American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ An Aperiodic Stable Chaos with Lyapunov Exponents in Time Series Saleem Iqbal a*, Farhana Sarwar b, Syed Mohsin Raza c , Abdul Rehman d, Yasmin Zahra Jafri e a,dDepartment of Mathematics, University of Balochistan, Quetta 87300, Pakistan bDepartment of Mathematics F.G.Girls Degree College, Madrissa Road , Quetta, Cantt, 87300, Pakistan. cDepartmen of Physics, University of Balochistan, Quetta 87300, Pakistan eDepartmen of Statistics, University of Balochistan, Quetta 87300, Pakistan aEmail: saleemiqbal81@yahoo.com bEmail: f_saleem10@yahoo.com cEmail: smraza7@yahoo.com dEmail: abdul_maths@yahoo.com eEmail: jafri_yasmin@yahoo.com Abstract A new formula is developed to reproduce the shape of energy profiles for aperiodic stable attractors with Lyapunov exponents, ±𝑛𝑓 by using the Fractional Fourier Transform (FRFT), .i.e. πœ“Β±π‘›π‘“(Β±πœ”πœŠπ‘‘π‘Žπ‘‘π‘‘) = οΏ½ πœ”0𝑑 βˆšπœ‹ οΏ½ 1 2 2Β± 𝑛𝑓 2 where πœ‹ 2 < πœ”πœŠπ‘‘ ≀ 2πœ‹, 0 ≀ πœ”πœŠπ‘‘ ≀ πœ‹ 2 , πœ”πœŠ is the initial angular frequency of the of the attractor and tatt , the time of flight of the attractor. With πœ”πœŠπ‘‘ = πœ‹ 2 , the energy profile for periodic unstable attractors at different values of Lyapunov exponents ±𝑛𝑓 is obtained, for πœ‹ 2 < πœ”πœŠπ‘‘ ≀ 2 πœ‹ π‘Žnd 0 ≀ πœ”πœŠπ‘‘ ≀ πœ‹ 2 aperiodic stable attraction at different values of Lyapunov exponents ±𝑛𝑓 are observed. The critical analysis about chaos is presented with emphasis to time series modeling and simulation. ------------------------------------------------------------------ * Corresponding author. 282 http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 15, No 1, pp 282-289 Keywords: Fractional Fourier Transform; Lyapunov Exponents; Aperiodic stable Chaos; Periodic instable attractors. 1. Introduction Deterministic nonlinear model techniques of complex data series, .i.e., uncorrelated series with a flat Fourier Spectrum have received a great deal of attention because it can be used to forecast the evaluation of a chaotic system [1]. The dynamic variables are usually unknown in chaotic systems. The cross-correlation function between observed and predicted values from nonlinear model techniques such as ARIMA (Auto Regressive Integrated Moving Average) provides estimated values of Lyapunov exponents of dynamic variables even for sparse time series (containing of the order of 103 data points). When the fit is achieved by using nonlinear modeling, we can say it is better than probalistic models provided a deterministic mechanism governs the process under study [2,3]. Most of the non-linear modeling techniques follow Non-Bayesian statistics and are grouped in to major classes: global and local. The local non-Bayesian statistics has more advantages as compared to global non-Bayesian statistics. In the Bayesian approach [4] one assumes that the prior uncertainty about unknown parameters which have to be inferred from random data or from a stochastic process, can also be encoded in a probability distribution, the so called prior. The problem of distinguishing chaos from correlated noise or combinations of deterministic and randomness is a more difficult task [5]. Jafri [6] exploited concepts of Bayesian and non-Bayesian statistics to prove