WANG HE-YUANAND CUI YAN,2

(1.School of Science,Liaoning University of Technology,Jinzhou,Liaoning,121001)

(2.Faculty of Information Science and Engineering,Northeastern University, Shenyang,110004)

A Nine-modes Truncation of the Plane Incompressible Navier-Stokes Equations?

WANG HE-YUAN1AND CUI YAN1,2

(1.School of Science,Liaoning University of Technology,Jinzhou,Liaoning,121001)

(2.Faculty of Information Science and Engineering,Northeastern University, Shenyang,110004)

In this paper a nine-modes truncation of Navier-Stokes equations for a two-dimensional incompressible fl uid on a torus is obtained.The stationary solutions, the existence of attractor and the global stability of the equations are fi rmly proved. What is more,that the force f acts on the mode k3and k7respectively produces two systems,which lead to a much richer and varied phenomenon.Numerical simulation is given at last,which shows a stochastic behavior approached through an involved sequence of bifurcations.

the Navier-Stokes equation,the strange attractor,Lyapunov function, bifurcation,chaos

1 Introduction

Navier-Stokesequations as very interesting complicated nonlinear equations have been widely studied in the last fi ve decades within many subjects.Such equations always exhibit a rich phenomenology as parameters go through certain values,and it attracts many scientists’attention.In the last century,the scientists like Lorenz and Franceshini did a lot of work on such equations and obtained many valuable achievements(see[1]–[7]).By studying their papers,recently we get a nine-modes model by extending their fi ve-mode equations(see[1]). The new equations exhibit a much richer and varied phenomenology with a large range of Reynolds number.It appears to us how the phenomenology changes with the addition of new modes.Moreover,by putting the force either on k3or k7,the systems both become to be chaos,which enrich the theory of Franceschini and Tebaldi[2].

2 Nine-modes Lorenz-like Equations

Consider the incompressible Navier-Stokes equations:

on the torus T2=[0,2π]×[0,2π],where u is the velocity field,p is the pressure and f is (periodic)volume force.We expand u,p,f in Fourier series on a torus T2=[0,2π]×[0,2π]:

where K=(k1,k2)is a“wave vector”,with integer components,K⊥=(k2,?k1),and the reality condition rK=??Kholds.Substituting(2.4),(2.5),(2.6)into(2.1)we get formally the following equations for{rK}(rK=rK(t)is a function of t)

where L is a set of wave vectors,such that if K∈L,also?K∈L.

In paper[2],a seven modes truncation of(2.7)was got.As the modes increase,it may lead to a much richer and varied phenomenon.But at the same time,with modes increasing, the calculations will be more and more complicated.

Suppose

and take L as the set of vectors

and their opposites.When ν=1,choose K as Ki(i=1,···,9)respectively in(2.7)and make the following transform:

Taking the force acting on each mode,after calculating we find that when it acts on mode k3and k7respectively,a complicated behavior about chaos exhibited.

Let

Then we obtain the following system(2.8)and(2.9):

where xi=xi(t)(i=1,2,···,9)is the spectral expanding coefficient.The nine-modes nonlinear differential equations are very like Lorenz system in the form,so we call it ninemodes Lorenz-like system.In the following,we choose the system(2.8)to be studied.And equations(2.8)have the following symmetry properties:

3 The Stationary Solution and Their Stability Properties

We consider the model(2.8)and let

Then we get a Lyapunov matrix as follows:

Suppose F(X,Re)=0,with a lot of complex calculation,we get the stationary solutions of the model(2.8).By changing Lyapunov characteristic exponent,we obtained the following properties:

which turns out to be stable and numerical evidence suggests that the above solution is a global attractor.

(ii)For R1＜Re＜R2=6.6489,there are 3 stationary solutions:the old one S1,which has become unstable(as a consequence of the crossing of the imaginary axis by one of the eigenvalues of the Lyapunov matrix)and two additional ones S2,S3,with components

Numerical evidences indicate that any randomly chosen initial data is attracted by either S2or S3,so S2,S3are global attractors.When Re=R2,a pair of complex conjugate eigenvalues become positive,so we have the following conclusion:

(iii)For Re＞R2,all the solutions of(2.8)become unstable.

