Comparison of Lorenz system and differential equations of induction motors^{1}

N.V. Kondrat’eva, G.A. Leonov^{2}, F.F. Rodyukov, A.I. Shepeljavyi shep@math.spbu.ru, leonov@math.spbu.ru, Fedor.Rodyukov@pobox.spbu.ru, ais@ais.usr.pu.ru

1 Introduction

In this paper the stability of the induction motor on the basic of its full mathematical model is investigated. The model is a system of differential equations of the sixth order which describes the dynamics of the induction motor with generally accepted idealizing assumptions, the latter are presented in detail in [1-4]. The main assumptions here are the following ones. In the first place the electromagnetic field in any cross-section of idealized physical model of the induction motor with the face effects neglected is indentical (the conjection of the flat model). Secondly we suppose that it is possible to describe the interaction of the electromagnetic processes in the stator winding and the rotor of the machine with the help of two symmetric linear electric circuits. Note that when investigating the static stability of the induction motor in practice often only its static mechanical perfomence is used. The latter gives the wide known Kloss formula in case the resistance of stator windings is neglected. Such approach is well agreed with the practical operation of the induction motor. But it is not rigorous enough from the point of view of mathematics.

To carry out the nonlocal analysis of complete mathematical model we exploite the special choice of mobile coordinates (the generalized Park transformation) and special choice of hybrid variables (the quasicurrents and quasiflux linkages of the stator windings). All this gives us the opportunity to reduce the system of differential equations to an autonomous system of the fifth order. For latter the Lyapunov functions are used. As a result, for various cases conditions for dissipativity, dichotomy and global stability are obtained and the domains of attraction for stable equilibria (dynamical stability) are evaluated. Similar functions were constructed in [5] for the well known Lorenz system.

2 Mathematical model

Let us consider the system of differential equations

(L_{s} \frac{d}{dt} + R_{s}) i_{α}^{s} + κ M \frac{d}{dt} (i_{α}^{r} cos γ − i_{β}^{r} sin γ) = −u_{m} sin ω_{s}t,

(L_{s} \frac{d}{dt} + R_{s}) i_{β}^{s} + κ M \frac{d}{dt} (i_{α}^{r} sin γ + i_{β}^{r} cos γ) = u_{m} cos ω_{s}t,

1Based on the lecture “Comparison of the Lorenz system and differential equation of induction motors”

hold at the DFG-Workshop “Dimensions of attractors”, St. Petersburg, Russia, October 11-12, 1999

2Partly supported by the Russian Research Fund 98-01-01005

M \frac{d}{dt} (i_{α}^{s} cos γ + i_{β}^{s} sin γ) + (L_{r} \frac{d}{dt} + R_{r}) i_{α}^{r} = 0,  (2.1)

M \frac{d}{dt}(−i_{α}^{s} sin γ + i_{β}^{s} cos γ) + (L_{r} \frac{d}{dt} + R_{r}) i_{β}^{r} = 0,

Jγ̈ = M[(i_{α}^{r}i_{β}^{s} − i_{β}^{r}i_{α}^{s}) cos γ − (i_{α}^{r}i_{α}^{s} + i_{β}^{r}i_{β}^{s}) sin γ] − M_{H}.

Here variables i_{α}^{s}, i_{β}^{s}, i_{α}^{r}, i_{β}^{r} are the currents of stator and rotor windings, γ is the angle of turn of the rotor; parameters R_{s}, L_{s}, R_{r}, L_{r} are resistances and inductances of appropriate windings, M is the peak significance of the mutual inductance between them, J is the moment of inertia of the rotor induction motor; ω_{s} = 2πf, f, u_{m} are the frequency and the amplitude of the voltage carried to the stator windings, M_{H} is the moment of the load on the shaft of the induction motor, t is the current time, κ is the parameter, characterizing a degree of influence of electromagnetic processes in a rotor on processes in the stator windings. The equations (2.1) coincide with the equations (8-1), (8-2) [1]. Neglecting the influence of electromagnetic processes in a rotor on processes in the stator windings, we assume in the system (2.1) κ = 0.

For further transformations of the equations (2.1) we need the expressions for the flux linkages of the windings. In case of the non-pole-salient electrical machine, which the induction motor is, we can receive these flux linkages from the formulas (3-3) - (3-3e) ([1]) with regard to the expressions for the inductances (3-40) - (3-49) ([1]). In the notations accepted above they have the form

ψ_{α}^{s} = L_{s}i_{α}^{s},
ψ_{β}^{s} = L_{s}i_{β}^{s},
ψ_{α}^{r} = M(i_{α}^{s} cos γ + i_{β}^{s} sin γ) + L_{r}i_{α}^{r},
ψ_{β}^{r} = M(−i_{α}^{s} sin γ + i_{β}^{s} cos γ) + L_{r}i_{β}^{r}.