mathematically that the chaotic time series is deterministic. She assumed that the optimum metric (used as tool to distinguish chaos from correlated noise) arises from a metric tensor whose components are 𝛿𝑖𝑗 = πœ•π‘–π‘—π‘’2𝑖𝑐 (1) where 𝛿𝑖𝑗 is the kronecker delta function and i and j run from i to 𝑑 (embedding dimension on the prediction interval) for input-output data pairs. If the parameter 𝑐 in equation (1) is varied to minimize the root mean square error of the forecast, then there is a single global minimum corresponding to a value of c closed to the most negative Lyapunov exponent of the dynamics. Chaos is having an impact on diverse discipline of knowledge including physics, biology, chemistry, economics and medicine [7,12]. The chaos may behave almost linearly in some part of phase space and highly non-linearly in other parts. Lorenz and Rossele [13,14], identified non-stable chaos approximately in to different (chaotic and non-chaotic) nonlinear dynamical system with their attractors. But, the recognition of attractors in a chaotic system is indeed difficult. Jafri [15,16,21], identified an aperiodic stable chaos with an attractor in Mackey- Glass simulation on hourly wind data and indeed followed a free fuzzy logic (FFL) design for prediction. The stable chaotic attractors do not influence the time series prediction. Jafri [16, 21], also studied the rule based on fuzzy logic time series prediction, feed forward back propagation neural networks (FFBPNN) and artificial neuro fuzzy information system (ANFIS). Her ARIMA and SARIMA (seasonal auto regressive integrated moving average) modeling techniques and statistical tests [16, 21], unraveled many hidden information with particular emphasis to aperiodic stable chaotic attractors. Lalarukh and Jafri [17] an ARMA process on hourly global radiation data, performed stochastic modeling through MTM (Markov Transition Matrix) and generated synthetic sequences of hourly global solar radiation. They found MTM approached relatively better as a simulator compared to ARMA (auto regressive moving average). But, their 283 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 15, No 1, pp 282-289 analysis for ARMA process to simulate and forecast hourly averaged wind speed yielded good results [18]. We observed with critical analysis [1,4,6,14], [18-20] and with experimental data modeling and simulation [15,18, 21] that the aperiodic attractors (usually with sparse time delays) produced stable chaos whereas the periodic attractors the unstable chaos in any nonlinear dynamical systems. Jafri [22] suggested methods for Kolmogorov- Sinai disorder in terms of β€œpoint stochastic process” and found that this stochastic process can be measured with a positive Lyapunov exponent. The point stochastic process deals with discrete, independent and identically distributed random data. The non-stationary stochastic process other than point stochastic process in dynamical systems depend on negative Lyapunov exponents. The positive Lyapunov exponents are preferably found in point stochastic processes [2,3,23,24] where disorder introduces chaos in dynamical systems. Such chaotic dynamical system have been studied by real valued tine series for random process [6,15,18, 21,22,17,18], Jafri [24] also studied