4 The Existence of Attractor and Analysis of Global Stability

4.1 The Existence of Attractor

For the system(2.8),we assume that H=R9,u(t)=(x1,···,x9),and do the following calculation:

If ρ is big enough,B(0,ρ)is not only a functional invariant set but also an absorbing set. As a result the system(2.8)has global attractors(see[10]).

4.2 The Global Stability

In order to verify its global stability,let us use Lyapunov’s second method(see[5]and[6]). At first,construct a Lyapunov function:

where k is a constant.So it is a sphere on H,which is denoted by E.Then in terms of (2.8),it would yield

5 Numerical Simulation

When R2=6.6489＜Re＜R7=650,the system exhibits a complicated behavior and often possesses several periodic and quasiperiodic solutions at the same time for rather large ranges of the parameters.In order to show them clearly,we choose the view of x1,x3,x7and x3,x4,x6at the similar period(with some plane picture)to study as follows:

(1)At Re=R2,the stationary solutions of S1become unstable because a pair of complex conjugate eigenvalues cross the imaginary axis,and the stable periodic orbits around the fixed points S1arise via a Hopf bifurcation as predicted by the general theory of bifurcation (see[8]).

(2)At R2＜Re＜R3=28.5,the stationary solutions S2and S3of the system(2.8)is stable,namely,the orbit of the system(2.8)in Fig.5.1 is stable and it tends to the limit loop in Fig.5.2.

Fig.5.1

Fig.5.2

(3)When Re=R3=28.5,the stationary solutions of S2and S3become unstable because a pair of complex conjugate eigenvalues cross the imaginary axis,and the stable periodic orbits around the fixed points S2,S3arise via a Hopf bifurcation.Its orbit is shown in Fig.5.2(a),and Fig.5.2(b)shows its projection with the view of x4,x6.

(4)At R3＜Re＜R4=33.93,the periodic orbits lose stability,there is a quasi-periodic close orbit(see Fig.5.3).

Fig.5.3

At R4＜Re＜R5=51.47,the computer simulation shows that the orbit considered in Fig.5.3 becomes a new quasi-periodic orbit(see Fig.5.4).

Fig.5.4

(5)At Re≥R6=54.5,the attractor shrinks into a symmetrical loop for a long time (see Fig.5.5).

Fig.5.5

(6)At Re≥R7=650,new orbits appear,and it becomes to be chaos(see Fig.5.6).

Fig.5.6

6 Conclusions

This article has presented and studied a nine-modes Lorenz-like system of the Navier-stokes equations for a two-dimensional incompressible fl uid on a torus.Dynamical behaviors of this new chaotic system,including some basic dynamical properties,bifurcations,and routes to chaos,etc.,have been investigated both theoretically and numerically by changing Reynolds number.Our purpose is to study how the phenomenology of the model changes when the number of modes in the truncation is slightly increased,or to find how to get a Navier-Stokes truncation.Even if the nine-mode model studied does not reproduce the interesting phenomena of the previous ones(see[1]and[2]),we think that it is also interesting by itself. Furthermore,the existence of attractor and the global stability of the equations have been fi rmly veri fi ed and these theories can be used in other similar systems.

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[3]Franceschini,V.and Zanasi,R.,Three-dimensional Navier-Stokes equations trancated on a torus,Nonlinearity,4(1992),189–209.

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[9]Liu,B.Z.and Peng,J.H.,Nonlinear Dynamics,Advanced Education Press,Beijing,2004.

[10]Li,K.T.and Ma,Y.C.,Hilbert Space Method for Equations of Mathematical Physics, Jiaotong University Press,Xi’an,1992.

[11]Xie,Y.Q.,Mathematical Method of Nonlinear Dynamic,Meteorology Press,Beijing,2001.

Communicated by Yin Jing-xue

76D05,58F13

A

1674-5647(2011)04-0297-10

date:March 2,2007.

The NSF(10571142)of China and the Science and Research Foundation(L2010178)of Liaoning Education Committee.