(2.2)

The equations (2.1) and the flux linkages (2.2) are written in so-called phase coordinates α − β. For a mathematical research they are inconvenient. But they allow for the simplification of the writing to use a nonholonomic transformation of coordinates Park ([3]). In case of the induction motor it can be written with the help of auxiliary orthogonal axes u − v, which rotate with arbitrary angular velocity and are at the angle γ_{k} to a magnetic axis, of the phase α of the stator of the induction motor. Such transformation is given in [1]. For our designations it takes the form

i_{u}^{s} = i_{α}^{s} cos γ_{k} + i_{β}^{s} sin γ_{k},
i_{v}^{s} = −i_{α}^{s} sin γ_{k} + i_{β}^{s} cos γ_{k},
i_{u}^{r} = i_{α}^{r} cos(γ_{k} − γ) + i_{β}^{r} sin(γ_{k} − γ),
i_{v}^{r} = −i_{α}^{r} sin(γ_{k} − γ) + i_{β}^{r} cos(γ_{k} − γ);
ψ_{u}^{s} = ψ_{α}^{s} cos γ_{k} + ψ_{β}^{s} sin γ_{k},
ψ_{v}^{s} = −ψ_{α}^{s} sin γ_{k} + ψ_{β}^{s} cos γ_{k},
ψ_{u}^{r} = ψ_{α}^{r} cos(γ_{k} − γ) + ψ_{β}^{r} sin(γ_{k} − γ),
ψ_{v}^{r} = −ψ_{α}^{r} sin(γ_{k} − γ) + ψ_{β}^{r} cos(γ_{k} − γ);

(2.3)

u_{u} = u_{m}(− sin ω_{s}t cos γ_{k} + cos ω_{s}t sin γ_{k}) = −u_{m} sin(ω_{s}t − γ_{k}),
u_{v} = u_{m}(sin ω_{s}t sin γ_{k} + cos ω_{s}t cos γ_{k}) = u_{m} cos(ω_{s}t − γ_{k}).

These transformations mean, that we go from real (phase) variables, written in two orthogonal coordinate systems, one of which is rigidly connected with a stator of the induction motor, and the other is connected with its rotor, over to projections of these variable (to the quasivariables) on the same orthogonal axes u − v. With respect to these axes the equations (2.1) and the expressions (2.2) linkages take the form:

\begin{matrix}L_{s}(i̇_{u}^{s} − γ̇_{k}i_{v}^{s}) = −u_{m} sin(ω_{s}t − γ_{k}), \\ L_{s}(i̇_{v}^{s} + γ̇_{k}i_{u}^{s}) = u_{m} cos(ω_{s}t − γ_{k}), \\ M[i̇_{u}^{s} − (γ̇_{k} − γ̇)i_{v}^{s}] + R_{r}i_{u}^{r} + L_{r}[i̇_{u}^{r} − (γ̇_{k} − γ̇)i_{v}^{r}] = 0, \\ M[i̇_{v}^{s} + (γ̇_{k} − γ̇)i_{u}^{s}] + R_{r}i_{v}^{r} + L_{r}[i̇_{v}^{r} + (γ̇_{k} − γ̇)i_{u}^{r}] = 0, \end{matrix}  (2.4)

Jγ̈ = M(i_{u}^{r}i_{v}^{s} − i_{v}^{r}i_{u}^{s}) − M_{H};

\begin{matrix}ψ_{u}^{s} = L_{s}i_{u}^{s},  ψ_{v}^{s} = L_{s}i_{v}^{s}, \\ ψ_{u}^{r} = Mi_{u}^{s} + L_{r}i_{u}^{r},  ψ_{v}^{r} = Mi_{v}^{s} + L_{r}i_{v}^{r}. \end{matrix}  (2.5)

Let us pass in the equations (2.4) and formulas (2.5) to the dimensionless form. For the purpose, we introduce dimensionless synchronous time τ, sliding s, dimensionless variables, parameters and the moment of load by the formulas

τ = ω_{s}t, \frac{dγ}{dτ} = 1 − s,

ψ^{s} = \frac{u_{m}}{ω_{s}} ψ̄_{s}, ψ^{r} = \frac{u_{m}}{ω_{s}} \frac{L_{r}}{M} ψ̄^{r}, i^{s} = \frac{u_{m}}{ω_{s}L_{s}} ī^{s}, i^{r} = \frac{u_{m}}{ω_{s}L_{s}} \frac{L_{r}}{M} ī^{r},

ε_{s} = \frac{R_{s}}{ω_{s}L_{s}} , ε_{r} = \frac{R_{r}}{ω_{s}L_{r}} , δ = \frac{u_{m}^{2}}{ω_{s}^{4}JL_{s}} ,

\overline{M}_{H} = \frac{M_{H}}{ω_{s}^{2}Jδ} μ = 1 − \frac{M^{2}}{L_{s}L_{r}} ,

where μ is the factor of electromagnetic scattering in air clearance of an induction motor.

Then, omitting the symbol of huphen over the dimensionless magnitudes, we can write the equations (2.4) and the relations (2.5) in the form

\begin{matrix}i̇_{u}^{s} − γ̇_{k}i_{v}^{s} + ε_{s}i_{u}^{s} = − sin(τ − γ_{k}), \\ i̇_{v}^{s} + γ̇_{k}i_{u}^{s} + ε_{s}i_{v}^{s} = cos(τ − γ_{k}), \\ (1 − μ)[i̇_{u}^{s} − (γ̇_{k} − 1 + s)i_{v}^{s}] + ε_{v}i_{u}^{r} + i̇_{u}^{r} − (γ̇_{k} − 1 + s)i_{v}^{r} = 0, \\ (1 − μ)[i̇_{v}^{s} + (γ̇_{k} − 1 + s)i_{u}^{s}] + ε_{v}i_{v}^{r} + i̇_{v}^{r} − (γ̇_{k} − 1 + s)i_{u}^{r} = 0, \\ ṡ = −δ[(i_{u}^{r}i_{v}^{s} − i_{v}^{r}i_{u}^{s}) − \overline{M}_{H}], \end{matrix}  (2.6)

\begin{matrix}ψ_{u}^{s} = i_{u}^{s}, ψ_{v}^{s} = i_{v}^{s}, \\ ψ_{u}^{r} = (1 − μ)i_{u}^{s} + i_{u}^{r}, ψ_{v}^{s} = (1 − μ)i_{v}^{s} + i_{v}^{r}, \end{matrix}  (2.7)

where a dot over a variable means its differentiation on synchronous time τ.