disorder in a stochastic system in the form of Fisher in- formation matrix in probability function, using the partition in the family of random variables for their corresponding degree of randomness for one-step prediction in time series, as a consequence of which, we obtained the disjoint sets of random variables. There is a literal difference between fractional exponents of Heaviside step function and the Lyapunove exponents. The Lyapunov exponents define attractors both in arbitrary positive and negative directions( hence positive and negative Lyapunov exponents) while the fractional exponents of exponential function in the denominator for the Heaviside step function show fragmented wavelet or wave fronts ( piecewise step function) and do not represent attractors. With this limitation, the Fourier Transform is changed in to Laplace transform. Using the generalized definitions of Heaviside step function. By Bracewell [25], we observe that 𝐻(π‘₯) changes between 1 and 1 2⁄ for π‘₯ β‰₯ 0 and changes between 0 and 1 2⁄ for π‘₯ ≀ 0. Few of the expressions of the Heaviside step function [25] are written to reflect our conjectures true in the above paragraph. 𝐻𝑛(π‘₯) = lim 𝑑→0 οΏ½ 1 2 𝑒π‘₯ 𝑑⁄ , if π‘₯ ≀ 0 1 βˆ’ 1 2 π‘’βˆ’π‘₯ 𝑑⁄ , if π‘₯ β‰₯ 0 (2) where π‘₯ 𝑑⁄ is a fractional exponent. The aperiodic stable chaos differs from periodic unstable chaos [20].The unstable periodic chaos [13], [14], [20] is associated with multiple iterations and for each iterations, periodicity is maintained whereas the stable aperiodic chaos [15] follow a singlet iteration (closed loop). We shall consider the time -frequency representation of Fractional Fourier Transform (FRFT), [26]. Such as the Wigner distribution (WD), Wiener Space (WS), the ambiguity function, the short time Fourier transform (STFT) and the Spectrograms used in speech processing, radar, image rotations, confocal microscopy, etc. with rotation vector Ξ± to define the aperiodic stable chaos with Lyapunov exponents in time series. 2. Results and Discussions Almeida [26] define the kernel of FRFT in time frequency plane, .i.e., 284 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 15, No 1, pp 282-289 𝐾𝛼(𝑑,𝑒) = ⎩ βŽͺ ⎨ βŽͺ ⎧�1βˆ’π‘— cot 𝛼 2πœ‹ 𝑒𝑗 𝑑2+𝑒2 2 cot π›Όβˆ’π‘—π‘’π‘‘ csc 𝛼 if 𝛼 β‰  πœ‹ 𝛿(𝑑 βˆ’ 𝑒) if 𝛼 = 2π‘›πœ‹ 𝛿(𝑑 + 𝑒) if 𝛼 = (2𝑛 βˆ’ 1)πœ‹ (3) where 𝑛 is an integer. With 𝛼 = πœ‹ 2 , the kernel in eq (3) coincides with the kernel of the Fourier transform (FT). Almeida [26] redefined the FRFT of a functionπ‘₯(𝑑), with an angle 𝛼, which is define in eq (4) ℱ𝛼[π‘₯(𝑑)] = 𝑋𝛼(𝑒) = οΏ½ π‘₯(𝑑) ∞ βˆ’βˆž 𝐾𝛼(𝑑,𝑒)𝑑𝑑 = ⎩ ⎨ ⎧�1βˆ’π‘— cot 𝛼 2πœ‹ 𝑒𝑗 𝑒2 2 cot 𝛼 ∫ π‘₯(𝑑)𝑒𝑗 𝑑2 2 cot π›Όβˆ’π‘—π‘’π‘‘ csc 𝛼𝑑𝑑 𝑖𝑓 𝛼 𝑖𝑠 π‘›π‘œπ‘‘ π‘Ž π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ π‘œπ‘“ πœ‹ ∞ βˆ’βˆž π‘₯(𝑑) 𝑖𝑓 𝛼 𝑖𝑠 π‘›π‘œπ‘‘ π‘Ž π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ π‘œπ‘“ 2πœ‹ π‘₯(βˆ’π‘‘) 𝑖𝑓 𝛼 + πœ‹ 𝑖𝑠 π‘›π‘œπ‘‘ π‘Ž π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ π‘œπ‘“ 2πœ‹ (4) Where 𝑒 ≑ πœ” (the angular frequency). We assume FRFT equivalent to 𝐻𝑛𝑓(π‘₯)𝑒π‘₯𝑝 οΏ½βˆ’ π‘₯2 2 οΏ½ and time periodic signal π‘₯ = 𝐴 sinπœ”πœŠ as Gaussian and exponentially correlated. A is real constant in π‘₯(𝑑) while 𝐻𝑛𝑓(π‘₯) is a Hermite polynomial for 𝑛𝑓 exponents (usually termed as