# Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow

Authors: Nejib Smaoui
Journal: Complexity
 (2018)

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## Abstract

The symmetries, dynamics, and control problem of the two-dimensional (2D) Kolmogorov flow are addressed. The 2D Kolmogorov flow is known as the 2D Navier-Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along thex-direction. First, using the Fourier Galerkin method on the original 2D Navier-Stokes equations, we obtain a seventh-order system of nonlinear ordinary differential equations (ODEs) which approximates the behavior of the Kolmogorov flow. The dynamics and symmetries of the reduced seventh-order ODE system are analyzed through computer simulations for the Reynolds number range 0<Re<26.41. Extensive numerical simulations show that the obtained system is able to display the different behaviors of the Kolmogorov flow. Then, we design Lyapunov based controllers to control the dynamics of the system of ODEs to different attractors (e.g., a fixed point, a periodic orbit, or a chaotic attractor). Finally, numerical simulations are undertaken to validate the theoretical developments.


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## Body

## 1. Introduction

In recent years, a lot of efforts have been devoted to construct dynamical systems that arise from solving the 2D Navier-Stokes equations. In the literature, the dynamics of the Navier-Stokes equations were approximated by using several reduced order models [1–20]. The concept of approximate inertial manifold (AIM) (see [1–11] and references therein) and Fourier Galerkin methods [12–20] is among the methods used to obtain such reduced order systems with the task of approximating the long-time behavior of the 2D Navier-Stokes equations. The behavior of the 2D Navier-Stokes equations depends on the nature of the forcing. When the force is of a single mode, the 2D N-S equations with periodic boundary conditions are known as the 2D Kolmogorov flow.During the last three decades, numerous numerical studies of the 2D Kolmogorov flow with different forcing terms have appeared [11–29]. In 1981, Franceschini et al. [14–17] constructed a system of ODEs approximating the dynamics of the Kolmogorov flow when the force acts on the mode (2,-1). In [15], a number of steady states and Hopf bifurcation have been observed up to Reynolds number 100. In 1996, Armbruster et al. [25] analyzed the dynamics of the 2D Navier-Stokes equation when the force acts on the mode (0,2) by exploiting as much as possible the symmetries in the problem and by using the Karhunen-Loeve decomposition (K-L). Later on and in 1997, Smaoui and Armbruster [27] described a computationally effective way to obtain a reduced order equivariant system of the 2D N-S equations using K-L decomposition with symmetries; the reduced order system obtained consists of 12 nonlinear ODEs. They showed that when Re = 16.6, the ODE system exhibits the same dynamics as the original simulation of the 2D N-S equations. However, the obtained Galerkin system does not show signs of a homoclinic behavior as observed in the original PDE simulation [25].On the other hand, the control problem of the Navier-Stokes equations and especially to the 2D Navier-Stokes equations has not been completely investigated (see [30–38] and the references therein). In 2003, using a global pinning coupling strategy, Guan et al. [34] designed an adaptive controller to control flow turbulence governed by the 2D Navier-Stokes equations. In 2009, Gambino et al. [35] designed an adaptive controller to drive the state of the system to the stationary solution. Recently, Smaoui and Zribi [36–38] constructed reduced order ODE models using the truncated Fourier expansion method for approximating the dynamics of the 2D N-S equations when the force acts on the mode (0,α). They showed that, for α=4, the dynamics of the reduced order models exhibit periodic doubling bifurcation leading to chaotic attractors. In addition, they designed Lyapunov based control laws to drive the states of the reduced order model to the basic state solution and to synchronize two reduced order ODE models having different Reynolds and starting from two different initial conditions.In this paper, we construct a system of seven ODEs that approximates the dynamics of Kolmogorov flow when the force acts on the mode(0,2). It should be noted that the approach used in this paper to construct the system of ODEs is different than the one used in [36–38]. In addition, unlike the system obtained earlier by Smaoui and Armbruster [27], this system shows a homoclinic gluing bifurcation similar to the one observed in the original PDE simulations [25]. Furthermore, Lyapunov based control laws are designed to control the dynamics of the system for a given Re.The paper is organized as follows. In Section2, the 2D Navier-Stokes equations are presented and a seventh-order nonlinear ODE system is obtained to approximate the behavior of these equations. The reduced order ODE system is also analyzed in Section 2. Section 3 presents the design of a control law which is used to regulate the states of the reduced order ODE system to a desired fixed point. Section 4 deals with the design of a control scheme to synchronize two reduced order systems obtained from the 2D N-S equations having the same or different Reynolds number but they start from different initial conditions. Finally, some concluding remarks are given in Section 5.