Let us remind, that in (2.6) and (2.7) the variables ψ_{u} = ψ_{u}(τ), ψ_{v} = ψ_{v}(τ), i_{u} = i_{u}(τ), i_{v} = i_{v}(τ) are the quasiflux linkages and the quasicurrents of appropriate windings, s = s(τ) is the the sliding, i.e. the relative difference of angular velocities of the rotor and the magnetic field of the stator, τ is the dimensionless time (angle of a turn of a magnetic field of a stator); parameters ε_{s}, ε_{r} are the resistances of the stator and the rotor windings, μ is the factor of electromagnetic scattering in an air clearance, δ is an electromechanical constant, inversely proportional to the moment of inertia of the rotor; γ_{k} is the angle of a rotation of the axes u − v, \overline{M}_{H} is the moment of the external load on the shaft of the rotor.

Let us assume in (2.6) γ_{k} = τ, i.e. we pass to so-called synchronous axes of coordinates x − y (which rotate synchronoasly with the magnetic field of the stator). Then we have

\begin{matrix}i̇_{x}^{s} − i_{y}^{s} + ε_{s}i_{x}^{s} = 0, \\ i̇_{y}^{s} + i_{x}^{s} + ε_{s}i_{y}^{s} = 1, \\ (1 − μ)(i̇_{x}^{s} − si_{y}^{s}) + ε_{v}i_{x}^{r} + i̇_{x}^{r} − si_{y}^{r} = 0, \\ (1 − μ)(i̇_{y}^{s} + si_{x}^{s}) + ε_{v}i_{y}^{r} + i̇_{y}^{r} + si_{x}^{r} = 0, \\ ṡ = −δ[(i_{x}^{r}i_{y}^{s} − i_{y}^{r}i_{x}^{s}) − \overline{M}_{H}], \end{matrix}  (2.8)

\begin{matrix}ψ_{x}^{s} = i_{x}^{s}, ψ_{y}^{s} = i_{y}^{s}, \\ ψ_{x}^{r} = (1 − μ)i_{x}^{s} + i_{x}^{r}, ψ_{y}^{s} = (1 − μ)i_{y}^{s} + i_{y}^{r}. \end{matrix}  (2.9)

The equations (2.8) take the form

\begin{matrix}i̇_{x}^{s} = −ε_{s}i_{x}^{s} + i_{y}^{s}, \\ i̇_{y}^{s} = −ε_{s}i_{y}^{s} − i_{x}^{s} + 1, \end{matrix}  (2.10)

\begin{matrix}ψ̇_{x}^{r} = −ε_{r}ψ_{x}^{r} + sψ_{y}^{r} + ε_{r}(1 − μ)x, \\ ψ̇_{y}^{r} = −ε_{r}ψ_{y}^{r} − sψ_{x}^{r} + ε_{r}(1 − μ)y, \\ ṡ = −δ[(ψ_{x}^{r}y − ψ_{y}^{r}x) − \overline{M}_{H}]. \end{matrix}  (2.11)

3 Statement of the results

Let us consider the autonomous system

ẋ = f(x), x ∈ R^{n}.  (3.1)

Let us introduce several definitions. Definition 3.1 A point p ∈ R^{n} is called stationary point (or the equilibrium state) of the system (3.1), if x(t) ≡ p is a solution of this system.

The set of all stationary points is called the stationary set.

Definition 3.2 We say that the system (3.1) is dychotomous, if any bounded solution, for t > 0, tends to the stationary set as t → +∞.

Definition 3.3 We call that the system (3.1) dissipative, if all its solutions are infinitely extendable to the right and there exists such number R > 0, that for any solution x(t, x_{0}) of this system with the initial data x(0, x_{0}) = x_{0} the following relation is true

\overline{lim_{t→+∞}}|x(t, x_{0})| < R.

It follows from Definition 3.3, that for each solution x(t, x_{0}) of (3.1) there exists a moment T_{x_{0}} such that

|x(t, x_{0})| < R for all t ≥ T_{x_{0}},

i.e.beginning with a moment T_{x_{0}}, the trajectory x(t, x_{0}) is located in an open ball of radius R. Definition 3.4 System (3.1) is called globally asymptotically stable, if any solution of this system tends to a certain equilibrium as t → +∞ .

With the help of the change of variables

\begin{matrix}x = \frac{ψ_{x}^{r} + ε_{s}ψ_{y}^{r}}{ε_{r}(1 − μ)} , y = \frac{ψ_{y}^{r} − ε_{s}ψ_{x}^{r}}{ε_{r}(1 − μ)} , \\ ψ_{x} = \frac{i_{x}^{s} + ε_{s}i_{y}^{s} − 1}{ε_{r}(1 − μ)} , ψ_{y} = \frac{i_{y}^{s} − ε_{s}i_{x}^{s}}{ε_{r}(1 − μ)} \end{matrix}  (3.2)

we can reduce the system (2.10), (2.11) to the form more convenient for research:

\begin{matrix}ψ̇_{x} = −α_{s}ψ_{x} + ψ_{y}, \\ ψ̇_{y} = −α_{s}ψ_{y} − ψ_{x}, \end{matrix}  (3.3)

\begin{matrix}ẋ = −α_{r}x + sy + α_{r}(1 − μ)ψ_{x} + 1, \\ ẏ = −α_{r}y − sx + α_{r}(1 − μ)ψ_{y}, \\ ṡ = δ[α_{r}by + α_{r}^{2}b(1 − μ)(ψ_{x}y − ψ_{y}x) + \overline{M}_{H}]. \end{matrix}  (3.4)

Here we put

α_{s} = ε_{s}, α_{r} = ε_{r}, M_{H} = \overline{M}_{H}, b = \frac{1 − μ}{1 + α_{s}^{2}} .

