0906.2794

arXiv:0906.2794v1 [math.DS] 15 Jun 2009

Mathematical analysis of stochasticmodels for tumor-immune systemsO. Chis, D. Opris

Euro University Dragan, Lugoj, RomaniaFaculty of Mathematics and Informatics, West University of Timisoara, RomaniaE-mail: [email protected], [email protected]

Abstract: In this paper we investigate some stochastic models for tumorimmune systems. To describe these models we used a Wiener process, as thenoise has a stabilization effect. Their dynamics are studied in terms of stochastic stability in the equilibrium points, by constructing the Lyapunov exponent,depending on the parameters that describe the model. We have studied andanalyzed a Kuznetsov-Taylor like stochastic model and a Bell stochastic modelfor tumor-immune systems. These stochastic models are studied from stabilitypoint of view and they were represented using the Euler second order scheme.MSC2000: 37L55, 65C30, 37H15, 60H20, 76M35.Keywords: stochastic model, stochastic stability, Wiener process, Lyapunov exponent, tumor-immune systems, Euler second order scheme.

1

Introduction

Cancer is a disease that may affect people at all ages. It causes about 13%of all human deaths. In the prognosis of cancer patients, it should be taken intoconsideration the type of cancer and the stage of the disease. Cancers may betreated or cured, depending on the specific type, location, and stage. We maysay that surgery and chemotherapies play an important role in treating cancer,but they do not represent a cure. What it is needed is a successful treatmentstrategies, one of these strategies is investigated through immunotherapy [13],by defining a model of differential equations that represents the interaction1

2between effector cells and tumor cells. This idea of immunotherapy is promising, but controversial from the point of view of the results obtained in medicalinvestigations.Stochastic modelling plays an important role in many branches of science.Because in practical situations we confront with instability and perturbations,we will express our mathematical models using white noise, represented bybrownian motion. We will study stochastic dynamical systems that are usedin medicine, in describing a tumor behavior. Cancer tumor may be destroyedusing some treatments, but a regression of the disease may appear. So, we neednot only preventative measures, but also more successful treatment strategies.Efforts along these lines are now being investigated through immunotherapy([5], [20], [22]). A simulating model is described by the existence of tumorfree equilibrium. A tumor size may tend to +, depending on the parametersof the model, and may exist a small tumor size equilibrium, which coexistswith the tumor free equilibrium [10].This tumor-immune study, from theoretical point of view, has been done fortwo cell populations: effector cells and tumor cells. It was predicted a threshold above which there is uncontrollable tumor growth, and below which thedisease is attenuated with periodic exacerbations occurring every 3-4 months.There was also shown that the model does have stable spirals, but the DulacBendixson criterion demonstrates that there are no stable closed orbits. It isconsider ODEs for the populations of immune and tumor cells and it is shownthat survival increases if the immune system is stimulated, but in some casesan increase in effector cells increases the chance of tumor survival.In the last years, stochastic growth models for cancer cells were developed.These models simulate the way tumors evolve with respect to a certain therapy,but also they show the interactions between tumor cells and immune cells. Wemention the papers of W.Y. Tan and C.W. Chen [19], N. Komarova, G. Albanoand V.Giorno [1], L. Ferrante, S. Bompadre, L. Possati and L. Leone [6], A.Boondirek Y. Lenbury, J. Wong-Ekkabut, W. Triampo, I.M. Tang, P. Picha[4].Our goal in this paper is to construct stochastic models and to analyzetheir behavior around the equilibrium point. In these points stability is studied by analyzing the Lyapunov exponent, depending of the parameters of themodels. Numerical simulations are done using a deterministic algorithm withan ergodic invariant measure. In this paper, the authors studied and analyzedtwo stochastic models. In Section 2, we considered a Kuznetsov and Taylorstochastic model. Beginning from the classical one, we have studied the caseof positive immune response. We gave the stochastic model and we analyzed

3it in the equilibrium points. Numerical simulations for this new model arepresented in Section 2.1. In Section 3 we presented a general family of tumorimmune stochastic systems and from this general representation, we analyzedBell model. We wrote this model as a stochastic model, using Annexe 1, andwe discussed its behavior around the equilibrium points. Numerical simulations were done using the software Maple 12 and we implemented the secondorder Euler scheme for a representation of the discussed stochastic models,described in Annexe 2.