## 2. The 2D Kolmogorov Flow

### 2.1. The Seven-Mode Reduced Order System of the 2D Kolmogorov Flow

The “basic 2D Kolmogorov flow”u→=(αsinαy,0) was introduced by Kolmogorov in 1958 as an example on which to study transition to turbulence [39]. This basic flow is the solution of the 2D Navier-Stokes equations with periodic boundary conditions in two directions 0≤x,y≤2π given by(1)ut→+u→·∇u→+∇p=ν∇2u→+f→,∇·u→=0,with u→=(u1,u2) and force f→=(α3νsinαy,0), which is assumed to be stationary and spatially biperiodic. The kinematic viscosity is ν=1/Re, where Re is the Reynolds number and the pressure is p.The perturbed nondimensional vorticityω=-Δϕ′ formulation of the Kolmogorov flow is(2)∂Δϕ′∂t~=Δ2ϕ′-Re∂∂xΔϕ′∂ϕ′∂y-∂∂yΔϕ′∂ϕ′∂x-αResin⁡αy∂∂xΔϕ′+α2ϕ′, where the scaled time t~=νt, u1=∂ϕ/∂y, u2=-∂ϕ/∂x, and ϕ′=ϕ+cosαy.In Smaoui and Zribi [36–38], the derivation of the reduced order system of ODEs was based on expanding the stream function ϕ′ in (2) as follows:(3)ϕ′x,y,t=∑n=1∞ϕ0,n′tcos⁡αny+∑m=12∑n=-∞∞ϕm,n′tcos⁡mx+my+αny.In this paper, we use a totally different approach than the one used in [36–38]. We derive a system of seven ODEs from the Navier-Stokes equations by expanding u→ in the following form:(4)u→x=∑k≠0exp⁡ik·xγkk⊥k,where k=(k1,k2) is a wave vector with integer components, k⊥=(k2,-k1), and the reality condition γk=-γ¯-k must hold.The equation for{γk}k≠0 is(5)γ˙k=-i∑k1+k2+k=0k1⊥·k2k22-k122k1k2kγ¯k1γ¯k2-νk2γk+fk,where γk=-γ¯-k and fk is the component of f→ with respect to k⊥/|k|exp(ik·x).Next, if the basic flowu→=(2sin2y,0), then one can consider the set of vectors, k1=(1,1), k2=(0,3), k3=(0,2), k4=(1,2), k5=(0,1), k6=(1,0), and k7=(1,-2), and their negatives in (5) to obtain the following system of ODEs:(6)γ˙k1=-i410γk4γ¯k5+-410γk2γk7-2νγk1+fk1,γ˙k2=-i310γk1γ¯k7-9νγk2+fk2,γ˙k3=-i-45γk4γ¯k6+45γk6γ¯k7-4νγk3+fk3,γ˙k4=-i110γk1γk5+-35γk3γk6-5νγk4+fk4,γ˙k5=-i-310γ¯k1γk4+-12γk1γ¯k6-νγk5+fk5,γ˙k6=-i12γk1γ¯k5+15γ¯k3γk4+15γk3γk7-νγk6+fk6,γ˙k7=-i-710γk1γ¯k2+-35γ¯k3γk6-5νγk7+fk7.Letγk1=ix1, γk2=x2, γk3=-ix3, γk4=x4, γk5=x5, γk6=x6, and γk7=x7; then system (6) becomes(7)x˙1=-410x4x5+410x2x7-2νx1-ifk1,x˙2=310x1x7-9νx2+fk2,x˙3=-45x4x6+45x6x7-4νx3+ifk3,x˙4=110x1x5+35x3x6-5νx4+fk4,x˙5=310x1x4-12x1x6-νx5+fk5,x˙6=12x1x5+15x3x4-15x3x7-νx6+fk6,x˙7=-710x1x2-35x3x6-5νx7+fk7.After changing the length scalexi=10νxi~ for i=1,…,7, the time scale t=t~/ν and ν=10/5ν~, and considering the forcing term to act on mode k3, system (7) becomes(8)x˙1=-2x1-4x4x5+4x2x7,x˙2=-9x2+3x1x7,x˙3=-4x3-42x4x6+42x6x7+4Re,x˙4=-5x4+x1x5+32x3x6,x˙5=-x5+3x1x4-5x1x6,x˙6=-x6+5x1x5+2x3x4-2x3x7,x˙7=-5x7-7x1x2-32x3x6.The system given by the equations in (8) can be written in the following form: (9)x˙=Ax+fx,where the the vector x is such that x=[x1x2x3x4x5,x6,x7]T and the diagonal matrix A is such that:(10)A=diag⁡-2,-9,-4,-5,-1,-1,-5,and the nonlinear vector f(x) is such that(11)f=f1f2f3f4f5f6f7Twith(12)f1=-4x4x5+4x2x7,f2=3x1x7,f3=-42x4x6+42x6x7+4Re,f4=x1x5+32x3x6,f5=3x1x4-5x1x6,f6=5x1x5+2x3x4-2x3x7,f7=-7x1x2-32x3x6.Remark 1.
It is noted that system (8) is invariant under the following symmetries:(13)rx:x1,x2,x3,x4,x5,x6,x7⟺x1,-x2,x3,-x4,-x5,-x6,-x7,ry:x1,x2,x3,x4,x5,x6,x7⟺-x1,-x2,x3,x4,-x5,x6,x7,ro:x1,x2,x3,x4,x5,x6,x7⟺-x1,x2,x3,-x4,x5,-x6,-x7,where rx,  ry, and ro are reflection symmetries across the x-axis, the y-axis, and the origin, respectively. Hence, rx, ry, ro with the identity transformation i form an Abelian group: G=rx,ry,ro,i.


### 2.2. The Dynamics of the Seven-Mode Truncation System

In this subsection, we analyze the dynamics of the seven-mode truncation ODE system presented in (8) for different Reynolds numbers. The DsTool software [40] is used in all numerical simulations presented in this section using the 4th-order Runge-Kutta method as the numerical integrator and with the time step Δt=0.0001.(i)
For0≤Re≤R1≃0.645, the basic fixed point P0=(0,0,Re,0,0,0,0) is the only stable solution and, numerically, it is globally attractive for all Re≤R1. This is a special case of the general results on the theory of Navier-Stokes equations [41].(ii)
ForR1<Re≤R2≃2.469 there are three fixed points: the old one P0, which becomes unstable as a result of the crossing of the imaginary axes by one of the eigenvalues of the Lyapunov matrix, and two more, P1 and P2, that bifurcated from P0 are stable and attracting. Numerical evidence shows that any randomly chosen initial condition is either attracted by P1 or P2.(iii)
ForR2<Re≤R3≃20.844, there are seven fixed points: four asymptotically stable and three unstable. Figure 1 presents the phase portrait of the four asymptotically stable fixed points at Re=18. The first asymptotically state fixed point (Figure 1, top left) was obtained using the following initial conditions:(14)x1x2x3x4x5x6x7T=-0.8275-0.47030.7234-2.3355-0.1220-3.59900.4255.The second asymptotically stable fixed point (Figure1, bottom right) was obtained by applying the reflection symmetry ro on the initial conditions used for the first asymptotically stable fixed point. That is,(15)x1x2x3x4x5x6x7T=ro-0.8275-0.47030.7234-2.3355-0.1220-3.59900.4255=0.8275-0.47030.72342.3355-0.12203.5990-0.4255.The third asymptotically stable fixed point (Figure1, top right) was generated by applying the reflection symmetry rx on the initial conditions used for the second asymptotically stable fixed point. That is,(16)x1x2x3x4x5x6x7T=rx0.8275-0.47030.72342.3355-0.12203.5990-0.4255=0.82750.47030.7234-2.33550.1220-3.59900.4255.The fourth asymptotically stable fixed point (Figure1, bottom left) was obtained by applying the reflection symmetry ry on the initial conditions used for the second asymptotically stable fixed point. That is,(17)x1x2x3x4x5x6x7T=ry0.8275-0.47030.72342.3355-0.12203.5990-0.4255=-0.82750.47030.72342.33550.12203.5990-0.4255.(iv)
ForR3<Re≤R4≃26.323 the four asymptotically stable fixed points become unstable because a pair of complex eigenvalues crosses the imaginary axes, and bifurcate via a Hopf bifurcation into a four stable periodic orbits around these four fixed points. The four stable periodic orbits remain stable up to Re=R4=26.323. The other three remain unstable (see Figure 2).(v)
ForR4<Re≤R5≃26.375. At Re=R4=26.323 each of the four periodic orbits loses stability and bifurcates into a new periodic orbit with double the period.(vi)
ForR5<Re≤R6≃26.41, a homoclinic gluing bifurcation similar to the one observed by Armbruster et al. [25] occurs (see Figure 3). It should be noted that the homoclinic gluing bifurcation observed leads to two chaotic or strange attractors each connecting a pair of two stable periodic orbits. Figure 3(b) shows that the dynamics of one of the chaotic attractors follow a quasiperiodic regime; then either bursts to the same attractor or to its symmetric counterpart; then other bursts follow. Numerical simulations indicate that intervals between bursts become shorter as the Reynolds number, Re, is increased up to Re=R6.Figure 1
The phase portrait of the four asymptotically stable fixed points atRe=18; the first asymptotically stable fixed point (top left) is obtained using the following initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (−0.8275, −0.4703, 0.7234, −2.3355, −0.122, −3.599, 0.4255); the second asymptotically stable fixed point (bottom right) is obtained using the initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (0.8275, −0.4703, 0.7234, 2.3355, −0.122, 3.599, −0.4255) = ro(−0.8275, −0.4703, 0.7234, −2.3355, −0.122, −3.599, 0.4255); the third asymptotically stable fixed point (top right) is generated using the initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (0.8275, 0.4703, 0.7234, −2.3355, 0.122, −3.599, 0.4255) = rx(0.8275, −0.4703, 0.7234, 2.3355, −0.122, 3.599, −0.4255); the fourth asymptotically stable fixed point (bottom left) is generated using the initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (−0.8275, 0.4703, 0.7234, 2.3355, 0.122, 3.599, −0.4255) = ry(0.8275, −0.4703, 0.7234, 2.3355, −0.122, 3.599, −0.4255).Figure 2
Phase portraits of the four stable periodic orbits arising from a Hopf bifurcation atRe=20.845 that remain stable for up to Re=26.323 and bifurcates again into a new four periodic orbits with double the period.
(a)
R e = 20.845 (b)
R e = 23 (c)
R e = 25 (d)
R e = 26.37Figure 3
(a) Phase portraits of the bursting phenomena resulting from a homoclinic gluing bifurcation connecting a pair of two stable periodic orbits. (b) Time series of the statex1 versus time at Re=26.377.
(a)
R e = 26.377 (b)
R e = 26.377