In this article the stability of the system (3.3), (3.4) is investigated for two cases - for the case of a constant moment M_{H} of the external load and for the case, when the moment M_{H} linearly depends on slidings of the rotor.

Theorem 3.1 Let M_{H}=const. Then the following statements take place:

1) If

0 < M_{H} < min \{\frac{b}{2}, \frac{2α_{r}^{2}}{δ} \} ,  (3.5)

then the system (3.3), (3.4) is dychotomous; 2) If

\frac{b}{2} < M_{H} < \frac{2α_{r}^{2}}{δ} ,  (3.6)

then all solutions of (3.3), (3.4) are unbounded.

Theorem 3.2 Let M_{H} = κ̄(1 − s), κ̄ = const. Then, if

0 < κ̄δ ≤ 3α_{r}\sqrt{(α_{r} + κ̄δ)(α_{r} − 2κ̄δ)}  (3.7)

the system (3.3), (3.4) is globally asymptotically stable.

4 Investigation of a model of the third order

The research of the system (3.3), (3.4) is largely based on the research of the system of the third order

\begin{matrix}ẋ = −α_{r}x + sy + 1, \\ ẏ = −α_{r}y − sx, \\ ṡ = δ[α_{r}by + M_{H}], \end{matrix}  (4.1)

obtained from (3.3), (3.4) by substitution of stationary solutions ψ_{x} = 0, ψ_{y} = 0 of the asymptotically stable linear system (3.3) in a system (3.4). Note, that (4.1) is of independent interest in the theory of induction motors. It can be obtained from a complete mathematical model of induction motor with the help of so-called quasistationary approach.

Note, that the structure of the system (4.1) in the case M_{H} = k(1 − s) is similar to a structure of the well known Lorenz system [6]. This circumstance allows to carry out the non local analysis of this system with the help of Lyapunov functions, considered in [5].

Theorem 4.1 Let M_{H}=const.

1) If the condition (3.5) is true then the system (4.1) dychotomous;

2) If M_{H} = 0, the system (4.1) is globally stable;

3) If the condition (3.6) is executed, all solutions of a system (4.1) unbounded.

Proof. The change of variables

σ = s, η = να_{r}y + γ, z = −α_{r}x − as + 1,  (4.2)

where

γ = δ M_{H}, ν = δ b, a = \frac{γ}{α_{r}ν} ,  (4.3)

reduces the system (4.1) to the form

\begin{matrix}σ̇ = η, η̇ = −α_{r}η + νσz − φ(σ), \\ ż = −α_{r}z − \frac{1}{ν} ση − aη, \end{matrix}  (4.4)

where φ(σ) = −aνσ^{2} + νσ − α_{r}γ.

Under condition (3.5) the stationary set Λ of (4.4) consists of two points, the stable equilibrium O_{1} and the saddle O_{2}:

O_{1} (\frac{α_{r}(ν − \sqrt{ν^{2} − 4γ^{2}} )}{2γ}, 0, 0) , O_{2} (\frac{α_{r}(ν + \sqrt{ν^{2} − 4γ^{2}} )}{2γ}, 0, 0) .

Let us introduce the function

V_{1} = \frac{ν^{2}}{2} z^{2} + \frac{1}{2} η^{2} + ∫_{σ_{1}}^{σ} φ(σ) dσ.  (4.5)

For solutions x(t) = (σ(t), η(t), z(t)) of the system (4.4) relation

\begin{matrix}V̇_{1}(x(t)) = −α_{r}ν^{2}z^{2}(t) − νσ(t)η(t)z(t)− \\ −aν^{2}η(t)z(t) − α_{r}η^{2}(t) + νσ(t)η(t)z(t)− \\ −η(t)φ(σ(t)) + φ(σ(t))η(t) = −α_{r}η^{2}(t)− \\ −aν^{2}η(t)z(t) − α_{r}ν^{2}z^{2}(t). \end{matrix}  (4.6)

takes place. It is not difficult to verify that under the condition of (3.5) the quadratic forms on right hand side of (4.6) are negatively defined by virtue of the Sylvester criterion [7]. Therefore it follows from (4.6), that the function V_{1} do not increase on some interval (τ, +∞) along any trajectory of the system (4.4). Hence for any bounded trajectory x(t, x_{0}) we receive by virtue of boundedness of function V_{1}(x(t, x_{0})) the existence of a finite limit lim_{t→+∞} V_{1}(x(t, x_{0})) = L.

From boundedness of the trajectory x(t, x_{0}) it follows that the ω limit set Ω of x_{0} is not empty. Let y ∈ Ω. It is known ([8]) that the x(t, y) ∈ Ω for all t ∈ R^{1}. Therefore V_{1}(x(t, y)) = L for all t ∈ R^{1}. Using (4.6), we receive the identities η(t, y) ≡ 0 and z(t, y) ≡ 0. From (4.4) and η(t, y) ≡ 0 we receive, that σ(t, y) ≡ const. Therefore Ω ⊂ Λ. Since the elements of Λ are isolated this inclusion implies the statement 1).