2

Kuznetsov and Taylor stochastic model

We will begin our study from the model of Kuznetsov and Taylor [13]. Thismodel describes the response of effector cells to the growth of tumor cells andtakes into consideration the penetration of tumor cells by effector cells, thatcauses the interaction of effector cells. This model can be represented in thefollowing way:

x(t)= a1 a2 x(t) + a3 x(t)y(t),(1)y(t) = b1 y(t)(1 b2 y(t)) x(t)y(t),where initial conditions are x(0) = x0 > 0, y(0) = y0 > 0 and a3 is the immuneresponse to the appearance of the tumor cells.In this paper we consider the case of a3 > 0, that means that immune response is positive. For the equilibrium states P1 and P2 , we study the asymptotic behavior with respect to the parameter a1 in (1). For b1 a2 < a1 , thesystem (1) has the equilibrium states P1 (x1 , y1 ) and P2 (x2 , y2 ), withx1 =x2 = (b1 (a3 b2 a2 ) +

a1, y1 = 0,a2

)/(2a3 ), y2 = (b1 (a3 + b2 a2 )

(2)

/(2b1 b2 a3 )

(3)

where = b21 (b2 a2 a3 )2 + 4b1 b2 a1 a3 .

In [13] it is shown that there is an a10 such that if a1 < a10 , the equilibriumstate P1 is asymptotical stable, for a1 > a10 the equilibrium state P1 is unstableand if a1 < a10 the equilibrium state P2 is unstable and for a1 > a10 theequilibrium state P2 is asymptotical stable.In the following, we associate a stochastic system of differential equationsto the classical system of differential equations (1).

4Let us consider (, Ft0 , P) a filtered probability space and (W (t))t0 astandard Wiener process adapted to the filtration (F)t0 . Let {X(t) = (x(t), y(t))}t0be a stochastic process.The system of Ito equations associated to system (1) is given byRtRtx(t) = x0 + 0 (a1 a2 x(s) + a3 x(s)y(s))ds + 0 g1 (x(s), y(s))dW (s),y(t) = y0 +

Rt

((b1 y(s)(1 b2 y(s)) x(s)y(s))ds +

Rt

g2 (x(s), y(s))dW (s),(4)where the first integral is a Riemann integral, and the second one is an Itointegral. {W (t)}t0 is a Wiener process [16].The functions g1 (x(t), y(t)) and g2 (x(t), y(t)) are given in the case when weare working in the equilibrium state. In P1 those functions have the followingformg1 (x(t), y(t)) = b11 x(t) + b12 y(t) + c11 ,(5)g2 (x(t), y(t)) = b21 x(t) + b22 y(t) + c21 ,0

0

wherec11 = b11 x1 b12 y1 , c21 = b21 x1 b22 y1 .

(6)

In the equilibrium state P2 , the functions g1 (x(t), y(t)) and g2 (x(t), y(t))are given byg1 (x(t), y(t)) = b11 x(t) + b12 y(t) + c12 ,(7)g2 (x(t), y(t)) = b21 x(t) + b22 y(t) + c22 ,wherec12 = b11 x2 b12 y2 , c22 = b21 x2 b22 y2 .

(8)

The functions g1 (x(t), y(t)) and g2 (x(t), y(t)) represent the volatilisationsof the stochastic equations and they are the therapy test functions.

2.1

The analysis of SDE (4). Numerical simulation.

Using the formulae from Annexe 1, Annexe 2, and Maple 12 software,we get the following results, illustrated in the figures below. For numericalsimulations, we use the following values for the parameters of the system (4):a1 = 0.1181, a2 = 0.3747, a3 = 0.01184, b1 = 1.636, b2 = 0.002.

