In the present paper we have considered the map  where  are parameters. The map was originally proposed by Maynard Smith [17] for study of population growth. We have shown how chaos creep into the model. We have used the techniques of Lyapunov exponent, time series analysis, Fourier spectra, Bifurcation diagram, correlation and embedding dimension etc. to draw our conclusions.  Further, we have shown how the ‘periodic proportional pulse’ method can be used to control the chaos generated in the system.

[1] B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52, 263 (1979).
[2] B. L. Hao, Chaos 2, World Scientific, Singapore, 1990.
[3] D. Whitley, On the Periodic Points of a Two –Parameter Family of Maps of the Plane, Acta Applicandae Mathematicae 5 (1986), pp 279-311
[4] D. G. Aronson, M. A. Chory, G. R. Hall, and R. P. McGehee : Bifurcations from an Invariant Circle for Two-Parameter Families of Maps of the Plane : A Computer Assisted Study, Commun. Math. Phys. 83 (1982), pp 303-354
[5] E. Ott, C. Grebogi and J. A. Yorke “Controlling Chaos”, Phys. Rev. Lett. 64, (1990), pp 1196-1199 [
[6] E. Ott, “Controlling chaos”, Physics Today, Vol. 48, Issue 5, (1995) pp 34-40.
[7] F. J. Romeiras, “Controlling Chaotic Dynamical Systems”, Physica D, vol. 58, (1992) pp 165-192,
[8] G. Casati and B. V. Chirikov, Quantum Chaos (Cambridge Univ. Press, 1995),
[9] H. G. Schuster, Deterministic Chaos (Physik-Verlag, Weinheim 1984),
[10] J. Guckenheimer, G. Oster, and A. Ipatchki, : The Dynamics of Density Dependent Population Models, J. Math. Biol. 4 (1977), pp 101-147
[11] J. Singer, “Controlling a Chaotic System”, Physical Review Leters, vol 66, (1991) pp 1230-1232,
[12] J. Guemez and M.A. Matias, Control of chaos in unidimensional maps, Physics Letters A 181, (1993) pp 29-32.
[13] L. M. Saha and M. Budhraja, The Largest Eigenvalue : An indicator of Chaos? Int. J. of Appl. Math and Mech. 3(1) (2007) pp 61-71.
[14] Maria I. Loffredo : Testing chaos and fractal properties in economic time series,www.internationalmathematicasymposium.org/IMS99/paper25/ims99.
[15] M.A. Matias and J. Guemez, Stabilization of chaos by proportional pulses in system variables, Physical Review Letters 72, (1994), pp 1455-1458.
[16] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations." J. Stat. Phys. 21, 669-706, 1979.
[17] M. J. Smith : Mathematical Ideas in Biology, CambridgeUniv. Press, 1968
[18] N.P. Chau, Controlling chaos by periodic proportional pulses, Phys. Lett. A, 234 (1997), pp 193-197
[19] P. Rohani, O. Miramontes and M. P. Hassell, Quasiperiodicity and chaos in population models, Proc. R. Soc. Lond. B (1994) 258, pp 17-22
[20] R. M. May : Biological Populations Obeying Difference Equations : Stable Points, Periodic Orbits and Chaos, J. Theor Biol, 51 (1975), pp 511-524
[21] R. C. Hilborn, Chaos and nonlinear dynamics, Oxford University Press 1994
[22] T. Kapitaniak, Controlling Chaos : Theoretical and Practical Methods in Non-Linear Dynamics, Academic Press, New York, 1996
[23] T. Shinbrot, “Using Small perturbations to Control Chaos”, Nature, Vol. 363 (1993) pp 411-417.
[24] W. L. Ditto, “Experimental control of Chaos”, Physical Review Letters, vol 65, No. 26, (1990) pp 3211-3214,
[25] Y. T. Li and J. A. Yorke : “Period Three Implies Chaos”, Amer. Math. Monthly 82 (1975), pp 985-992
[26] Z. Gills, “Tracking Unstable Steady States: Extending the Stability Regime of a Multimode Lazer System”, Physical Review Letters, vol. 69, No. 22, (1992) pp 3169-3172.

@article{hemanta02021001,
title = "Chaotic behavior and its control in a two parameter map with variable Jacobian ",
journal = "International Journal of Science and Engineering Applications (IJSEA)",
volume = "2",
number = "2",
pages = "26 - 33",
year = "2013",
author = "Hemanta Kr. Sarmah, Ranu Paul ",
}