2) In the case M_{H} = 0 (γ = 0) the stationary set of (4.4) consist of a single point (0, 0, 0), φ(σ) = νσ. In this case we have V_{1} = \frac{ν^{2}}{2} z^{2} + \frac{η^{2}}{2} + \frac{ν}{2} σ^{2}, V̇_{1} ≤ 0, and V̇_{1} = 0 on a set, not containing whole trajectories, except for a stationary point (0, 0, 0). So all conditions of the Barbashin- Krasovsky theorem ([9]) are valid and the system (4.4) is globally stable.

3) Assume that the condition (3.5) is satisfied and x(t) is a bounded solution of (4.4). The condition (3.6) is a special case of (3.5). Therefore, as it was shown in item 1), x(t) tends to a certain equilibrium. However under the condition of (3.5) the stationary set of (3.10) is empty. We have received a contradiction, which proves the statement 3). Theorem 4.1 is proved.

For further investigation of stability of (4.1) we shall prove an analog of the Barbashin - Krasovsky theorem [8].

Let us consider the system

ẋ = f(x), x ∈ R^{n}, f ∈ C.  (4.7)

Lemma 4.1 Suppose that an unbounded positively invariant set G of (4.7) contains only one stationary point p of this system. Suppose also that in the phase space of (4.7) a certain function v is determined which satisfies the following conditions: 1) v(x) > 0 for any x ∈ G, x ≠ p; 2) v (p) = 0; 3) v̇(x)) ≤ 0 for all x ∈ G, 4) The set {x : v̇(x) = 0} does not contain whole trajectories, except for a point p; 5) lim v(x_{n}) = +∞ for any sequence {x_{n}} such that |x_{n}| → ∞ for n → ∞ and x_{n} ∈ G for all n. Then any trajectory x(t, x_{0}) of system (4.7) such that x_{0} ∈ G tends for t → +∞ to the point p.

Proof. Let us consider the trajectory x(t, x_{0}) of the point x_{0} ∈ G for t > 0. Because of the properties of v̇ and the positive invariancy of G it is true that x(t, x_{0}) ∈ G_{1} = {x : v(x) ≤ v(x_{0})} ∩ G for all t > 0. Let us prove the boundedness of the set G_{1}. Let us assume the opposite. Let {x_{n}} be a sequence of x_{n} such that for any n x_{n} ∈ G_{1} and lim |x_{n}| = ∞ for n → ∞. By virtue of a condition 3) lim_{n→∞} v(x_{n}) ≤ v(x_{0}), which contradicts to the condition 5).

Thus, G_{1} is abounded positively invariant set, p ∈ G_{1} and G_{1} does not contain other stationary points.

Let us consider the trajectory x(t, x_{0}) of x_{0} ∈ G_{1} for t > 0. By virtue of condition 5) and properties of G_{1}, function v(x(t, x_{0})) does not increase and is bounded on (0, +∞). Therefore for t → +∞ there is a finite lim v(x(t, x_{0})) = L.

The trajectory x(t, x_{0}) is bounded on (0, +∞), therefore the set Ω of its ω - limit points is not empty. Let q ∈ Ω. It is known ([8]), that x(t, q) ∈ Ω for all t ∈ R^{1}. Therefore v((x(t, q)) ≡ L for all t ∈ R^{1} and therefore v̇(x(t, q)) ≡ 0 for all t ∈ R^{1}. By virtue of condition 4) of the lemma x(t, q) ≡ p. Then q = p. Lemma 4.1 is proved.

Using (4.2), (4.3) and (4.5) we can present function V_{1} in coordinates and parameters of (4.1) as

V_{1} = \frac{δ^{2}α_{r}^{2}b^{2}}{2} [(x − \frac{α_{r}b − M_{H}s}{α_{r}^{2}b})^{2} + (y − \frac{M_{H}}{α_{r}b} )^{2}] + ∫_{s_{1}}^{s} φ(s) ds,

where φ(s) = δ( − M_{H}/s^{2}α_{r} + bs − α_{r}M_{H}),

s_{1,2} = \frac{α_{r}}{2M_{H}} (b ∓ \sqrt{b^{2} − 4M_{H}^{2}} ) .

Let us introduce the function

W_{1} = \frac{α_{r}^{2}}{2} x^{2} + \frac{α_{r}^{2}}{2} y^{2}

and the sets

D_{0} = {X : W_{1}(X) ≤ Γ_{1}},
D_{1} = {X : 2V_{2}(X) < p_{0}^{2}, s < s_{2}},
D_{2} = {X : W_{1}(X) ≤ Γ_{1}, s ≤ s_{1}},

where

\begin{matrix}p_{0} = \sqrt{2 ∫_{s_{1}}^{s_{2}} φ(s) ds}, \\ Γ_{1} = \frac{α_{r}^{2}}{8λ(α_{r} − λ)}, λ ∈ [0, α_{r}]. \end{matrix}

Theorem 4.2 Let M_{H} = const and assume that the condition (3.5) and the condition

Γ_{0} = \frac{p_{0}}{δb} − \frac{M_{H} \sqrt{α_{r}^{2} + s_{2}^{2}}}{α_{r}b} ≥ 1  (4.8)

are fulfilled. If Γ_{1} = Γ_{0}^{2}/2, then the set G = D_{1} ⋃ D_{2} is contained in the domain of attraction of a stable equilibrium of the system (4.1).

Proof. Let us prove, that the set D_{0} is positively invariant for solutions of (4.1). For this purpose we shall show, that for any solutions X(t) = (x(t), y(t), s(t)) of (4.1) the inequality

\overline{lim_{t→+∞}}W_{1}(X(t)) ≤ Γ_{1}  (4.9)

is valid. Really,

Ẇ_{1} + 2λW_{1} = −α_{r}^{2}(α_{r} − λ) x^{2} + α_{r}^{2}x − α_{r}^{2}(α_{r} − λ) y^{2} ≤ \frac{α_{r}^{2}}{4(α_{r} − λ)} = 2λΓ_{1}.

