Synchronization of Chaos in a Food Web in Ecological Systems

Authors: Anuraj Singh, Sunita Gakkhar

Abstract:

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.

Keywords: Lyapunov Exponent, Bidirectional Coupling, ChaosSynchronization, Synchronization Manifold

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1335198

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.