## 2.1. The Seven-Mode Reduced Order System of the 2D Kolmogorov Flow

The “basic 2D Kolmogorov flow”u→=(αsinαy,0) was introduced by Kolmogorov in 1958 as an example on which to study transition to turbulence [39]. This basic flow is the solution of the 2D Navier-Stokes equations with periodic boundary conditions in two directions 0≤x,y≤2π given by(1)ut→+u→·∇u→+∇p=ν∇2u→+f→,∇·u→=0,with u→=(u1,u2) and force f→=(α3νsinαy,0), which is assumed to be stationary and spatially biperiodic. The kinematic viscosity is ν=1/Re, where Re is the Reynolds number and the pressure is p.The perturbed nondimensional vorticityω=-Δϕ′ formulation of the Kolmogorov flow is(2)∂Δϕ′∂t~=Δ2ϕ′-Re∂∂xΔϕ′∂ϕ′∂y-∂∂yΔϕ′∂ϕ′∂x-αResin⁡αy∂∂xΔϕ′+α2ϕ′, where the scaled time t~=νt, u1=∂ϕ/∂y, u2=-∂ϕ/∂x, and ϕ′=ϕ+cosαy.In Smaoui and Zribi [36–38], the derivation of the reduced order system of ODEs was based on expanding the stream function ϕ′ in (2) as follows:(3)ϕ′x,y,t=∑n=1∞ϕ0,n′tcos⁡αny+∑m=12∑n=-∞∞ϕm,n′tcos⁡mx+my+αny.In this paper, we use a totally different approach than the one used in [36–38]. We derive a system of seven ODEs from the Navier-Stokes equations by expanding u→ in the following form:(4)u→x=∑k≠0exp⁡ik·xγkk⊥k,where k=(k1,k2) is a wave vector with integer components, k⊥=(k2,-k1), and the reality condition γk=-γ¯-k must hold.The equation for{γk}k≠0 is(5)γ˙k=-i∑k1+k2+k=0k1⊥·k2k22-k122k1k2kγ¯k1γ¯k2-νk2γk+fk,where γk=-γ¯-k and fk is the component of f→ with respect to k⊥/|k|exp(ik·x).Next, if the basic flowu→=(2sin2y,0), then one can consider the set of vectors, k1=(1,1), k2=(0,3), k3=(0,2), k4=(1,2), k5=(0,1), k6=(1,0), and k7=(1,-2), and their negatives in (5) to obtain the following system of ODEs:(6)γ˙k1=-i410γk4γ¯k5+-410γk2γk7-2νγk1+fk1,γ˙k2=-i310γk1γ¯k7-9νγk2+fk2,γ˙k3=-i-45γk4γ¯k6+45γk6γ¯k7-4νγk3+fk3,γ˙k4=-i110γk1γk5+-35γk3γk6-5νγk4+fk4,γ˙k5=-i-310γ¯k1γk4+-12γk1γ¯k6-νγk5+fk5,γ˙k6=-i12γk1γ¯k5+15γ¯k3γk4+15γk3γk7-νγk6+fk6,γ˙k7=-i-710γk1γ¯k2+-35γ¯k3γk6-5νγk7+fk7.Letγk1=ix1, γk2=x2, γk3=-ix3, γk4=x4, γk5=x5, γk6=x6, and γk7=x7; then system (6) becomes(7)x˙1=-410x4x5+410x2x7-2νx1-ifk1,x˙2=310x1x7-9νx2+fk2,x˙3=-45x4x6+45x6x7-4νx3+ifk3,x˙4=110x1x5+35x3x6-5νx4+fk4,x˙5=310x1x4-12x1x6-νx5+fk5,x˙6=12x1x5+15x3x4-15x3x7-νx6+fk6,x˙7=-710x1x2-35x3x6-5νx7+fk7.After changing the length scalexi=10νxi~ for i=1,…,7, the time scale t=t~/ν and ν=10/5ν~, and considering the forcing term to act on mode k3, system (7) becomes(8)x˙1=-2x1-4x4x5+4x2x7,x˙2=-9x2+3x1x7,x˙3=-4x3-42x4x6+42x6x7+4Re,x˙4=-5x4+x1x5+32x3x6,x˙5=-x5+3x1x4-5x1x6,x˙6=-x6+5x1x5+2x3x4-2x3x7,x˙7=-5x7-7x1x2-32x3x6.The system given by the equations in (8) can be written in the following form: (9)x˙=Ax+fx,where the the vector x is such that x=[x1x2x3x4x5,x6,x7]T and the diagonal matrix A is such that:(10)A=diag⁡-2,-9,-4,-5,-1,-1,-5,and the nonlinear vector f(x) is such that(11)f=f1f2f3f4f5f6f7Twith(12)f1=-4x4x5+4x2x7,f2=3x1x7,f3=-42x4x6+42x6x7+4Re,f4=x1x5+32x3x6,f5=3x1x4-5x1x6,f6=5x1x5+2x3x4-2x3x7,f7=-7x1x2-32x3x6.Remark 1.
It is noted that system (8) is invariant under the following symmetries:(13)rx:x1,x2,x3,x4,x5,x6,x7⟺x1,-x2,x3,-x4,-x5,-x6,-x7,ry:x1,x2,x3,x4,x5,x6,x7⟺-x1,-x2,x3,x4,-x5,x6,x7,ro:x1,x2,x3,x4,x5,x6,x7⟺-x1,x2,x3,-x4,x5,-x6,-x7,where rx,  ry, and ro are reflection symmetries across the x-axis, the y-axis, and the origin, respectively. Hence, rx, ry, ro with the identity transformation i form an Abelian group: G=rx,ry,ro,i.