5The matrices A and B are given, in the equilibrium point P1 ( aa21 , 0) by

10 2a2 + a3 y1a3 x1A=, B=.y1b1 2b2 y1 x12 10In a similar way, matrices A and B are defined in the other equilibriumpoint (b (b a a ) + ) (b (b a + a ) ) 1 2 231 2 23,,P22a32b1 b2 a3with = b21 (b2 a2 a3 )2 + 4b1 b2 a1 a3 .Using the second order Euler scheme for the ODE system (1), respectivelySDE system (4), we get the following orbits.0.316

3.00.314

0.3122.5

0.310

2.00.308

0.3061.5

0.304

1.0

0.302

0.3000.50.2980

200

400

600

800

1,000

0

100

200

300

400

Fig 1: (n, x(n)) in P1 for ODE (1)

Fig 2: (n, x(n, )) in P1 for SDE (4)

optimal behavior of tumor cells


for ODE(1) in P1

for ODE(4) in P1

500

1200.11

0.10100

0.09

800.08

0.0760

0.06

40

0.05

0.0420

0.03

0

200

400

600

800

1,000

0

100

200

300

400

Fig 3: (n, y(n)) in P1 for ODE (1)

Fig 4: (n, y(n, )) in P1 for SDE (4)

optimal behavior of effector cells for

optimal behavior of effector cells

for ODE(1) in P1

for ODE(4) in P1

500

61200.11

0.10100

0.09

800.08

0.0760

0.06

40

0.05

0.0420

0.03

0.5

1.0

1.5

2.0

2.5

3.0

Fig 5: (x(n), y(n)) in P1 for ODE (1)

0.300

0.305

0.310

0.315

0.320

0.325

Fig 6: (x(n, ), y(n, )) in P1 for SDE (4)



vs effector cells for ODE(1) in P1

vs effector cells for ODE(4) in P10.4

3.00.3

2.5

0.2

0.1

2.0

0500

1.5

1,000

1,500

2,000

K

0.1

K

1.0

0.2

K

0.3

0.5

0

200

400

600

800

1,000

K

0.4

Fig 7: (n, x(n)) in P2

Fig 8: (n, x(n, )) in P2



for ODE(1) in P2

for ODE(4) in P225

120

100

20

80

1560

1040

520

00

200

400

600

800

1,000

100

200

300

400

Fig 9: (n, y(n)) in P2

Fig 10: (n, y(n, )) in P2



for ODE(1) in P2

for ODE(4) in P2

500

71203

100

2

80

1

K

60

0.4

K

0.3

K

0.2

K

0

0.1

0.1

0.2

0.3

0.4

K

1

40

K

2

20

K

3

0.5

1.0

1.5

2.0

2.5

3.0

Fig 11: (x(n), y(n)) in P2

K

2

Fig 12: (x(n, ), y(n, )) in P2





a

K

1

0

1

2

0.2

K

2.00.00.2

K

2.5

K

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

a

0.2

K

0.4

K

3.0

K

3.5

K

0.6

K

0.8

K

1.0

K

4.0

K

1.2

Fig 13: (, ()) in P1

Fig 14: (, ()) in P2

The Lyapunov exponent, for the equilibrium point P1 is negative, so P1is asymptotically stable for each R. For the equilibrium point P2 , it isasymptotically stable for all values of from the interval (1.8, 1.8), thatmeans that P2 is unstable for all (, 1.8) (1.8, ).

