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Populations oscillate regularly and periodically in phase, but with irregular and chaotic peaks together in abundance—twin realistic features that are not found in standard ecological models. Peak population abundances, however, remain chaotic and largely uncorrelated. Although synchronization is often perceived as being detrimental to spatially structured populations 14phase synchronization leads to the emergence of complex chaotic travelling-wave structures which may be crucial for species persistence.
Chaos20 401 Dec Cited by: 1 article PMID: Cited by: 6 articles PMID: Phys Rev Lett94 1307 Apr Cited by: 48 articles PMID: To arrive at the top five similar articles we use a word-weighted algorithm to compare words from the Title and Abstract of each citation.
In this paper, the problem of complete synchronization for reaction-diffusion systems is investigated. Linear and nonlinear synchronization control schemes have been proposed to exhibit synchronization between coupled reaction-diffusion systems. Synchronization behaviors of coupled Lengyel-Epstein systems are obtained to demonstrate the effectiveness and feasibility of the proposed control techniques.
Synchronization of chaos is a phenomenon that may occur when two, or more, chaotic systems adjust a given property of their motion to a common behavior due to a coupling or to a forcing. This phenomenon has attracted the interest of many researchers from various fields due to its potential applications in physics, biology, chemistry, and engineering sciences since the pioneering work by Pecora and Carroll [ 1 ].
Various synchronization types have been presented, such as complete synchronization, phase synchronization, lag synchronization, anticipated synchronization, function projective synchronization, generalized synchronization, and Q-S synchronization.
Most of the research efforts have been devoted to the study of chaos control and chaos synchronization problems in low-dimensional nonlinear dynamical systems [ 2 — 10 ]. Synchronizing high dimensional systems in which state variables depend not only on time but also on the spatial position remains a challenge.
These high dimensional systems are generally modelled in spatial-temporal domain by partial differential systems. Recently, the search for synchronization has moved to high dimensional nonlinear dynamical systems. Over the last years, some studies have investigated synchronization of spatially extended systems demonstrating spatiotemporal chaos such as the work presented in [ 11 — 32 ].
Synchronization dynamics of reaction-diffusion systems has been studied in [ 1112 ] using phase reduction theory. It has been shown that reaction-diffusion systems can exhibit synchronization in a similar way to low-dimensional oscillators. A general approach for synchronizing coupled partial differential equations with spatiotemporally chaotic dynamics by driving the response system only at a finite number of space points has been introduced in [ 1314 ].
Synchronization and control for spatially extended systems based on local spatially averaged coupling signals have been presented in [ 17 ]. The effect of asymmetric couplings in the synchronization of spatially extended chaotic systems has been investigated in [ 19 ].
The effect of time-delay autosynchronization on uniform oscillations in a general model described by the complex Ginzburg-Landau equation has been presented in [ 20 ]. Furthermore, generalized synchronization [ 21 ], complete-like synchronization [ 22 ], the backstepping synchronization approach [ 26 ], the graph-theoretic synchronization approach [ 27 ], pinning impulsive synchronization [ 30 ], and impulsive type synchronization strategy [ 31 ] for coupled reaction-diffusion systems have been introduced.
The main aim of the present paper is to study the problem of complete synchronization in coupled reaction-diffusion systems. Linear and nonlinear control schemes have been proposed to realize complete synchronization for partial differential systems. As a special case, we investigate complete synchronization behaviors of coupled Lengyel-Epstein systems. Reaction-diffusion systems have shown important roles in modelling various spatiotemporal patterns that arise in chemical and biological systems [ 3334 ].
Reaction-diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals.
The most familiar way to study synchronization is to use a controller to make the output of the slave response system copy in some manner the master drive system one.
In this case, we design the controller in which the difference of states of synchronized systems converges to zero. This phenomenon is called complete synchronization.
Consider the master and the slave reaction-diffusion systems as and where and are the corresponding states, is a bounded domain in with smooth boundaryis the Laplacian operator onare the diffusivity constants,and are nonlinear continuous functions, and and are controllers to be designed.
We impose the homogeneous Neumann boundary conditions where is the unit outer normal to. The aim of the synchronization process is to force the error between the master and slave systems, defined as to zero. We assume that the diffusivity constants satisfy and the error system satisfies the homogeneous Neumann boundary condition To realize complete synchronization between the master system given in 1 and the slave system given in 2we discuss the asymptotical stable of zero solution of synchronization error system given in 4.
That is, in the following sections, we find the controllers andin linear and nonlinear forms, such that the solution of the error system go to, as goes to.In previous chapter the chaos suppression was discussed. However, there is one more interesting problem in chaos control: the synchronization. Synchronize means to share the same time and signifies that two or more events occurs at same time. In nonlinear science diverse synchronization phenomena have been found in chaotic systems.