## 2.2. The Dynamics of the Seven-Mode Truncation System

In this subsection, we analyze the dynamics of the seven-mode truncation ODE system presented in (8) for different Reynolds numbers. The DsTool software [40] is used in all numerical simulations presented in this section using the 4th-order Runge-Kutta method as the numerical integrator and with the time step Δt=0.0001.(i)
For0≤Re≤R1≃0.645, the basic fixed point P0=(0,0,Re,0,0,0,0) is the only stable solution and, numerically, it is globally attractive for all Re≤R1. This is a special case of the general results on the theory of Navier-Stokes equations [41].(ii)
ForR1<Re≤R2≃2.469 there are three fixed points: the old one P0, which becomes unstable as a result of the crossing of the imaginary axes by one of the eigenvalues of the Lyapunov matrix, and two more, P1 and P2, that bifurcated from P0 are stable and attracting. Numerical evidence shows that any randomly chosen initial condition is either attracted by P1 or P2.(iii)
ForR2<Re≤R3≃20.844, there are seven fixed points: four asymptotically stable and three unstable. Figure 1 presents the phase portrait of the four asymptotically stable fixed points at Re=18. The first asymptotically state fixed point (Figure 1, top left) was obtained using the following initial conditions:(14)x1x2x3x4x5x6x7T=-0.8275-0.47030.7234-2.3355-0.1220-3.59900.4255.The second asymptotically stable fixed point (Figure1, bottom right) was obtained by applying the reflection symmetry ro on the initial conditions used for the first asymptotically stable fixed point. That is,(15)x1x2x3x4x5x6x7T=ro-0.8275-0.47030.7234-2.3355-0.1220-3.59900.4255=0.8275-0.47030.72342.3355-0.12203.5990-0.4255.The third asymptotically stable fixed point (Figure1, top right) was generated by applying the reflection symmetry rx on the initial conditions used for the second asymptotically stable fixed point. That is,(16)x1x2x3x4x5x6x7T=rx0.8275-0.47030.72342.3355-0.12203.5990-0.4255=0.82750.47030.7234-2.33550.1220-3.59900.4255.The fourth asymptotically stable fixed point (Figure1, bottom left) was obtained by applying the reflection symmetry ry on the initial conditions used for the second asymptotically stable fixed point. That is,(17)x1x2x3x4x5x6x7T=ry0.8275-0.47030.72342.3355-0.12203.5990-0.4255=-0.82750.47030.72342.33550.12203.5990-0.4255.(iv)
ForR3<Re≤R4≃26.323 the four asymptotically stable fixed points become unstable because a pair of complex eigenvalues crosses the imaginary axes, and bifurcate via a Hopf bifurcation into a four stable periodic orbits around these four fixed points. The four stable periodic orbits remain stable up to Re=R4=26.323. The other three remain unstable (see Figure 2).(v)
ForR4<Re≤R5≃26.375. At Re=R4=26.323 each of the four periodic orbits loses stability and bifurcates into a new periodic orbit with double the period.(vi)
ForR5<Re≤R6≃26.41, a homoclinic gluing bifurcation similar to the one observed by Armbruster et al. [25] occurs (see Figure 3). It should be noted that the homoclinic gluing bifurcation observed leads to two chaotic or strange attractors each connecting a pair of two stable periodic orbits. Figure 3(b) shows that the dynamics of one of the chaotic attractors follow a quasiperiodic regime; then either bursts to the same attractor or to its symmetric counterpart; then other bursts follow. Numerical simulations indicate that intervals between bursts become shorter as the Reynolds number, Re, is increased up to Re=R6.Figure 1
The phase portrait of the four asymptotically stable fixed points atRe=18; the first asymptotically stable fixed point (top left) is obtained using the following initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (−0.8275, −0.4703, 0.7234, −2.3355, −0.122, −3.599, 0.4255); the second asymptotically stable fixed point (bottom right) is obtained using the initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (0.8275, −0.4703, 0.7234, 2.3355, −0.122, 3.599, −0.4255) = ro(−0.8275, −0.4703, 0.7234, −2.3355, −0.122, −3.599, 0.4255); the third asymptotically stable fixed point (top right) is generated using the initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (0.8275, 0.4703, 0.7234, −2.3355, 0.122, −3.599, 0.4255) = rx(0.8275, −0.4703, 0.7234, 2.3355, −0.122, 3.599, −0.4255); the fourth asymptotically stable fixed point (bottom left) is generated using the initial conditions: (x1,x2,x3,x4,x5,x6,x7) = (−0.8275, 0.4703, 0.7234, 2.3355, 0.122, 3.599, −0.4255) = ry(0.8275, −0.4703, 0.7234, 2.3355, −0.122, 3.599, −0.4255).Figure 2
Phase portraits of the four stable periodic orbits arising from a Hopf bifurcation atRe=20.845 that remain stable for up to Re=26.323 and bifurcates again into a new four periodic orbits with double the period.
(a)
R e = 20.845 (b)
R e = 23 (c)
R e = 25 (d)
R e = 26.37Figure 3
(a) Phase portraits of the bursting phenomena resulting from a homoclinic gluing bifurcation connecting a pair of two stable periodic orbits. (b) Time series of the statex1 versus time at Re=26.377.
(a)
R e = 26.377 (b)
R e = 26.377


