauka”, Moscow, 1987, Title trans.: Elements of mathematical theory of multiphase oscilla- tion. MR 928806

[3] N.V. Stepanenko, Some properties of a set of Lyapunov functions in the theory of linear extensions of dynamical systems on a torus., Nonlinear Oscil. 4 (2001), no. 4, 539–546, Zbl 1070.34072.

[4] Ewa Tkocz-Piszczek, Sign-changing Lyapunov functions in regularity of linear extensions of dynamical systems on a torus., Opusc. Math. 28 (2008), no. 1, 93–101.

O pewnych klasach liniowych rozszerzeń systemów dynamicznych na torusie

Marek Ryszard Morawiak

Streszczenie W pracy przedstawiono metodę konstrukcji funkcji Lapunowa dla pewnych klas liniowych rozszerzeń układów dynamicznych na torusie. Pierwsza część artykułu zawiera wstęp teoretyczny, w którym przedstawione zostały m.in. definicje funkcji Greena-Samojlenki oraz regularności układu równań różniczkowych. W dru- giej części udowodniono twierdzenie, które umożliwia ustalenie regularności układu poprzez konstrukcję funkcji Lapunowa. Przedstawione zostały także przykłady, które pokazują jak wielkie możliwości daje to twierdzenie przy badaniu regularności ukła- dów równań różniczkowych.

2010 Klasyfikacja tematyczna AMS (2010): 34D99;34D10.

Słowa kluczowe: systemy dynamiczne ^• systemy regularne ^• równania różniczkowe

• torus.

Marek Ryszard Morawiak is the PhD student of mathe- matics at Silesian University of Technology in Gliwice. He graduated from mathematics with distinction at Silesian University of Technology in 2015. His main academic in- terests are: differential equations and the qualitative the- ory of differential equations. In the free times he likes to travel and spend time actively.

Marek Ryszard Morawiak Silesian University of Technology Faculty of Applied Mathematics

ul. Kaszubska 23, 44-100 Gliwice, Poland E-mail: Marek.Morawiak@polsl.pl

Communicated by: Jerzy Klamka

(Received: 1st of March 2017; revised: 3rd of October 2017)