Abstract
The dynamical behavior of pulse and traveling hole in a one-dimensional system depending on the boundary conditions, obeying the complex Ginzburg-Landau (CGL) equation, is studied numerically using parameters near a subcritical bifurcation. In a spatially extended system, the criterion of Benjamin-Feir-Newell (BFN) instability near the weakly inverted bifurcation is established, and many types of regimes such as laminar regime, spatiotemporal regime, defect turbulence regimes, and so on are observed. In finite system by using the homogeneous boundary conditions, two types of regimes are detected mainly the convective and the absolute instability. The convectively unstable regime appears below the threshold of the parameter control, and beyond, the absolute regime is observed. Controlling such regimes remains a great challenge; many methods such as the nonlinear diffusion parameter control are used. The unstable traveling hole in the one-dimensional cubic-quintic CGL equation may be effectively stabilized in the chaotic regime. In order to stabilize defect turbulence regimes, we use the global time-delay auto-synchronization control; we also use another method of control which consists in modifying the nonlinear diffusion term. Finally, we control the unstable regimes by adding the nonlinear gradient term to the system. We then notice that the chaotic system becomes stable under strong nonlinearity.
Keywords
- Benjamin-Feir-Newell instability
- subcritical bifurcation
- complex Ginzburg-Landau equation
- unstable traveling hole
1. Introduction
Many complex systems evolve in a non-equilibrium environment. Further out of the equilibrium [1], these systems tend to display progressively more complicated dynamics. The non-chaotic patterned state and spatiotemporal chaos are observed in the system. In the domain of the envelope equations, the quintic complex Ginzburg-Landau (CGL) equation is appropriate to obtain stable localized solutions (pulses, holes) [1, 2]. Among physical applications of the quintic CGL equation, one can mention binary fluid convection [3], spiral waves in the Couette-Taylor flow between counterrotating cylinders [4], wave propagation in nonlinear optical fibers with gain and spectral filtering [5], the oscillatory chemical reaction [6], hydrodynamic turbulence [7], chemical turbulence [8, 9], and electrical turbulence in the cardiac muscle [10]. Our work focuses on two types of systems: the spatially extended system and the finite system. In the case of spatially extended systems, we use as initial conditions a traveling-hole solution with periodic boundary conditions [11, 12, 13, 14, 15]. All the dynamical regimes obtained during our work are summarized in a phase diagram. In the case of the finite domain, we use as initial condition a pulse solution. Wave patterns are described by CGL equation in which the amplitude of the wave pattern vanishes at the lateral boundaries of the domain in order to retrieve numerically some coherent structures observed experimentally, in the case of absolute or convective instabilities [16, 17, 18].
Over the past decade, problems of chaos control and synchronization started to play a central role in the studies of chaotic dynamics [19] in many different areas such as chemistry [20], laser physics [21], electronic circuits [22], plasma [23], and mechanical systems. Since the pioneering work of Ott et al. [24] on the control of low-dimensional chaos in nonlinear systems based on Floquet theory, chaos control techniques have been well developed [25, 26]. Up to date, many control techniques have been suggested to control low-dimensional chaos by stabilizing unstable periodic orbits or fixed points. The realization of chaos control mainly includes feedback and non-feedback methods, both of which have advantages and disadvantages. Pyragas is one of the first to work on a delayed feedback loop called time-delay auto-synchronization (TDAS) [25]. Another part of our works is to control turbulence regimes observed, in particular the defect turbulence regime by employing the methods already successfully used in the cubic case, namely, the nonlinear diffusion technique [20], the feedback method [27], and the lower-order complex Ginzburg-Landau (LOCGL) equation [28, 29, 30, 31]. The LOCGL equation which describes a system exhibiting a subcritical bifurcation to traveling waves must contain a quintic nonlinearity. It is obtained by adding nonlinear terms to the system. The effects of the nonlinear gradient terms are confirmed by using some indicators such as the Lyapunov exponent and the energy bifurcation diagram. Most of the results related to these different aspects are presented in the rest of this work.