## 3. Controlling the Dynamics of the Reduced Order System to a Fixed Point

This section deals with the design of a control scheme to drive the states of the system to a stable or unstable desired fixed point. Hence, control inputs are added to the system in (8). The controlled system of ODEs with control inputs added to the first, the third, the fifth, and the sixth ODE of (8) is as follows.(18)x˙1=-2x1-4x4x5+4x2x7+u1,x˙2=-9x2+3x1x7,x˙3=-4x3-42x4x6+42x6x7+4Re+u2,x˙4=-5x4+x1x5+32x3x6,x˙5=-x5+3x1x4-5x1x6+u3,x˙6=-x6+5x1x5+2x3x4-2x3x7+u4,x˙7=-5x7-7x1x2-32x3x6.Let the constant desired fixed point be such thatxd=[x1dx2dx3dx4dx5dx6dx7d]T. Since xd is an equilibrium point of system (18) then it must satisfy the following algebraic equations:(19)-9x2d+3x1dx7d=0,-5x4d+x1dx5d+32x3dx6d=0,-5x7d-7x1dx2d-32x3dx6d=0.Define the errors ei(i=1,…,7) such that(20)e1=x1-x1d,e2=x2-x2d,e3=x3-x3d,e4=x4-x4d,e5=x5-x5d,e6=x6-x6d,e7=x7-x7d.Using equations (18), (19), and (20), the model for the error system can be written as follows:(21)e˙1=-2e1-4e4e5-4x5de4-4x4de5+4e2e7+4x7de2+4x2de7+m1+u1,e˙2=-9e2+3e1e7+3x7de1+3x1de7,e˙3=-4e3-42e4e6-42x6de4-42x4de6+42e6e7+42x7de6+42x6de7+m2+u2,e˙4=-5e4+e1e5+x5de1+x1de5+32e3e6+32x3de6+32x6de3,e˙5=-e5+3e1e4+3x1de4+3x4de1-5e1e6-5x6de1-5x1de6+m3+u3,e˙6=-e6+5e1e5+5x5de1+5x1de5+2e3e4+2x3de4+2x4de3-2e3e7-2x3de7-2x7de3+m4+u4,e˙7=-5e7-7e1e2-7x2de1-7x1de2-32e3e6-32x6de3-32x3de6,where the parameters m1, m2, m3, and m4 are such that(22)m1=-2x1d-4x4dx5d+4x2dx7d,m2=4Re-4x3d-42x4dx6d+42x6dx7d,m3=-x5d+3x1dx4d-5x1dx6d,m4=-x6d+5x1dx5d+2x3dx4d-2x3dx7d.Let the gainsci¯(i=1,…,4) be positive scalars and let the control gains ai(i=1,…,7) be positive scalars such that(23)3a2-7a7=0.Theorem 2.
The control law,(24)u1=4e4e5+4x5de4+4x4de5-4e2e7-4x7de2-4x2de7-3a2a1e2e7-3a2a1x7de2-a4a1e4e5-a4a1x5de4+7a7a1e2e7+7a7a1x2de7-m1-c1¯e1,u2=42e4e6+42x4de6+42x6de4-42e6e7-42x6de7-42x7de6-32a4a3e4e6-32a4a3x6de4+32a7a3e6e7+32a7a3x6de7-m2-c2¯e3,u3=-3e1e5-3x1de4-3x4de1+5e1e6+5x1de6+5x6de1-a4a5x1de4-m3-c3¯e5,u4=-5e1e5-5x5de1-5x1de5-2e3e4-2x3de4-2x4de3+2e3e7+2x3de7+2x7de3-32a4a6x3de4+32a7a6x3de7-m4-c4¯e6,when applied to the error model (21) guarantees the convergence of the errors ei(i=1,…,7) to zero as t tends to infinity. Therefore, system (18) asymptotically converges to a desired fixed point xd as t tends to infinity.Proof.
Consider the Lyapunov function candidateV such that(25)V=12a1e12+12a2e22+12a3e32+12a4e42+12a5e52+12a6e62+12a7e72.
Using the model of the errors system given in (21), the control law given by (24), and the constraints on the control parameters a2 and a7 given by (23), the derivative of V with respect to time is such that(26)V˙=-2a1e12-9a2e22-4a3e32-5a4e42-a5e52-a6e62-5a7e72-c1¯a1e12-c2¯a3e32-c3¯a5e52-c4¯a6e62.
Since the design parametersai(i=1,…,7) and ci¯(i=1,…,4) are positive constants, then it can be concluded that V˙ is negative definite. Therefore, the Lyapunov function V in (25) is positive definite, its derivative with respect to time V˙ is negative definite, and V is radially unbounded. Hence, it can be concluded that the errors ei(t)(i=1,…,7) in (20) asymptotically converge to zero as t tends to infinity. Thus, the states of system (18) asymptotically converge to a desired constant fixed point xd as t tends to infinity.The simulation results of system (18) when controlled using the proposed control scheme given by (24) are presented below. The simulations are carried out for the Reynolds number Re=26. Note that the dynamics at Re=26 corresponds to four stable periodic solutions of the system of ODEs. These periodic solutions are a result of Hopf bifurcations that took place at Re=20.845. The values of the control gains ci¯ are such that c1¯=10, c2¯=10, c3¯=10, and c4¯=10. The control gains a1–a7 are taken to be a1=1, a2=7, a3=1, a4=1, a5=1, a6=1, and a7=3. The simulation results correspond to an initial condition,(27)x0=-2.260430.680260.8938572.9772-0.173334.0286-0.90282T,and the four desired fixed points(28)xd1=2.26043-0.680260.8938572.9772-0.173334.0286-0.90282T;xd2=-2.26043-0.680260.8938572.9772-0.173334.02860.90282T;xd3=-2.26043-0.680260.893857-2.9772-0.17333-4.02860.90282T;xd4=-2.26043-0.680260.893857-2.9772-0.17333-4.02860.90282T.These desired fixed points are the asymptotically stable fixed points that became unstable as the result of Hopf bifurcations. At the beginning of the simulations, the controllers ui(i=1,…,4) are set to zero for the first 200,000Δt. Then the control law given by (24) is switched on and off and the desired fixed point xd is set to one of the four desired fixed points each time the control law is switched on.The simulation results are presented in Figures4 and 5. Figure 4 presents the controlled states x7(t) versus x1(t), and Figure 5 depicts the time evolution of the controlled states xi(t)(i=1,…,7). The figures clearly show how the controlled states xi(t) converge to one of the desired fixed points each time the control law is switched on and to one of the four stable periodic solutions when the control law is switched off. Hence, it can be concluded that the proposed controller given by (24) drives the states of the system in (18) to their desired states.Figure 4
The statex7 versus x1 at Re=26 when the controllers are switched on and off and when the desired fixed point is one of the four unstable fixed points.Figure 5
The statesxi(i=1,…,7) versus time at Re=26 when the controllers are switched on and off and when the desired fixed point is one of the four unstable fixed points.