That is why

(W_{1} − Γ_{1})^{•} + 2λ(W_{1} − Γ_{1}) ≤ 0,

whence it follows, that for all t ≥ 0

W_{1}(X(t)) − Γ_{1} ≤ [W_{1}(X(0)) − Γ_{1}]e^{−2λt},

i.e. (4.9) is true. From (4.9) the positive invariance of the set D_{0} follows.

Let us prove, that the set D_{1} is positively invariant for solutions of (4.1). It is obvious, that there is a number s = s_{3} such that s_{3} < s_{1} and

∫_{s_{3}}^{s_{2}} φ(s) ds = 0.

The level surfaces of function V_{1} are defined by the equations

δ^{2}α_{r}^{2}b^{2} [(x − \frac{α_{r}b − M_{H}s}{α_{r}^{2}b})^{2} + (y + \frac{M_{H}}{α_{r}b} )^{2}] = Φ(s, C),

where Φ(s, C) is the one-parameter family of functions

Φ(s, C) = C − 2 ∫_{s_{1}}^{s} φ(s) ds.

The domain D_{1} is bounded by the closed part of the surface Φ(s, p_{0}^{2}) and contains a stable stationary point O_{1}. Under the condition of (3.5) we have V̇_{1} ≤ 0, and V̇_{1} < 0 in all points ∂D_{1}, except for the points

O_{i} (\frac{α_{r}b − M_{H}s_{i}}{α_{r}^{2}b} − \frac{M_{H}}{α_{r}b} s_{i}) , i = 1, 3.

Hence by virtue of continuous dependence of solutions on initial data the positive invariance of the set D_{1} follows.

Let us consider the cuts S_{1} and S_{2} of sets D_{1} and D_{2} respectively by the plane s = s_{1}

\begin{matrix}S_{1} = \{X : δ^{2}b^{2}α_{r}^{2} (x − \frac{Ms_{2}}{α_{r}^{2}b} )^{2} + δ^{2}b^{2}α_{r}^{2}(y − M/α_{r}b)^{2} ≤ p_{0}^{2}\} , \\ S_{2} = \{X : α_{r}^{2}x^{2} + α_{r}^{2}y^{2} ≤ 2Γ, s = s_{1}\}. \end{matrix}

Under the condition (4.8) there exists Γ_{1} = \frac{Γ_{0}^{2}}{2} such that the circle S_{2} touches the boundary of the circle S_{1} from within. Then we have inclusion S_{2} ⊂ S_{1} ⊂ D_{1}, from which using the positive invariancy of the sets D_{0} and D_{1} we receive a positive invariance of the set G.

So, G is positively invariant, and G contains only one stationary point of (4.1) — the point O_{1}. For the function V_{1} in domain G the conditions 1) - 5) are satisfied. Really, V_{1} > 0 in all points of G, except for the point O_{1}, in which V_{1} = 0, i.e. the conditions 1) and are satisfied 2). From (4.6) we have conditions 3) and 4). Further, it is easy to

see, that lim_{s→−∞} ∫_{s_{1}}^{s} φ(s) ds = +∞. Therefore condition 5) is satisfied. Consequenly, any

trajectory X(t, X_{0}) of (4.1) such that X_{0} ∈ G, tends to the point O_{1}. Theorem 4.2 is proved.

Theorem 4.3 Let M_{H} = κ̄(1 − s), κ̄ = const. Then, if the condition (3.7) is satisfied, the system (4.1) is globally asymptotically stable.

Proof. Let us make the change of variables

x̄ = −α_{r}x + 1, ȳ = α_{r}y, s̄ = s

and preserve their names. Then system (4.1) is reduced to the form

\begin{matrix}ẋ = −α_{r}x + sy, \\ ẏ = −α_{r}y + s − xs, \\ ṡ = −νy − κs + κ, \end{matrix}  (4.10)

where ν = δb, κ = κ̄δ.

Let us introduce the function

W_{2} = \frac{1}{2}[x − (1 − ν)]^{2} + \frac{1}{2} y^{2} + \frac{1}{2} s^{2}

and the designation

Γ_{2} = \frac{(α_{r} − 2λ)^{2}(1 − ν)^{2}}{8λ(α_{r} − λ)} + \frac{κ^{2}}{8λ(κ − λ)},

where λ ∈ [0, λ_{0}], λ_{0} = min{α_{r}, κ}. Let us show that for any solution X(t) = (x(t), y(t), s(t)) of (4.10) the inequality

\overline{lim_{t→+∞}}W_{2}(X(t)) ≤ Γ_{2}.  (4.11)

is valid. Really,

Ẇ_{2} + 2λW_{2} = −α_{r}x^{2} + α_{r}(1 − ν)x − κs^{2} + κs + λx^{2}+

+λ(1 − ν)^{2} − 2λ(1 − ν)x + λy^{2} + λs^{2} = −(α_{r} − λ)x^{2} + (α_{r} − 2λ)(1 − ν)x−

−(κ − λ)s^{2} + κs − (α_{r} − λ)y^{2} = (α_{r} − λ) [x^{2} − \frac{(α_{r} − 2λ)(1 − ν)}{α_{r} − λ} x] −

−(κ − λ) [s^{2} + \frac{κs}{κ − λ}] − (α_{r} − λ)y^{2} =

= −(α_{r} − λ) [x^{2} − \frac{(α_{r} − 2λ)(1 − ν)}{2(α_{r} − λ)} x]^{2} + \frac{(α_{r} − 2λ)^{2}(1 − ν)^{2}}{4(α_{r} − λ)} −

−(κ − λ) [s + \frac{κ}{2(κ − λ)}]^{2} + \frac{κ^{2}}{4(κ − λ)} − (α_{r} − λ)y^{2} ≤

≤ \frac{(α_{r} − 2λ)^{2}(1 − ν)^{2}}{4(α_{r} − λ)} + \frac{κ^{2}}{4(κ − λ)} = 2λΓ_{2}.