2. Dynamics of traveling hole in one-dimensional systems near subcritical bifurcation
2.1 Model description
We consider a subcritical Hopf bifurcation, with a one-dimensional complex amplitude
where
where
The precise form of the initial condition is not important here as long as we have a one-parameter family of localized phase-gradient peaks. This is because the left moving and right moving coherent holes for fixed
2.2 Results of numerical simulation
The parameters
2.2.1 Plane wave regime
The plane wave regime is a laminar state where no chaos is observed. The plane wave is localized below the BFN line in a zone called stable zone. The spatial profile of the wave patterns in the plane wave regime is shown in Figure 2. We notice that by the growing of the time, the regime still stable, and laminar regime is observed.
2.2.2 Spatiotemporal intermittency regime
It consists of space-time regions of stable plane waves separated by localized objects evolving and interacting in a complex manner [33]. It represents a special scenario of transition to turbulence in extended systems: it is characterized by the coexistence of laminar (ordered) and turbulent (disordered) domains that occur randomly in different places of the system for the same values of the control parameters [13, 33]. It has been observed in many experiments such as plane Couette flow, counterrotating Taylor-Couette flow, and Taylor-Dean system. In 1D extended systems, spatiotemporal intermittency has been observed in rectangular and annular Rayleigh-Bénard cells at large values of the Rayleigh number [15]. This spatiotemporal intermittency occurs via a subcritical bifurcation from purely laminar state, and the coexistence of two different stable states can be described phenomenologically using an amplitude equation derived from a Lyapunov function. We have plotted in Figure 3 the characteristic pattern of a spatiotemporal intermittency in which a global mode coexists with a chaotic attractor: the state consists of patches of plane waves, which are separated y various holes. Figure 3b and c shows in detail how a hole generates a phase defect and in turn generates two daughter holes close-up of the amplitude
2.2.3 Phase turbulence regime
Just above the BFN line, the phase turbulence regime is observed (Figure 4). It is best defined by the absence of space-time defects. In this regime, the region is a weakly disordered one in which
where
2.2.4 Weak turbulence regime
Beyond the BFN line, we observe that for the parameter equations (
2.2.5 Defect turbulence
Father away from the BFN line a spatiotemporally disordered regime called amplitude or defect turbulence is observed (see Figure 7). The behavior in this region is characterized by defects. The defect turbulence regime is the dynamical regime wherein the fluctuations of
3. Nonlinear structures of traveling waves in subcritical systems with finite geometries
3.1 The cubic-quintic complex Ginzburg-Landau equation in a finite domain
The one-dimensional cubic-quintic CGL equation in this case is given by:
This equation describes the envelope of a traveling wave propagating at the group velocity
Figure 8 illustrates the deterministic evolution of wave pattern amplitude for convective instability and absolute instability regimes. In the case of convective regime, the wave patterns disappear with the time, while in the case of the absolute instability, they propagate in the whole system.
3.2 Stability of wave patterns of the 1D cubic-quintic CGLE
Let us note that, in the convective regime, the localized disturbances of the basic state are growing but step away from the source. This is why we have restricted the study to the dynamics of pattern for parameters corresponding to the absolute instability regime. When the criticality parameter
3.3 Numerical simulations of the 1D cubic-quintic CGLE
We investigate the effects of the quintic nonlinear dispersion coefficient
4. Controlling spatiotemporal chaos in one-dimensional systems near subcritical bifurcation
4.1 The dynamical model
The modified cubic-quintic CGLE is given by [20]:
where
4.2 Numerical simulation
We start by assuming that the system is in a deeply chaotic region, i.e., parameters are chosen from defect turbulence area in order to verify the results obtained from the linear stability analysis. Figure 16 plots the trajectory in phase space, in which the system parameters are exemplified as
5. Time-delay auto-synchronization control of defect turbulence in cubic-quintic complex Ginzburg-Landau equation
5.1 Model equation
To control the different turbulence regimes observed in the domain, a global feedback term can be introduced. The modified cubic-quintic CGL eqiuation is given by [27, 41, 42].
where
denotes the spatial average of
where
5.2 Numerical simulation
In this section, we present the results of a numerical study of Eq. (11). We analyze the stability of the unstable wave patterns observed inside a defect turbulence regime by using global feedback term [27]. We study the effects of the feedback term on the system. A sufficient strong feedback can suppress spatiotemporal chaos and establishes uniform oscillations. We show that for certain values of the global feedback term and the delay time, the system which initially was chaotic becomes completely stable. The dynamic regimes observed during the numerical study are summarized in the state diagram of Figure 18. The five regimes observed are defect turbulence, spatiotemporal intermittency, phase turbulence, standing waves, and plane waves. We remark that as the feedback intensity is increased starting from zero, global oscillations set in, and defect turbulent regimes are replaced by other interesting regimes until the appearance of the laminar state.