## 4. Synchronization of Two ODE Systems Having the Same or Different Reynolds Numbers

In this section, we design a control law to synchronize two ODE systems obtained from the 2D Navier-Stokes equation using the Fourier Expansion truncation method having the same or different Reynolds numbers.


### 4.1. Model of the Error System

In this subsection, we first derive the error system between the two ODE systems to be synchronized. The model of the first ODE system is(29)x˙1=-2x1-4x4x5+4x2x7,x˙2=-9x2+3x1x7,x˙3=-4x3-42x4x6+42x6x7+4Re1,x˙4=-5x4+x1x5+32x3x6,x˙5=-x5+3x1x4-5x1x6,x˙6=-x6+5x1x5+2x3x4-2x3x7,x˙7=-5x7-7x1x2-32x3x6,where Re1 is the Reynolds number of the first ODE system. The model of the second ODEs system is(30)y˙1=-2y1-4y4y5+4y2y7+u1,y˙2=-9y2+3y1y7,y˙3=-4y3-42y4y6+42y6y7+4Re2+u2,y˙4=-5y4+y1y5+32y3y6,y˙5=-y5+3y1y4-5y1y6+u3,y˙6=-y6+5y1y5+2y3y4-2y3y7+u4,y˙7=-5y7-7y1y2-32y3y6,where yi(i=1,…,7) are the state variables of the second ODE and Re2 is its Reynolds number. Define the errors ei(t)(i=1,…,7) between the first ODE system in (29) and the second ODE system in (30) as follows:(31)e1=x1-y1,e2=x2-y2,e3=x3-y3,e4=x4-y4,e5=x5-y5,e6=x6-y6,e7=x7-y7.Using the equations defined in (29)–(31), the model of the error system can be written as follows:(32)e˙1=-2e1-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7-u1,e˙2=-9e2+3e1e7+3y7e1+3y1e7,e˙3=-4e3-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+4R-u2,e˙4=-5e4+e1e5+y5e1+y1e5+32e3e6+32y6e3+32y3e6,e˙5=-e5+3e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6-u3,e˙6=-e6+5e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7-u4,e˙7=-5e7-7e1e2-7y2e1-7y1e2-32e3e6-32y6e3-32y3e6,where R=Re1-Re2.


### 4.2. A Lyapunov Based Controller to Synchronize Two Systems Having the Same or Different Reynolds Number

In this subsection, a Lyapunov based controller is designed to drive the states of the system in (30) to asymptotically converge to the states of the system in (29).Let the gainsgi(i=1,…,4) be positive scalar. In addition, let the control gains bi(i=1,…,7) be positive scalars such that(33)3b2-7b7=0.The following theorem gives the second control result of the paper.Theorem 3.
The control law,(34)u1=-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7+3b2b1e2e7+3b2b1y7e2+b4b1e4e5+b4b1y5e4-7b7b1e2e7-7b7b1y2e7+g1e1,u2=-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+32b4b3e4e6+32b4b3y6e4-32b7b3e6e7-32b7b3y6e7+4R+g2e3,u3=3e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6+b4b5y1e4+g3e5,u4=5e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7+32b4b6y3e4-32b7b6y3e7+g4e6,when applied to the error model (32) guarantees the convergence of the errors ei(i=1,…,7) to zero as t tends to infinity. Therefore, the states of system (30) asymptotically converge to the states of system (29) as t tends to infinity.Proof.
Consider the Lyapunov function candidateV such that(35)V=12b1e12+12b2e22+12b3e32+12b4e42+12b5e52+12b6e62+12b7e72.
Using the model of the errors system given in (32), the control law given by (34), and the constraints on the control parameters b2 and b7 given by (33), the derivative of V with respect to time is such that(36)V˙=b1e1e1˙+b2e2e2˙+b3e3e3˙+b4e4e4˙+b5e5e5˙+b6e6e6˙+b7e7e7˙=b1e1-2e1-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7-u1+b2e2-9e2+3e1e7+3y7e1+3y1e7+b3e3-4e3-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+4R-u2+b4e4-5e4+e1e5+y5e1+y1e5+32e3e6+32y6e3+32y3e6+b5e5-e5+3e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6-u3+b6e6-e6+5e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7-u4+b7e7-5e7-7e1e2-7y2e1-7y1e2-32e3e6-32y6e3-32y3e6.or(37)V˙=-2b1e12-9b2e22-4b3e32-5b4e42-b5e52-b6e62-5b7e72+b1e1-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7-u1+3b2b1e2e7+3b2b1y7e2+b4b1e4e5+b4b1y5e4-7b7b1e2e7-7b7b1y2e7+b3e3-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+4R-u2+32b4b3e4e6+32b4b3y6e4-32b7b3e6e7-32b7b3y6e7+b5e53e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6-u3+b4b5y1e4+b6e65e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7-u4+32b4b6y3e4-32b7b6y3e7+3b2-7b7y1e2e7=-2b1e12-9b2e22-4b3e32-5b4e42-b5e52-b6e62-5b7e72-g1b1e12-g2b3e32-g3b5e52-g4b6e62.Since the design parameters bi(i=1,…,7) and gi(i=1,…,4) are positive constants, then it is concluded that the Lyapunov function V defined by (35) is positive definite and its derivative V˙ is negative definite. Therefore, the error functions in (31) asymptotically converge to zero as t tends to infinity. Thus, the states of system (30) asymptotically converge to the states of system (29) as t tends to infinity.The performance of the controller in (34) when applied to the ODE system in (18) is simulated. The control gains g1,g2,g3, and g4 are taken to be 50. Three cases are simulated. For the first case, we choose Re=26, and the initial condition   x0 = [-0.9171  -0.3868  0.6297  -2.4110  -0.0780  -4.1250  1.4072]T for the system in (29) and Re=23 and the initial condition y(0) = [3.1934-1.11941.51193.2126-0.66013.9139-1.0963]T for the system in (30). Note that these two states correspond to two stable periodic orbits with different periods. For the second case, we choose Re=23, and x(0)=[-0.8861-0.37280.6196-2.48230.0167-4.17251.3078]T for the system in (29), and Re=26.377 and y(0)=[-0.02180.01780.63902.2250-0.01084.1041-2.2075]T for the system in (30). The first state in the second case corresponds to a periodic orbit, and the second state corresponds to a chaotic attractor or a strange attractor connecting a pair of two stable periodic orbits. For the third case, we choose Re=26.377 and x(0) = [-0.0634  -0.05960.6172  -2.24330.0501  -4.19402.0967]T for the system in (29), and Re=26.377 and y(0)=[0.0219-0.02660.63052.22670.03184.1382-2.1713]T for the system in (30). Note that, in this case, the two states correspond to two different chaotic or strange attractors each connecting a pair of two stable periodic orbits. Moreover, at the beginning of simulations in each case, the controllers u1,u2,u3, and u4 are set to zero for the first 30 seconds. Then the control law presented in (34) is switched on to force system (30) to synchronize with system (29).The simulation results are presented in Figure6 for case 1. Figure 6(a) depicts the L2 norm of the error e=∑i=17xi-yi2 versus time. Also, the states x1(t) and y1(t) versus time and x7(t) and y7(t) versus time are plotted in Figures 6(b) and 6(c), respectively. Figure 6(d) plots the states x7(t) and y7(t) versus x1(t) and y1(t); the figure shows the effectiveness of the controller to drive the dynamics from one attractor to a different attractor. Figure 7 shows the simulation results for case 2, and Figure 8 shows the results for case 3. In the three cases, it is shown how the error converges to zero. Hence, it can be concluded that the designed control law in (34) is able to synchronize the ODE systems in (29) and (30) when these systems have the same or different Reynolds numbers but they start from two different initial conditions.Figure 6
Control of a periodic orbit to a different periodic orbit whenRe=26 in system (29) and Re=23 in system (30); the control is switched on at t=30. (a) The L2 norm of the error e versus time. (b) The states x1(t) and y1(t) versus time. (c) The states x7(t) and y7(t) versus time. (d) The states x7(t) and y7(t) versus x1(t) and y1(t) showing how the controller drives the states from one attractor to another attractor.
(a) (b) (c) (d)Figure 7
Control of a chaotic attractor to a periodic orbit whenRe=23 in system (29) and Re=26.377 in system (30); the control is switched on at t=30. (a) The L2 norm of the error e versus time. (b) The states x1(t) and y1(t) versus time. (c) The states x7(t) and y7(t) versus time. (d) The states x7(t) and y7(t) versus x1(t) and y1(t) showing how the controller drives the states from one chaotic attractor to a periodic orbit.
(a) (b) (c) (d)Figure 8
Control of a chaotic attractor to a different chaotic attractor whenRe=26.377 in system (29) and Re=26.377 in system (30); the control is switched on at t=30. (a) The L2 norm of the error e versus time. (b) The states x1(t) and y1(t) versus time. (c) The states x7(t) and y7(t) versus time. (d) The states x7(t) and y7(t) versus x1(t) and y1(t) showing how the controller drives the states from one chaotic attractor to a different chaotic attractor.
(a) (b) (c) (d)