Thus, such a problem results in very interesting dynamical phenomena and has technological applications, as in communication , and scientific impact as, for example, in animal gait , or cells of human organs . A continuation path for synchronization is in spatially extended systems  where synchronization phenomena are already being studied. Other interesting issues on synchronization is, on the one hand, the cost of synchronizing chaotic systems ; that is, to measure the energy required to achieve chaotic synchronization.
On the other, the geometrical properties of synchronization are also a raising theme , . Here, geometrical control theory can be used to compute the invariant manifolds . This Chapter is related to the robust synchronization, and is centred on the robust analysis and some interpretations about robustness in synchronization.
Complex dynamics and phase synchronization in spatially extended ecological systems
To this end we exploit the simpler controller in Chapter 2: the Proportional-Integral feedback and some approaches.
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Wu, C. Collins, J. Holstein-Rathlou, N. Bragard, J. Sarasola, C. Zhou, K. Martens, M. Nijmeijer, H.
Synchronization of spatially extended chaotic systems in the presence of asymmetric coupling.
Perez, G. Pyragas, K. Rabinovich, M. Kapitaniak, T.To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Synchronization of spatially extended chaotic systems in the presence of asymmetric coupling Physical Review E, Jean Bragard.
Synchronization of spatially extended chaotic systems in the presence of asymmetric coupling. Bragard,1 S. Boccaletti,2 and H. Fermi, 6, Florence, Italy Received 23 December ; published 8 August We analyze the effects of asymmetric couplings in setting different synchronization states for a pair of unidimensional fields obeying complex Ginzburg-Landau equations.
Novel features such as asym- metry enhanced complete synchronization, limits for the appearance of phase synchronized states, and selection of the final synchronized dynamics are reported and characterized. DOI: Jn, Outside this range, PWS become unstable through focused on external forcings and bidirectional symmetric the Eckhaus instability . When crossing from below or unidirectional master-slave coupling schemes.
Above this tional or perfectly symmetrical coupling configurations. For Therefore, our intention in this Letter is to address the the scope of this work, we mainly concentrate on phase effects of asymmetries in the coupling of space-extended turbulence PT and amplitude turbulence AT or defect continuous fields.
We refer to a pair of unidimensional fields obeying PT is a regime where the chaotic behavior of the field is complex Ginzburg-Landau equations. Simu- rameter accounting for asymmetries in the coupling. The results indicate where T denotes a transient time and K is a suitable real that the threshold for the appearance of a CS state de- number. Condition 2 implies that the maximum relative pends crucially on the asymmetry in the coupling. In phase difference remains bounded for all times. In 2we particular, Fig.
Even Let us now move to discuss how asymmetry influences though formally Eq. Looking at indicator for PS only within phase turbulent regimes, we Fig. This suring intermediate PS states displaying AT. In practice, because FIG. It is well known that N is an extensive solid line. Time is increasing upwards.
Synchronization of spatially extended chaotic systems in the presence of asymmetric coupling
Other parameters are as in the caption of Fig. More insight on the limits for PS 0. The upper left plot 1 1 of Fig. The lower left are as in the caption of Fig. The density of space-time defects plot of Fig. There, one can see that the system shows rather long epochs of phase locked states, system. Figure 4 reports the tially leads to a degradation of frequency locking.
This anomaly is the asymmetry, and both synchronization features are intimately related to the preference of the system for a PT enhanced for 1 [see Figs.In spite of the early discovery, the phenomenon was fully understood much later with the experiments and theoretical analysis of E.
Appleton and B. Nowadays we know that suitable coupled nonlinear systems can synchronize and that the synchronization can be realized also in chaotic systems.
Figure: Synchronization of spatio-temporal chaotic systems when the prevailing instability mechanism is of nonlinear type.
No document with DOI "10.1.1.537.876"
Note that for short range we recover the usual picture of Directed Percolation DP. For details see Ref. Remarkably chaotic synchronization can occur also in high dimensional chaotic systems, and in particular in systems with a spatial extension, i. The presence of many degrees of freedom and, in particular, the spatial extension of these systems convert the bifurcation transition toward the synchronized regime into a non-equilibrium phase-transition, very similar to, e.
In a series of works we extended such analogy to the case of spatially chaotic systems with long range interactions . In particular, we first analysed the influence of nonlinear mechanisms for the establishment of self-synchronization of all the elements of a system , which manifests with the onset of supertransients exponentially long in the size of the system that drive the asymptotic behavior in the thermodynamic limit.
Finally, we completed our program examining generic systems with long range interactions and with linear and nonlinear instabilities at play , in this case we identified new phenomena whose theoretical understanding is still lacking. We also studied the case of one and two dimensional systems iwith short range interactions , including also the possibility of errors between replicas of the system to approach, on the theoretical side, situations closer to possible experimental realizations of synchronization in spatio-temporal chaotic systems.
We have shown that the presence of mismatch is akin to the effect of an external field, see .
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