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 the
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 [
During the last three decades, numerous numerical studies of the 2D Kolmogorov flow with different forcing terms have appeared [
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 [
In this paper, we construct a system of seven ODEs that approximates the dynamics of Kolmogorov flow when the force acts on the mode
The paper is organized as follows. In Section
The “basic 2D Kolmogorov flow”
The perturbed nondimensional vorticity
In Smaoui and Zribi [
In this paper, we use a totally different approach than the one used in [
The equation for
Next, if the basic flow
Let
After changing the length scale
The system given by the equations in (
It is noted that system (
In this subsection, we analyze the dynamics of the seven-mode truncation ODE system presented in ( For For For The second asymptotically stable fixed point (Figure The third asymptotically stable fixed point (Figure The fourth asymptotically stable fixed point (Figure For For For
The phase portrait of the four asymptotically stable fixed points at
Phase portraits of the four stable periodic orbits arising from a Hopf bifurcation at
(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 state
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 (
Let the constant desired fixed point be such that
Using equations (
Let the gains
The control law,
Consider the Lyapunov function candidate
Using the model of the errors system given in (
Since the design parameters
The simulation results of system (
The simulation results are presented in Figures
The state
The states
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.
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
Using the equations defined in (
In this subsection, a Lyapunov based controller is designed to drive the states of the system in (
Let the gains
The following theorem gives the second control result of the paper.
The control law,
Consider the Lyapunov function candidate
Using the model of the errors system given in (
The performance of the controller in (
The simulation results are presented in Figure
Control of a periodic orbit to a different periodic orbit when
Control of a chaotic attractor to a periodic orbit when
Control of a chaotic attractor to a different chaotic attractor when
This paper investigates the symmetries, dynamics, and control of the 2D Kolmogorov flow with a forcing in the mode
Future work will address the design of adaptive control schemes for the reduced order ODE model of the 2D Kolmogorov flow.
The author declares that there are no conflicts of interest regarding the publication of this paper.
This research was supported and funded by the Research Sector, Kuwait University, under Grant no. SM07/16.