## 4.1. Model of the Error System

In this subsection, we first derive the error system between the two ODE systems to be synchronized. The model of the first ODE system is(29)x˙1=-2x1-4x4x5+4x2x7,x˙2=-9x2+3x1x7,x˙3=-4x3-42x4x6+42x6x7+4Re1,x˙4=-5x4+x1x5+32x3x6,x˙5=-x5+3x1x4-5x1x6,x˙6=-x6+5x1x5+2x3x4-2x3x7,x˙7=-5x7-7x1x2-32x3x6,where Re1 is the Reynolds number of the first ODE system. The model of the second ODEs system is(30)y˙1=-2y1-4y4y5+4y2y7+u1,y˙2=-9y2+3y1y7,y˙3=-4y3-42y4y6+42y6y7+4Re2+u2,y˙4=-5y4+y1y5+32y3y6,y˙5=-y5+3y1y4-5y1y6+u3,y˙6=-y6+5y1y5+2y3y4-2y3y7+u4,y˙7=-5y7-7y1y2-32y3y6,where yi(i=1,…,7) are the state variables of the second ODE and Re2 is its Reynolds number. Define the errors ei(t)(i=1,…,7) between the first ODE system in (29) and the second ODE system in (30) as follows:(31)e1=x1-y1,e2=x2-y2,e3=x3-y3,e4=x4-y4,e5=x5-y5,e6=x6-y6,e7=x7-y7.Using the equations defined in (29)–(31), the model of the error system can be written as follows:(32)e˙1=-2e1-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7-u1,e˙2=-9e2+3e1e7+3y7e1+3y1e7,e˙3=-4e3-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+4R-u2,e˙4=-5e4+e1e5+y5e1+y1e5+32e3e6+32y6e3+32y3e6,e˙5=-e5+3e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6-u3,e˙6=-e6+5e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7-u4,e˙7=-5e7-7e1e2-7y2e1-7y1e2-32e3e6-32y6e3-32y3e6,where R=Re1-Re2.