3

A general family of tumor-immune stochastic systems

A Volterra-like model was proposed in [17], for the interaction between apopulation of tumor cells (whose number is denoted by x) and a population

8of lymphocyte cells (y), and it is given by

x(t)= ax(t) bx(t)y(t),y(t) = dx(t)y(t) f y(t) kx(t),

(9)

where the tumor cells are supposed to be in exponential growth (which is, however, a good approximation only for the initial phases of the growth) and thepresence of tumor cells implies a decrease of the input rate of lymphocytes.A general representation for such models can be considered in the formgiven by dOnofrio in [5]:

x(t)= f1 (x(t), y(t)), y(t) = f2 (x(t), y(t)),(10)x(0) = x0 , y(0) = y0 ,where x is the number of tumor cells, y the number of effector cells of immunesystem andf1 (x(t), y(t)) = x(t)(h1 (x(t)) h2 (x(t))y(t)),

(11)

f2 (x(t), y(t)) = (h3 (x(t)) h4 (x(t)))y(t) + h5 (x(t)).The functions h1 , h2 , h3 , h4 , h5 are given such that the system (10) admits theequilibrium points P1 (x1 , y1 ), with x1 = 0, y1 > 0, and P2 (x2 , y2), with x2 6=0, y2 > 0.Particular cases, that will be discussed here, are the following:Volterra model [21] if h1 (x(t)) = a1 , h2 (x(t)) = a2 x(t), h3 (x(t)) = b3 x(t), h4 (x(t)) =b2 and h5 (x(t)) = b1 x(t);Bell model [3] h1 (x(t)) = a1 x(t), h2 (x(t)) = a2 x(t), h3 (x(t)) = b1 x(t), h4 (x(t)) =b3 and h5 (x(t)) = b2 x(t) + b4 ;Stepanova model [18] with h1 (x(t)) = a1 , h2 (x(t)) = 1, h3 (x(t)) = b1 x(t), h4 (x(t)) =b and h5 (x(t)) = b2 x(t) + b4 ;Vladar-Gonzalez model [20] if in (10) we consider h1 (x(t)) = log(K/x(t)), h2 (x(t)) =1, h3 (x(t)) = b1 x(t), h4 (x(t)) = b2 + b3 x2 (t) and h5 (x(t)) = 1;Exponential model [22] if in (10) we consider h1 (x(t)) = 1, h2 (x(t)) = 1,h3 (x(t)) = b1 x(t), h4 (x(t)) = b2 + b3 x2 (t), and h5 (x(t)) = 1;a1, h2 (x(t)) = 1,Logistic model [14] if in (10) we consider h1 (x(t)) = 1 x(t)2h3 (x(t)) = b1 x(t), h4 (x(t)) = b2 + b3 x (t), and h5 (x(t)) = 1.

9For a considered filtered probability space (, Ft0 , P) and a standardWiener process (W (t))t0 , we consider the stochastic process in two dimensional space (F)t0 .The system of Ito equations associated to system (10) is given, in theequilibrium point P (x0 , y0 ), byRtRtx(t) = x0 + 0 [x(s)(h1 (x(s)) h2 (x(s))y(s)]ds + 0 g1 (x(s), y(s))dW (s),Rty(t) = y0 + 0 [(h3 (x(s)) h4 (x(s)))y(s) + h5 (x(s))]ds+Rt+ 0 g2 (x(s), y(s))dW (s),(12)where the first integral is a Riemann integral, and the second one is an Itointegral. {W (t)}t0 is a Wiener process [16].The functions g1 (x(t), y(t)) and g2 (x(t), y(t)) are given in the case when weare working in the equilibrium state Pe , and they are given byg1 (x(t), y(t)) = b11 x(t) + b12 y(t) + c1e ,(13)g2 (x(t), y(t)) = b21 x(t) + b22 y(t) + c2e ,wherecie = bi1 xe bi2 ye , i = 1, 2,

(14)

and bij R, i, j = 1, 2.

3.1

Analysis of Bell model. Numerical simulations.

Following the algorithm for determining the Lyapunov exponent (A1) andthe description of the second order Euler scheme (A2) in Maple 12 software,we get the following results, illustrated in the figures below. For numericalsimulations we use the following values of parameters:a1 = 2.5, a2 = 1, b1 = 1, b2 = 0.4, b3 = 0.95, b4 = 2.The matrices A and B are given, in the equilibrium point P1 by

a2 y1 + a1 a2 x1,, B=A= b2 + b1 y1 b1 x1 b3with = a R, = 2. In a similar way the matrices A and B are defineda1 b3 a2 b4 a1in the equilibrium point P2 a1 b1 a2 b2 , a2 .

