Synchronization of Chaos in a Food Web in Ecological Systems

The three-species food web model proposed and investigated by Gakkhar and Naji is known to have chaotic behaviour for a choice of parameters. An attempt has been made to synchronize the chaos in the model using bidirectional coupling. Numerical simulations are presented to demonstrate the effectiveness and feasibility of the analytical results. Numerical results show that for higher value of coupling strength, chaotic synchronization is achieved. Chaos can be controlled to achieve stable synchronization in natural systems.





References:
[1] M. A. Aziz-Alaoui, "Synchronization of Chaos", Encyclopedia of
mathematical physics, 2006.
[2] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A
universal Concept in Nonlinear Science. Cambridge: Cambridge
University Press..
[3] L. Pecora and T. Carroll, "Synchronization in chaotic systems,"
Physics Review Letters, vol. 64, No. 8, pp. 821-824, 1990.
[4] S. Gakkhar and R. K. Naji, "Order and chaos in a food web
consisting of a predator and two Independent preys,"
Communications in Nonlinear Science and Numerical Simulation,
vol. 10, pp. 105-120, 2005.
[5] X.J. Wu, J. Lie and R. K. Upadhayay, "Chaos control and
synchronization of a three-species food chain model via Holling
functional response," International Journal of Computer
Mathematics, pp.1- 16, 2008.
[6] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano,
"Determining Lyapunov exponents from a time series," Physica D,
vol. 16, pp. 285-317, 1985.
[7] J. L. Kaplan and J. Yorke, "Chaotic behaviour of multidimensional
difference equations", Functional Differential Equations and
Approximations of Fixed points, edited by H. O. Walter and H-O.
Peitgen, vol. 730 of Lectures Notes in Mathematics, Springer,
Berlin, 1979, pp. 204-227.
[8] L. Pecora and T. Carroll, "Master Stability Functions for
Synchronized Coupled System," Physics Review Letters, vol. 64,
no. 8, pp. 821-824, 1990.
[9] J. Heagy, L. Pecora and T. Carroll, "Short wavelength Bifurcations
and Size instabilities in Coupled Oscillator Systems," Physical
Review Letters, vol. 74, no. 21, pp. 4185-4188, 1995.
[10] J. Heagy, T. Carroll, and L. Pecora, "Synchronous Chaos in Coupled
Oscillator Systems," Physical Review E, vol. 50, no. 3, pp. 1874-
1884, 1994.
[11] G. Chen and X. Dong, From Chaos to Order, Singapore: World
Scientific, 1998.