## 4.2. A Lyapunov Based Controller to Synchronize Two Systems Having the Same or Different Reynolds Number

In this subsection, a Lyapunov based controller is designed to drive the states of the system in (30) to asymptotically converge to the states of the system in (29).Let the gainsgi(i=1,…,4) be positive scalar. In addition, let the control gains bi(i=1,…,7) be positive scalars such that(33)3b2-7b7=0.The following theorem gives the second control result of the paper.Theorem 3.
The control law,(34)u1=-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7+3b2b1e2e7+3b2b1y7e2+b4b1e4e5+b4b1y5e4-7b7b1e2e7-7b7b1y2e7+g1e1,u2=-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+32b4b3e4e6+32b4b3y6e4-32b7b3e6e7-32b7b3y6e7+4R+g2e3,u3=3e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6+b4b5y1e4+g3e5,u4=5e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7+32b4b6y3e4-32b7b6y3e7+g4e6,when applied to the error model (32) guarantees the convergence of the errors ei(i=1,…,7) to zero as t tends to infinity. Therefore, the states of system (30) asymptotically converge to the states of system (29) as t tends to infinity.Proof.
Consider the Lyapunov function candidateV such that(35)V=12b1e12+12b2e22+12b3e32+12b4e42+12b5e52+12b6e62+12b7e72.
Using the model of the errors system given in (32), the control law given by (34), and the constraints on the control parameters b2 and b7 given by (33), the derivative of V with respect to time is such that(36)V˙=b1e1e1˙+b2e2e2˙+b3e3e3˙+b4e4e4˙+b5e5e5˙+b6e6e6˙+b7e7e7˙=b1e1-2e1-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7-u1+b2e2-9e2+3e1e7+3y7e1+3y1e7+b3e3-4e3-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+4R-u2+b4e4-5e4+e1e5+y5e1+y1e5+32e3e6+32y6e3+32y3e6+b5e5-e5+3e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6-u3+b6e6-e6+5e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7-u4+b7e7-5e7-7e1e2-7y2e1-7y1e2-32e3e6-32y6e3-32y3e6.or(37)V˙=-2b1e12-9b2e22-4b3e32-5b4e42-b5e52-b6e62-5b7e72+b1e1-4e4e5-4y5e4-4y4e5+4e2e7+4y7e2+4y2e7-u1+3b2b1e2e7+3b2b1y7e2+b4b1e4e5+b4b1y5e4-7b7b1e2e7-7b7b1y2e7+b3e3-42e4e6-42y6e4-42y4e6+42e6e7+42y7e6+42y6e7+4R-u2+32b4b3e4e6+32b4b3y6e4-32b7b3e6e7-32b7b3y6e7+b5e53e1e4+3y4e1+3y1e4-5e1e6-5y6e1-5y1e6-u3+b4b5y1e4+b6e65e1e5+5y5e1+5y1e5+2e3e4+2y4e3+2y3e4-2e3e7-2y7e3-2y3e7-u4+32b4b6y3e4-32b7b6y3e7+3b2-7b7y1e2e7=-2b1e12-9b2e22-4b3e32-5b4e42-b5e52-b6e62-5b7e72-g1b1e12-g2b3e32-g3b5e52-g4b6e62.Since the design parameters bi(i=1,…,7) and gi(i=1,…,4) are positive constants, then it is concluded that the Lyapunov function V defined by (35) is positive definite and its derivative V˙ is negative definite. Therefore, the error functions in (31) asymptotically converge to zero as t tends to infinity. Thus, the states of system (30) asymptotically converge to the states of system (29) as t tends to infinity.The performance of the controller in (34) when applied to the ODE system in (18) is simulated. The control gains g1,g2,g3, and g4 are taken to be 50. Three cases are simulated. For the first case, we choose Re=26, and the initial condition   x0 = [-0.9171  -0.3868  0.6297  -2.4110  -0.0780  -4.1250  1.4072]T for the system in (29) and Re=23 and the initial condition y(0) = [3.1934-1.11941.51193.2126-0.66013.9139-1.0963]T for the system in (30). Note that these two states correspond to two stable periodic orbits with different periods. For the second case, we choose Re=23, and x(0)=[-0.8861-0.37280.6196-2.48230.0167-4.17251.3078]T for the system in (29), and Re=26.377 and y(0)=[-0.02180.01780.63902.2250-0.01084.1041-2.2075]T for the system in (30). The first state in the second case corresponds to a periodic orbit, and the second state corresponds to a chaotic attractor or a strange attractor connecting a pair of two stable periodic orbits. For the third case, we choose Re=26.377 and x(0) = [-0.0634  -0.05960.6172  -2.24330.0501  -4.19402.0967]T for the system in (29), and Re=26.377 and y(0)=[0.0219-0.02660.63052.22670.03184.1382-2.1713]T for the system in (30). Note that, in this case, the two states correspond to two different chaotic or strange attractors each connecting a pair of two stable periodic orbits. Moreover, at the beginning of simulations in each case, the controllers u1,u2,u3, and u4 are set to zero for the first 30 seconds. Then the control law presented in (34) is switched on to force system (30) to synchronize with system (29).The simulation results are presented in Figure6 for case 1. Figure 6(a) depicts the L2 norm of the error e=∑i=17xi-yi2 versus time. Also, the states x1(t) and y1(t) versus time and x7(t) and y7(t) versus time are plotted in Figures 6(b) and 6(c), respectively. Figure 6(d) plots the states x7(t) and y7(t) versus x1(t) and y1(t); the figure shows the effectiveness of the controller to drive the dynamics from one attractor to a different attractor. Figure 7 shows the simulation results for case 2, and Figure 8 shows the results for case 3. In the three cases, it is shown how the error converges to zero. Hence, it can be concluded that the designed control law in (34) is able to synchronize the ODE systems in (29) and (30) when these systems have the same or different Reynolds numbers but they start from two different initial conditions.Figure 6
Control of a periodic orbit to a different periodic orbit whenRe=26 in system (29) and Re=23 in system (30); the control is switched on at t=30. (a) The L2 norm of the error e versus time. (b) The states x1(t) and y1(t) versus time. (c) The states x7(t) and y7(t) versus time. (d) The states x7(t) and y7(t) versus x1(t) and y1(t) showing how the controller drives the states from one attractor to another attractor.
(a) (b) (c) (d)Figure 7
Control of a chaotic attractor to a periodic orbit whenRe=23 in system (29) and Re=26.377 in system (30); the control is switched on at t=30. (a) The L2 norm of the error e versus time. (b) The states x1(t) and y1(t) versus time. (c) The states x7(t) and y7(t) versus time. (d) The states x7(t) and y7(t) versus x1(t) and y1(t) showing how the controller drives the states from one chaotic attractor to a periodic orbit.
(a) (b) (c) (d)Figure 8
Control of a chaotic attractor to a different chaotic attractor whenRe=26.377 in system (29) and Re=26.377 in system (30); the control is switched on at t=30. (a) The L2 norm of the error e versus time. (b) The states x1(t) and y1(t) versus time. (c) The states x7(t) and y7(t) versus time. (d) The states x7(t) and y7(t) versus x1(t) and y1(t) showing how the controller drives the states from one chaotic attractor to a different chaotic attractor.
(a) (b) (c) (d)


## 5. Conclusion

This paper investigates the symmetries, dynamics, and control of the 2D Kolmogorov flow with a forcing in the mode(0,2). First, a seventh-order system of ODEs is constructed using the Fourier series expansion. A computer analysis shows that the constructed system exhibits a chaotic behavior for a certain range of the Reynolds number. In addition, the ODE system shows a homoclinic gluing bifurcation similar to the original PDE simulation. Then, Lyapunov based controllers are designed to control the dynamics of ODEs to a fixed point, a periodic orbit, or a chaotic or strange attractor. Extensive numerical studies are presented to show the effectiveness of the proposed controllers.Future work will address the design of adaptive control schemes for the reduced order ODE model of the 2D Kolmogorov flow.



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*Source: 4602485-2018-05-13.xml*