Hence we have the estimate

W_{2}(X(t)) − Γ_{2} ≤ [W_{2}(X(0)) − Γ_{2}] e^{−2λt},

which implies (4.11). From (4.11) the dissipativity of (4.10) follows. Consequently the system (4.1) is dissipative as well.

Using the change of variables

σ = s, η = −νy − κs + κ, z = x − \frac{κ}{ν} (s − s^{2})

we reduce the system (4.10) to the system

\begin{matrix}σ̇ = η, η̇ = −(α_{r} + κ)η + νσz − ψ(σ), \\ ż = −α_{r}z + (\frac{2κ}{α_{r}ν} − \frac{1}{ν}) ση − \frac{κ}{α_{r}ν} η, \end{matrix}  (4.12)

where ψ(σ) = \frac{κ}{α_{r}} (σ_{3} − σ) + (α_{r}κ + ν)σ − α_{r}κ.

The stationary set of (4.12) consists of one, two or three points.

Let us introduce the function

V_{2} = \frac{α_{r}ν^{2}}{2(α_{r} − 2κ)} z^{2} + \frac{1}{2} η^{2} + ∫_{0}^{σ} ψ(σ) dσ

and calculate the derivative of function V_{2} by virtue of (4.12)

\begin{matrix}V̇_{2} = −\frac{α_{r}ν^{2}}{α_{r} − 2κ} z^{2} + \frac{α_{r}ν^{2}}{α_{r} − 2κ} (\frac{2κ}{α_{r}ν} − \frac{1}{ν}) σηz− \\ −\frac{νκ}{α_{r} − 2κ} ηz − (α_{r} + κ)η^{2} + νσηz − ψ(σ)η+ \\ +ψ(σ)η = −\frac{α_{r}ν^{2}}{α_{r} − 2κ} z^{2} − \frac{νκ}{α_{r} − 2κ} ηz − (α_{r} + κ)η^{2}. \end{matrix}  (4.13)

If the conditions (3.7) are fulfilled then the quadratic form in right hand side of (4.13) is negatively definite. Repeating the argument used when proving Theorem 4.2 after (4.6) was obtained, we come to the conclusion that (4.12) is dychotomous and consequently (4.1) is dychotomous as well. Since (4.1) is dissipative and dychotomous it is globally asymptotically stable. The Theorem 4.3 is proved.

5 Proof of the Theorems 3.1 and 3.2

Proof of the Theorem 3.1.

1) Under the condition (3.5) the stationary set of the system (3.3), (3.4) consists of two points - the stable point O_{1} and the saddle O_{2}: O_{i} (0, 0, x_{i}, y_{i}, s_{i}), i = 1, 2,

x_{i} = \frac{α_{r}}{α_{r}^{2} + s_{i}^{2}} , y_{i} = −\frac{s_{i}}{α_{r}^{2} + s_{i}^{2}} ,

s_{1,2} = \frac{α_{r}}{2} (b ∓ \sqrt{B^{2} − 4M_{H}^{2}} ) .

Let the trajectory X(t, p) = (ψ_{x}(t, p), ψ_{y}(t, p), x(t, p), y(t, p), s(t, p)) of the systems (3.3), (3.4) be bounded on (0,+∞). Then the set Ω_{p} of ω - limit points of p is not empty. Let q ∈ Ω_{p}, q = (q_{i}), i = 1, …, 5. By virtue of asymptotic stability of the linear system (3.3) we have that q_{1} = q_{2} = 0.

Let us introduce the set Ψ = {X = X(t, q) : t ∈ R^{1}}. It is known [8], that Ψ ⊂ Ω_{p}, i.e. all points X(t, q) are ω - limit points for the point p. Obviously, system (3.3), (3.4) on the set Ψ can be represented as (4.1).

By virtue of (4.13) the functions V_{2}(x(t)) does not increase with respect to t on some interval (τ, +∞). From here and from the boundedness of V_{2}(X(t, q)) the existence of the finite limits

\begin{matrix}lim_{t→+∞} V_{2}(X(t, q)) = L_{1}, \\ lim_{t→−∞} V_{2}(X(t, q)) = L_{2}. \end{matrix}

follows. It follows from the boundedness of the trajectory X(t, q), that there exists an ω - limit point q_{ω} of X(t, q). As it was marked above X(t, q_{ω}) ∈ Ω_{q} for all , t ∈ R^{1} (Ω_{q} is the ω - limit set of the point q). Therefore V_{2}(X(t, q_{ω})) = L_{1} for all t ∈ R^{1}. Then, using (4.13) and (4.12), we receive the identities η(t, q_{ω}) ≡ 0, z(t, q_{ω}) ≡ 0. From (4.12) and η(t, q_{ω}) ≡ 0 it follows, that σ(t, q_{ω}) ≡ const. So, lim_{t→+∞} X(t, q) = c_{1}, where c_{1} is a stationary point.