10

0

0.20200

400

600

800

1,000

1,200

1,400

1,600

1,800

K

0.2

K

0.19

0.4

K

0.60.18

K

0.8

K

0.17

1.0

K

1.20.160

100

200

300

400

500

600

700





for ODE(11) in P1

for ODE(12) in P1

800

2.42.45

2.2

2.402.0

1.82.35

1.6

2.301.4

1.2

2.25

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

0

100

200

300

400

500

600

700





for ODE(11) in P1

for ODE(12) in P1

800

2.42.45

2.2

2.402.0

1.82.35

1.6

2.301.4

1.2

K

1.2

K

1.0

K

0.8

K

0.6

K

0.4

K

0.2

2.25

0.0

0.16

0.17

0.18

0.19

0.20







11

01,000

2,000

3,000

4,000

5,000

6,000

0.200

K

0.50.195

K

1.00.190

K

0.185

K

0.180

1.5

2.0

K

2.5

0

100

200

300

400

500

600

700





for ODE(11) in P2

for ODE(12) in P2

800

2.4

2.44

2.2

2.02.43

1.8

2.421.6

2.41

1.4

1.22.40

1.0

2.390

1,000

2,000

3,000

4,000

5,000

6,000

0

100

200

300

400

500

600

700





for ODE(11) in P2

800

for ODE(12) in P22.4

2.44

2.2

2.02.43

1.8

2.421.6

2.41

1.4

1.22.40

1.0

K

2.5

K

2.0

K

1.5

K

1.0

K

0.5

2.390.0

0.180

0.185

0.190

0.195

0.200







12The Lyapunov exponent variation, with b11 = a variable parameter,is given in Figure 27 for the equilibrium point P1 , and in Figure 28 for theequilibrium point P2 .11

K3

K2

K1

0

K31

a

K1K2

2

K2

K1

0

3

1

a

2

3

K1K2K3

Fig 27: (, ()) in P1

Fig 28: (, ()) in P2

From the figures above, the equilibrium points P1 and P2 are asymptoticallystable for all such that the Lyapunov exponents () < 0, and unstableotherwise. So, P1 is asymptotically stable for (, 1.78) (2.02, )and P2 is asymptotically stable for (, 1.62) (1.88, ).

4

Conclusions

As considered in this paper, we used established conceptual models, but itis also very important to consider the model through all its aspects, as we havedone in this case by imposing the positivity of its solutions. Even if the initialmodel violates the positivity rule, it is valuable because it may be read as amodel which takes into account a disease-induced depression in the influx oflymphocytes. Then, instead of proposing another specific model, we preferredto add this new feature to a family of equations, and so, in particular to ourmodels chosen for study.We have focused on two important tumor-immune systems, presented fromstochastic point of view: a Kuznetsov-Taylor model and Bell model, that belongs to a general family of tumor-immune stochastic systems. We have determined the equilibrium points and we have calculated the Lyapunov exponents.A computable algorithm is presented in A1. These exponents help us to decidewhether the stochastic model is stable or not. For numerical simulations we

13have used the Euler scheme presented in detail in A2 and the implementationof this algorithm was done in Maple 12. In a similar way other models given by(11) can be studied. The model given by the SDE (12) allows the control of themodel given by ODE (1) with a stochastic process. This model is dependenton initial conditions. These are very difficult to find for a concrete case, thatis why it is quite impossible to plan an anticancer therapy based only on thismethod. This is the ony disadvantage for the immunotherapy.In our further work, we will consider the tumor-immune model with delay,and also another technique used for a successful therapy, using synchronizationof the coupled tumor-immune model of repressilators in tumor cells aggregations.