For certain values of
6. Discussion
Let us now introduce to the CGL equation the global time-delay feedback and study its effects on the system. The new CGL equation with time-delay auto-synchronization is given by:
where
and
with the amplitude and frequency given by
and
A cubic CGL equation with a similar feedback scheme has been investigated in Refs. [13, 35]. In their work, they have shown how the strongly disordered state can be stabilized in the system. The initially unstable system undergoes several transformations successively and become stable; we have in order defect turbulence, phase turbulence, standing wave state, and uniform oscillations. The results of our numerical study of Eq. (12) are given in Figure 22. This figure shows the progressive transition from defect turbulence to plane wave state. In the absence of feedback (see Figure 22a, with
7. Effects of nonlinear gradient terms on the defect turbulence regime in weakly dissipative systems
The LOCGL equation which describes a system exhibiting a subcritical bifurcation to traveling waves must contain a quintic nonlinearity; at this order, it is necessary to include the lower-order nonlinear gradient terms:
with
The aim is to see the impact of these nonlinear gradient terms on a defect turbulence regime. We use the indicators such as the Lyapunov exponent and the energy bifurcation diagram to confirm the nature of the regime.
7.1 Numerical simulations
7.1.1 Dynamical indicators
We will essentially characterize the different types of dynamical behavior of the system by the energy function
which is frequently used to characterize non-regular dynamics in optics [48], localized patterns in fluids, and other physical systems, respectively [49]. The one-dimensional system is assumed to be of length 2
where
where
7.1.2 Numerical results
We will present here some of data obtained for systems with the presence of the nonlinear gradient terms. The results are summarized in Figures 23–28. In particular, we study the influence of the nonlinear gradient terms in the defect turbulence regime. Figure 23 shows the wave patterns and the energy as a function of time corresponding to the laminar regime. We notice that the system that was initially chaotic becomes completely stable by the presence of the nonlinear gradient terms. The chaos has been eliminated. By changing the values of nonlinear gradient terms, the dynamics of the system also change; it is confirmed by Figures 24 and 25 which represent the oscillating patterns. The corresponding largest Lyapunov exponent is zero.
The plot observed in Figure 24 is the running waves. They are quasi-periodic states; they move in one direction with constant speed, according to its initial condition; and this is the so-called oriental symmetry breaking. A double periodicity in time and in space is observed. In Figure 25, we have another type of the oscillating patterns in a color-coded space-time plot (see Figure 25a). After a transient time, the waves propagate uniformly, with a well-defined wave number and constant amplitude. We note also the presence of an attractor into the system which annihilate the wave patterns (see Figure 25b). The drop observed near
The Lyapunov exponent shown in Figure 26b indicates the dynamical behavior of the system and confirms the results. Figure 27 is obtained for increasing values of the nonlinear gradient terms. We observe several transitions between regular and chaotic states. In particular, there is a small stability part of the system in the range
8. Conclusion
The numerical investigation of the 1D quintic CGL equation in such systems represents a big topic in the understanding of many physical systems with pattern formation. Concerning the extended system, we have summarized in a phase diagram all the regimes that have been observed. On the phase diagram, we have the BFN line that divides the regions in two regions: the stable zone which contains laminar state and spatiotemporal intermittency regime and the unstable zone with chaotic regimes as phase turbulence, weak turbulence, and defect turbulence [49]. For the case of finite system, we have described from the 1D cubic-quintic CGL equation the effects of the boundaries on the waves traveling in a preferred direction. We have used the homogeneous boundary conditions, and the waves were nonlinear dissipative waves. We have studied the nature of convective or absolute instabilities of wave patterns. In our simulations, we have found new states that were similar from those obtained in the cubic CGL equation with homogeneous boundary conditions or with the Neumann boundary conditions. The presence of the quintic term has a large influence in the wave pattern. All the dynamic regimes observed have been summarized in a state diagram. The regimes as the global mode regime, turbulence regime observed in the secondary structures, and spatiotemporal intermittency regimes have been detected, but their shape and behavior are different depending on the sign of
Acknowledgments
The work by CBT is supported by the Botswana International University of Science and Technology under the grant
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