ISSN 1028�3358, Doklady Physics, 2010, Vol. 55, No. 9, pp. 471–473. © Pleiades Publishing, Ltd., 2010.Original Russian Text © G.A. Leonov, N.V. Kuznetsov, S.M. Seledzhi, and K.E. Starkov, 2010, published in Doklady Akademii Nauk, 2010, Vol. 434, No. 3, pp. 342–343.

471

The problem of localization of the attractor of athree�dimensional solar wind–magnetosphere–iono�sphere model (the WINDMI model [1, 2]) has stimu�lated the development of the method of conical nets tosolve this problem. In this communication, wedescribe this development and obtain analytical esti�mates for the attractor of the model.

Consider a system

(1)

where P is a constant singular (n × n) matrix, q and rare n�dimensional vectors, * is the transposition oper�ation, and ϕ(σ) is a differentiable scalar function thatsatisfies the sector conditions

(2)

(3)

Here, μ, α, and β are some numbers such that

We will assume that the pair (P, q) is fully control�lable, the pair (P, r) is fully observable, and system (1)has only one equilibrium state.

Theorem 1. Let r*q ≤ 0. There exists a number λ > 0such that the matrix P + λI has n – 1 eigenvalues withnegative real parts and the following inequality holds:

(4)

dxdt���� Px qϕ r*x( ), x+ �

n,∈=

ϕ σ( ) μ σ α–( ), σ∀ α,≥<

μ σ β–( ) ϕ σ( ), σ∀ β.≤<

μ 0, α β.<>

ReW iω λ–( ) μ W iω λ–( ) 2 0, ω∀ �.∈≤+

Then, for any solution x(t) of system (1), there existsa number T such that

(5)

Here, W(p) = r*(P – pI)–1q is the transfer function ofsystem (1), I is a unit (n × n) matrix, and i is the imagi�nary unit.

Below, we present a scheme for proving Theorem 1.The existence of a symmetric n × n matrix H for whichthe following conditions are met follows from condi�tion (4) (a detailed proof is available in [3]):

(1) the matrix H has one negative and (n – 1) posi�tive eigenvalues;

(2) for all z ∈ �n and ξ ∈ �, the following inequal�ity holds:

(6)

Note that the equality

follows from (6).

It follows from this and from the condition of thetheorem that r*H–1r = 2μr*q ≤ 0 and, hence [3],

(7)

Let d ∈ �n such that d ≠ 0 and Pd = 0, r*d = 1. Itthen follows from (6) that d*Hd < 0.

Let us now consider Lyapunov functions

It follows from Eqs. (2), (3), and (6) that

r*x t( ) α β,( ), t∀ T.>∈

2z*H P λI+( )z qξ+[ ] r*z r*z μ 1– ξ–( ) 0.≤+

2Hq μ 1– r=

z*Hz 0, z∀ z*r 0={ }.∈≥

V1 x( ) V x αd–( ) x αd–( )*H x αd–( ),= =

V2 x( ) V x βd–( ) x βd–( )*H x βd–( ).= =

V· 1 x t( )( ) 2λV1 x t( )( )+ 0 for r*x t( ) α,><

V· 2 x t( )( ) 2λV2 x t( )( )+ 0 for r*x t( ) β.< <

Localization of the Attractor of the Differential Equations for the Solar Wind–Magnetosphere–Ionosphere Model

Corresponding Member of the RAS G. A. Leonova, N. V. Kuznetsova, S. M. Seledzhia, and K. E. Starkovb

Received May 14, 2010

DOI: 10.1134/S1028335810090120

a St. Petersburg State University, Universitetskii pr. 28, St. Petersburg, 198504 Russiab Center for Research and Development of Technologies in Digital Systems, Av. del Parque 1310, Mesa de Otay, Tijuana, BC, Mexicoe�mail: nkuznetsov239@mail.ru

MECHANICS

472

DOKLADY PHYSICS Vol. 55 No. 9 2010

LEONOV et al.