AnnexeA1 Lyapunov exponents and stability in stochastic 2dimensional structures.The behavior of a deterministic dynamical system which is disturbed bynoise may be modelled by a stochastic differential equation (SDE). In manypractical situations, perturbations are generated by wind, rough surfaces orturbulent layers are expressed in terms of white noise, modelled by brownianmotion. The stochastic stability has been introduced by Bertram and Sarachik[12] and is characterized by the negativeness of Lyapunov exponents. But itis not possible to determine this exponents explicitly. Many numerical approaches have been proposed, which generally used simulations of stochastictrajectories.Let (, F, P) a probability space. It is assumed that the algebra Ft (t 0) such thatFs Ft F, s t, s, t I,where I = [0, T ], T (0, ).Let {x(t) = (x1 (t), x2 (t))}t0 be a stochastic process. The system of Itoequationsdxi (t, ) = fi (x(t, ))dt + gi (x(t, ))dW (t, ), i = 1, 2,

(15)

with initial condition x(0) = x0 is interpreted in the sense thatZ tZ txi (t, ) = xi0 (t, ) +fi (x(s, ))ds +gi (x(s, ))dW (s, ), i = 1, 2, (16)0

0

14for almost all and forR t each t > 0, where fi (x) is a drift function,gi (x) is a diffusion function, 0 fi (x(s))ds, i = 1, 2 is a Riemann integral andRtg (x(s))dW (s), i = 1, 2 is an Ito integral. It is assumed that fi and gi , i =0 i1, 2 satisfy the conditions of existence of solutions for this SDE with initialconditions x(0) = a0 Rn .Let x0 = (x01 , x02 ) R2 be a solution of the systemfi (x0 ) = 0, i = 1, 2.

(17)

The functions gi are chosen such thatgi(x0 ) = 0, i = 1, 2.In the following, we will considergi (x) =

2Xj=1

bij (xj x0j ), i = 1, 2,

where bij R, i = 1, 2.The liniarized of system (16) in x0 is given byZ tZ tX(t) =AX(s)ds +BX(s)dW (s),0

where

x(t, )X(t) =,y(t, )

(18)

(19)

0

a11 a12b11 b12A=, B=(20)a21 a22b21 b22gi fi (21)aij = , bij = .xj x0xj x0The Oseledec multiplicative ergodic theorem [15] asserts the existence oftwo non-random Lyapunov exponents 2 1 = . The top Lyapunov exponent is given byp1 = lim sup log x(t)2 + y(t)2.(22)t tApplying the change of coordinatesx(t) = r(t) cos (t), x(t) = r(t) sin (t),by writing the Ito formula for1log(x2 + y 2) = log(r),2y,h2 (x, y) = arctanx

h1 (x, y) =

results

15Proposition 1Z t r(t) Z t122=q1 ((s)) + (q4 ((s)) q2 ((s)) )ds +q2 ((s))dW (s),logr(0)200(23)ZZt

(t) = (0) +

0

where

t

(q3 ((s)) q2 ((s))q4 ((s)))ds +

q4 ((s))dW (s),

(24)

0

q1 () = a11 cos2 + (a12 + a21 ) cos sin + a22 sin2 ,q2 () = b11 cos2 + (b12 + b21 ) cos sin + b22 sin2 ,q3 () = a21 cos2 + (a22 a11 ) cos sin a12 sin2 ,q4 () = b21 cos2 + (b22 b11 ) cos sin b12 sin2 .

(25)

As the expectation of the Ito stochastic integral is null,Z tEq2 ((s))dW (s) = 0,0

the Lyapunov exponent is given byZ t r(t) 111= lim E[q1 ((s)) + (q4 ((s))2 q2 ((s)))]ds. = lim logtt tr(0)t20Applying the Oseledec theorem, if r(t) is ergodic, results thatZ t1=[q1 () + (q4 ()2 q2 ())]p()d,20where p() is the probability distribution of the process .