The similar reasoning implies that lim_{t→−∞} X(t, q) = c_{2}, where c_{2} is also a stationary

point. If c_{1} ≠ c_{2}, then X(t, q) is a double asymptotic trajectory, going out from a saddle O_{2} and entering either the stable stationary point O_{1}, or the saddle O_{2}. Let q̂ ∈ Ψ. It means, that there is a sequence {t_{n}}, t_{n} → +∞, such that lim_{t→∞} X(t_{n}, p) = q̂, i.e. for any ε > 0 there exists N, such that for all n > N the inequality |X(t_{n}, p) − q̂| < ε is true. Then by virtue of continuous dependence of solutions on initial data we have lim_{n→∞} X(t_{n}, p) = O_{i}, i = 1, 2. Hence it follows, that q̂ = O_{i}, i = 1, 2. So, Ω_{p} ⊂ Λ, i.e. Ω_{p} consists of acnodes.

The statement 2) of the theorem can be proved just in the same way as the statement 3) of the Theorem 4.1. The Theorem 3.1 is proved.

Proof of Theorem 3.2.

Under the conditions of the theorem the stationary set of the system (3.3), (3.4) can consist of one, two or three points.

Repeating the argument, used in Theorem 3.1 when proving that (3.3), (3.4) is dychotomous, and using Theorem 4.2 and function the V_{2}, we establish that (3.3), (3.4) is dychotomous.

Let us prove the dissipativity of (3.3), (3.4) for M_{H} = κ̄(1 − s). Let us introduce the function

W = \frac{1 + ν}{2} (x^{2} + y^{2}) + \frac{1}{2} s^{2} − x.

Let us prove, that for any solution X(t) = (ψ_{x}(t), ψ_{y}(t), x(t), y(t), s(t)) of (3.3), (3.4) the inequality

\overline{lim_{t→+∞}}W (X(t)) ≤ Γ,  (5.1)

where Γ is a constant, is true. Really

\begin{matrix}Ẇ + 2λW = −[(α_{r} − λ)x^{2} + (α_{r} − 2λ + α_{r}(1 − μ)(1 + ν)ψ_{x}x]− \\ −[(α_{r} − λ)y^{2} − α_{r}(1 + ν)(1 − μ)ψ_{y}y] − [(α_{r} − λ)s^{2} − (κ − νφ_{2}(z))s]− \\ − [ν(α_{r} − λ)x^{2} − α_{r}ν(1 − μ)ψ_{y}xs + \frac{λ_{1}}{2} s^{2}] − \\ − [ν(α_{r} − λ)y^{2} − α_{r}ν(1 − μ)ψ_{x}ys + \frac{λ_{1}}{2} s^{2}] − \\ −α_{r}^{2}(1 − μ)ψ_{x} ≤ 2λφ(ψ_{x}, ψ_{y}) − (x, s)H_{x}(x, s)^{T} − (y, s)H_{y}(y, s)^{T} , \end{matrix}  (5.2)

where, we remind, ν = δb, k = δ\overline{κ}. Here

H_{i} = (\begin{matrix}ν(α_{r} − λ) & & −\frac{α_{r}ν(1−μ)ψ_{i}}{2} \\ & −\frac{α_{r}ν(1−μ)ψ_{i}}{2} & \frac{λ_{1}}{2} \end{matrix} ) , i = x, y,

\begin{matrix}φ(ψ_{x}, ψ_{y}) = \frac{(α_{r} − 2λ + α_{r}(1 + ν)(1 − μ)ψ_{x})^{2}}{8λ(α_{r} − λ)} + \\ +\frac{α_{r}^{4}(1 − μ)^{2}(1 + ν)^{2}ψ_{y}^{2}}{8λ(α_{r} − λ)} + \frac{(κ − α_{r}ν(1 − μ)ψ_{y})^{2}}{8λ(κ − λ_{2})} + \frac{α_{r}^{2}(1 − μ)ψ_{x}}{2λ} , \end{matrix}

λ = λ_{1} + λ_{2}, λ_{1} > 0, λ_{2} > 0, λ ∈ [0, λ_{0}], λ_{0} = min{α_{r}, k}.

Then

det H_{i} = \frac{λ_{1}ν(α_{r} − λ)}{2} − \frac{α_{r}^{2}ν^{2}(1 − μ)^{2}ψ_{i}^{2}}{4} , i = x, y.

By virtue of asymptotic stability of (3.3) there exists a number M > 0 such that

|ψ_{x}(t)| ≤ Me^{−α_{s}t}, |ψ_{y}(t)| ≤ Me^{−α_{s}t} for all t > 0.

It is easy to see, that there exists a number T > 0 such that for all t ≥ T we have

det H_{i}(t) > 0, i = x, y.

Hence and from (5.2) it follows, that for all t ≥ T the relation

Ẇ + 2λW ≤ 2λψ(t),  (5.3)

where ψ(t) = φ(Me^{−α_{s}t}, Me^{−α_{s}t}), takes place. It is obvious, that there is a finite lim_{t→+∞} ψ(t) = Γ. Therefore numbers T_{0} ≥ T, k_{0} λ_{0} exist such that for all t ≥ T_{0}

ψ(t) ≤ Γ + k_{0}e^{−λ_{0}t}.

Therefore from (5.2) and (5.3) the inequality

W (X(t)) − Γ ≤ [W (X(0)) − Γ] e^{−2λt} + \frac{2λk_{0}}{α_{0} − 2λ} (e^{−2λt} − e^{−α_{0}t}) ,

follows, whence (5.1) follows.

From (5.1) and asymptotic stability of the linear system (3.3) the dissipativity of (3.3), (3.4) follows.

Since the system (3.3),(3.4) is dissipative and dychotomous it is globally asimptotically stable. Theorem 3.2 is proved.

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