It follows from this and from (7) that the sets

are positively invariant [3]. It is easy to see that the clo�

sures (α) and (β) are then also positively invari�ant. In this case, the boundaries ∂Ω(α) and ∂Ω(β)contain no whole trajectories and are almost every�where transversal to the vector field of system (1).These boundaries form a continuum set of surfaces(conical net) in the phase space of system (1) that isshown in Fig. 1. The structure constructed here“drives in” any solution in the set Ω(α) ∩ Ω(β) as thetime t increases. The latter proves the theorem.

Note that the estimate obtained is unimprovable inthe class of nonlinearities under consideration,because if ϕ(σ) = 0 for all σ ∈ (α, β), then x = νd forν ∈ (α, β) is a stationary solution of the system.

Consider the WINDMI model

(8)

It describes the energy flow dynamics in the solarwind–magnetosphere–ionosphere model and is usedto analyze geomagnetic storms and substorms [1, 2].

Let us apply the theory developed above to investi�gate Eq. (8). Here, the maximum value of coefficientμ can be calculated [6] from the formula

Let us define a point x0 > 0 such that ϕ'(x0) = μ,

Ω α( ) x αd–( )*H x αd–( ) 0, r*x α≥<{ },=

Ω β( ) x βd–( )*H x βd–( ) 0, r*x β≤<{ }=

Ω Ω

x··· bx·· c1x· ϕ x( )+ + + 0, ϕ x( ) c2 c3 x( )tanh+( ),= =

b 0, c1 0, c3 c2 0.> > > >

μ

b3�� c1

29��b2–⎝ ⎠

⎛ ⎞ , b2 3c1,≤

b3�� c1

29��b2–⎝ ⎠

⎛ ⎞ 2 b2

9����

c1

3���–⎝ ⎠

⎛ ⎞3/2

, b2 3c1.>+⎩⎪⎪⎨⎪⎪⎧

=

x0c3

�� for

c3

�� 1,>cosharg=

and, given the equality ϕ'(x0) = ϕ'(–x0) = μ, we willobtain constraints (2) and (3) for the nonlinearity ϕ(x)(Fig. 2). Denote

According to (2), (3), and (5), we will then obtain

(9)

To estimate and , note that, in view of the con�ditions for the coefficients being positive, the charac�teristic polynomial of the linear part of Eq. (8) haseigenvalues to the left of the imaginary axis and thenonlinearity ϕ(x) is limited:

Using the Cauchy formula, estimates can then beobtained [7–9] for | (t)|,

and for | (t)|,

Together with estimate (9) for |x(t)|, they localize theattractor of Eq. (8).

ACKNOWLEDGMENTS

This work was performed within the framework ofthe “Scientific and Scientific�Pedagogical Personnelof Innovative Russia” Federal Goal�Oriented Pro�gram for 2009–2013 and the CONACYT projectno. 78890.

α0ϕ x0( )

μ�����������– x0,

ϕ x0–( )μ

�������������– x0–+ β0.= =

α0 infx t( ), supx t( )t +∞→

lim β0.≤t +∞→

lim≤

x· x··

ϕ x( ) c2 c3.+≤

x·

yatt

sup x· t( )t +∞→

lim 1c1

��� c2 c3+( ), b2≤ 4c1,≥

sup x· t( )t +∞→

lim 2

b c1b2

4����–

������������������ c2 c3+( ), b2 4c1,<≤

⎩⎪⎪⎨⎪⎪⎧

=

x··

sup x·· t( )t +∞→

limc1yatt c2 c3+ +( )

b�������������������������������.≤

Fig. 1. Conical net.

μ(x – x0) + ϕ(x0)

μ(x + x0) + ϕ(–x0)

α0 β0 xx0–x0

ϕ(x)

Fig. 2. Nonlinearity estimation.

DOKLADY PHYSICS Vol. 55 No. 9 2010

LOCALIZATION OF THE ATTRACTOR OF THE DIFFERENTIAL EQUATIONS 473

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Translated by V. Astakhov