(26)2

An approximation of this distribution is calculated by solving the FokkerPlanck equation. Associated with equation (24) for p = p(t, ) we getp1 2+ (q3 () q2 ()q4 ()p) (q4 ()2 p) = 0.t 2 2

(27)

From (27) results that the solution p() of the Fokker-Planck equation isthe solution of the following first order equation1(q3 () + q1 ()q4 () + q2 ()q5 ())p() + q4 ()2 p () = p0 ,2where p () =

dpd

(28)

and

q5 () = (b12 + b21 ) sin 2 (b22 b11 ) cos 2.

(29)

16Proposition 2 If q4 () 6= 0, the solution of equation (28) is given byZ Kp() =(1 + D(u)du),D()q4 ()20where K is determined by the normality conditionZ 2p()d = 1,

(30)

(31)

0

and

D(2) 1 = R 2.D(u)du0

The function D is given by

D() = exp 2

Z

0

q3 (u) q2 (u)q4 (u) q4 (u)q5 (u) du .q4 (u)2

(32)

(33)2

A numerical solution of the phase distribution could be performed by asimple backward difference scheme.Let N R+ and h = N . Letq1 (i) = a11 cos2 (ih) + (a12 + a21 ) cos(ih) sin(ih) + a22 sin2 (ih),q2 (i) = b11 cos2 (ih) + (b12 + b21 ) cos(ih) sin(ih) + b22 sin2 (ih),q3 (i) = a21 cos2 (ih) + (a22 a11 ) cos(ih) sin(ih) a12 sin2 (ih),q4 (i) = b21 cos2 (ih) + (b22 b11 ) cos(ih) sin(ih) b12 sin2 (ih),q5 (i) = (b12 + b21 ) sin(2ih) (b22 b11 ) cos(2ih).The p(i), i = 0, ..., N is given by the following relationsp(i) = (p(0) +whereF (i) =

q4 (i)2 p(i 1))F (i),2h

2h.2h(q3 (i) + q2 (i)q4 (i) + q4 (i)q5 (i)) + q4 (i)2

The Lyapunov function is = (N), where(N) =

NX

1(q1 (i) + (q4 (i)2 q2 (i)2 ))p(i)h.2i=1

(34)

17Proposition 3 If the matrix B is given byb11 = , b12 = , b21 = , b22 = ,probability distribution p() is given by111Kexp{((aa)+(aa)cos2+(a21 a12 ) sin 2},211211222222

p() =

K = R 20

2exp{ 12 ((a21 a12 ) + 12 (a11 a22 ) cos 2 + 12 (a21 a12 ) sin 2}

d,

and the Lyapunov exponent is given by111 = (a11 + a22 + 2 2 ) + (a11 a22 )c2 + (a21 + a12 )s2 ,222whereZ2Z2c2 = cos(2)p()d, s2 = sin(2)p()d.0

0

2

A2 The Euler scheme.In general 2-dimensional case, the Euler scheme has the form:xi (n + 1) = xi (n) + fi (x(n))h + gi (x(n))Gi (n), i = 1, 2,

(35)

with Wiener process incrementGi (n) = Wi ((n + 1)h) Wi (nh), n = 0, ..., N 1, i = 1, 2,and (n) = (nh, ), Gi (n) are generated using boxmuller method.It is shown that Euler scheme has the order for weak convergence 1, forsufficiently regular drift and diffusion coefficients.We assume that fi and gi in (35) are sufficiently smooth such that thefollowing schemas are well defined.The second order Euler scheme is defined by the relationsGi (n)2 hgi (x(n))+xi (n)2h 2 fi (x(n)) i h2 hfi (x(n))fi (x(n)) 1+ (gi (x(n))2+ gi(x(n))+ fi (x(n))xi (n)2xi (n)xi (n) 2xi (n)i2gi (x(n)) 1 gi (x(n)) hGi (n)+ fi (x(n))+ (gi(x(n))2, i = 1, 2,xi (n)2xi (n)xi (n)2

xi (n + 1) = xi (n) + fi (x(n))h + gi (x(n))Gi (n) + gi (x(n))

18where we used the random variables Gi (n), i = 1, 2. In [11], it is shown thatthese schemes converge weakly with order 2.

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