An Aperiodic Stable Chaos with Lyapunov Exponents in Time Series
Keywords:
Fractional Fourier Transform, Lyapunov Exponents, Aperiodic stable Chaos, Periodic instable attractors.Abstract
A new formula is developed to reproduce the shape of energy profiles for aperiodic stable attractors with Lyapunov exponents, by using the Fractional Fourier Transform (FRFT), .i.e.
where, is the initial angular frequency of the of the attractor and , the time of flight of the attractor. With, the energy profile for periodic unstable attractors at different values of Lyapunov exponents is obtained, for aperiodic stable attraction at different values of Lyapunov exponents are observed. The critical analysis about chaos is presented with emphasis to time series modeling and simulation.
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[2] J.H Lefebvre, D.A. Goodings, M.V. Kannath, and E.L. Fallen, “Predictability of normal heart rhythms and
deterministic chaos,” Chaos, vol. 3, pp. 267–277, 1993.
[3] D.M. Rubin, “Use of forecasting signatures to help distinguish periodicity, randomness, and chaos in ripples
and other spatial patterns” Chaos, vol. 2, pp. 525–535, 1992.
[4] J.D.Berger, “StatiRstical decision theory and [4Bayesian analysis,” Springer Verlag ,Berlin, 1982.
[5] M. Casdagli and J. Roy, “Chaos and deterministic versus stochastic nonlinear modeling,” Stat. Soc, B., vol.
54, pp. 303–328, 1991.
[6] Y.Z. Jafri, “Is chaotic time series deterministic?” Sci International (LHR), vol. 8, no.1, pp. 13–14, 1996.
[7] H. Casdagli, “A dynamical system approach to modeling input-output system in non-linear modeling and
forecasting,” SFI (Scienti?c Forum International) studies in the Series of Complexity Process, vol. 12,
pp. 265–281, 1992.
[8] S. Neil Rasband, “Chaotic dynamics of non-linear systems,” Wiley, 1997
[9] J.D. Farmer, “Chaotic attractors of in?nite dimensional dynamic systems,” Physica D: Nonlinear
Phenomena vol, 4. no.3 pp. 366-393 ,1982
[10] E. Canessa and A. Calmetta, “Physics of biological random process,” Phy Rev.E.50, 1994.
[11] P.V. Baylay and N.L. Virgin, “Practical considerations in the control of chaos,” Physical Review E
50, no. 1, pp. 604, 1994.
[12] L.X. Wang, “Adaptive fuzzy system and control, design and stability analysis,” Prentice Hall, Englewood
cliffs, 1994.
[13] E. Lorenz, “Deterministic no periodic ?ow,” Atmospheric Science, vol. 20, pp. 130, 1963.
[14] O.E. Rossler, “An equation for continuous chaos,” Physical Review Letters, vol. 57A, pp. 397–398, 1976. [15] Y.Z Jafri, L. Kamal, and S. Massod Raza, “Chaotic time series prediction and Meckey Glass simulation
with fuzzy logic,” International Journal of the Physical Sciences, vol. 7, no. 17, pp. 2596–2606, 2012.
[16] Y.Z Jafri, L. Kamal, and S. Massod Raza, “Rule based fuzzy logic time series prediction,” International
Journal of the Physical Sciences, vol. 7, no. 11, pp. 1862–1873, 2012.
[17] L.Kamal and Y.Z.Jafri, “Stochastic modelling and generation of synthetic sequences of hourly global solar
radiation at Quetta, Pakistan,” Renewable Energy, vol. 18, pp. 565–572, 1999.
[18] L. Kamal and Y.Z. Jafri, “Time series models to simulate and forecast hourly averaged wind speed in
Quetta, Pakistan,” Renewable Energy, vol. 61, no. 1, pp. 23–32, 1997.
[19] M. Casdagli, J.and Roy, “Chaos and deterministic versus stochastic nonlinear modeling” Jr. Royal
Stat.Soc.B, vol. B54, pp. 303–328, 1991.
[20] B. Kosko, “Fuzzy engineering,” 1997.
[21] Y.Z Jafri, “Resents trends in time series modelling, simulation and prediction: Statistical and fuzzy logic
approach, PH.D thesis,” Ph. D thesis, University of Balochistan, Quetta, Pakistan, 2007.
[22] L. Kamal and Y.Z. Jafri, “Information theory in stochastic process,” Science International Lahore, vol. 9,
no. 3, pp. 329–330, 1997.
[23] L.E. Reided, “The transition to chaos,” no. 3, 1992.
[24] L. Kamal and Y.Z. Jafri, “Disorder in a stochastic processes,” vol. 7, no. 4, pp. 465–467, 1995.
[25] R.N. Bracewell, “The Fourier transform,” Scienti?c American, vol. 260, no. 6, pp. 86–95, 2000.
[26] L.B.Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Transactions
on signal processing, vol. 42, no. 11, pp. 3084–3091, 1994.
[27] S. Iqbal, F. Sarwar, S.M.Raza, A.Rehman “How fractional charge on an electron in the momentum space
is quantized? American Scientific Research Journal for Engineering, Technology, and Sciences vol. 14(2),
pp.265-272, Nov 2015.
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Published
2016-01-23
How to Cite
Iqbal, S., Sarwar, F., Raza, S. M., Rehman, A., & Jafri, Y. Z. (2016). An Aperiodic Stable Chaos with Lyapunov Exponents in Time Series. American Scientific Research Journal for Engineering, Technology, and Sciences, 15(1), 282–289. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/1305
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