Lyapunov exponents). The exponent of 𝑋𝛼(𝑒 ≑ πœ”) in eq (4) is 𝛼. We assume the exponent 𝛼 of 𝑋𝛼(𝑒) to become Lyapunov exponents 𝑋𝛼(𝑒 ≑ πœ”) ~𝐻𝑛𝑓(π‘₯)exp (βˆ’π‘₯2 2 ) β€˜ (5) where 𝛼 = π‘Ž πœ‹ 2 . Using our recent results on how fractional charge on an electron in the momentum space is quantized? [27], we find 𝐻𝑛𝑓(πœ‰) = 2𝑛𝑓 (6) Where 𝐻𝑛𝑓(πœ‰) are polynomials with 𝑛𝑓, Lyaponov exponent of 2. Eq (6) is consistent with the following definition of Hermite polynomials: 𝐻𝑛𝑓(πœ‰) = (βˆ’1)π‘›π‘“π‘’πœ‰2 𝑑𝑛𝑓 π‘‘πœ‰π‘›π‘“ π‘’βˆ’πœ‰2 (7) 𝐻𝑛𝑓(πœ‰) = (βˆ’1)π‘›π‘“π‘’πœ‰2 οΏ½πœ‰ βˆ’ 𝑑 π‘‘πœ‰ οΏ½ 𝑛𝑓 π‘’βˆ’πœ‰2 (8) Eqs (7) and (8) show fractional exponents of 𝑑 π‘‘πœ‰ and οΏ½πœ‰ βˆ’ 𝑑 π‘‘πœ‰ οΏ½. The fractional exponents, 𝑛𝑓 can be envisaged 285 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 15, No 1, pp 282-289 with Lyapunov exponents for attractors. The Lyaponov exponents are both positive and negative fractional exponents. Hence, eq(6) is modified as 𝐻𝑛𝑓(πœ‰) = 2±𝑛𝑓 (9) With the Hermite generating function, 𝐺(πœ‰, 𝑠) = π‘’βˆ’π‘ 2+2π‘ πœ‰ βˆ‘ 𝐻𝑛𝑓+1(πœ‰)𝑠𝑛 𝑛! ∞ 𝑛=0 (10) and for 𝑛 = 0, we can prove that the Hermite polynomials satisfy the recursion relations: οΏ½ 𝐻𝑛𝑓+1(πœ‰) βˆ’ 2πœ‰π»π‘›π‘“ + π»π‘›π‘“βˆ’1(πœ‰) = 0 𝑑 π‘‘πœ‰ 𝐻𝑛𝑓(πœ‰) = 2π‘›π‘“π»π‘›π‘“βˆ’1(πœ‰) (11) Eq(10) shows that if the function π‘’βˆ’π‘ 2+2π‘ πœ‰ is expanded in a power series in 𝑠, the coefficients of successive powers of 𝑠 are just 1 𝑛! times the Hermite polynomials, 𝐻𝑛𝑓. Using eq(11) for each of the fractional exponents,𝑛𝑓 on πœ†π‘“ = 2𝑛𝑓+1 for 𝐸𝑛𝑓 = �𝑛𝑓 + 1 2οΏ½β„πœ”, there is only one physically acceptable solution for energy profile of the attractor πœ“π‘›π‘“(πœ‰) = 𝑁𝑛𝑓𝑒 βˆ’π›Ό2πœ‰ 2 2 𝐻𝑛𝑓(πœ‰) (12) where πœ‰ ≑ π‘₯(𝑑) and 𝑠 ≑ πœ” this implies that 𝑑~π›Όπœ‰. Considering the Hermite generating function eq(10) and equating the t coefficient of equal parts of 𝑠 ≑ πœ” and 𝑑~π›Όπœ‰ , the normalized eigen functions for Lyaponov exponents are given by πœ“π‘›π‘“οΏ½π‘₯(𝑑)οΏ½ = οΏ½ 𝛼 βˆšπœ‹2 𝑛𝑓𝑛! οΏ½ 1 2 π‘’βˆ’ 𝛼2π‘₯2(𝑑) 2 𝐻𝑛𝑓(π›Όπœ‰) (13) For Lyaponov exponents , ±𝑛𝑓 , 𝛼 = πœ”πœŠπ‘‘ , 𝑛! = 0! = 1 and π‘₯(𝑑) = 𝐴 sinπœ”πœŠπ‘‘ as Gaussian, using eq(9) in eq(13). Eq (13) is modified as πœ“π‘›π‘“(𝐴 sinπœ”πœŠπ‘‘ ) = οΏ½ πœ”πœŠπ‘‘ βˆšπœ‹2 ±𝑛𝑓� 1 2 π‘’βˆ’ 𝛼2(𝐴sinπœ”πœŠπ‘‘)2 2 Γ— 2±𝑛𝑓 = οΏ½ πœ”πœŠπ‘‘ βˆšπœ‹2 ±𝑛𝑓� 1 2 2 ±𝑛𝑓 2 π‘’βˆ’π›Ό 2�𝐴sinπœ”πœŠπ‘‘ √2 οΏ½ 2 (14) the term π‘’βˆ’ 𝛼2(𝐴sinπœ”πœŠπ‘‘)2 2 ~π‘’βˆ’οΏ½ πœ”πœŠπ‘‘ 𝐴sinπœ”πœŠπ‘‘ √2 οΏ½ 2 is a convergent series and tends to unity in eq (14). With sinπœ”πœŠπ‘‘~πœ”πœŠπ‘‘ = 𝛼 , eq (14) becomes 286 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 15, No 1, pp 282-289 πœ“π‘›π‘“(πœ”πœŠπ‘‘) = οΏ½πœ”πœŠπ‘‘ βˆšπœ‹ οΏ½ 1 2 2 ±𝑛𝑓 2 (15) In a time series following aperiodic stable chaos (stationary stochastic process), the coefficient 𝐴 of sinπœ”πœŠπ‘‘ is a real constant for an attractor with frequencyπœ”πœŠ. At any time in a time series in the plane (𝑑,πœ”) [Wiener space], .i.e., 𝑑 = π‘‘π‘Žπ‘‘π‘‘ (time of flight for an attractor), a phase change of πœ”πœŠπ‘‘π‘Žπ‘‘π‘‘ will cause a change in energy profile and hence an aperiodic stable chaotic attractors will be produced on a time series. The coefficient 𝐴, being a real constant, can be quantized with simulation. Equation (15) can be modified as πœ“Β±π‘›π‘“(Β±πœ”πœŠπ‘‘π‘Žπ‘‘π‘‘) = οΏ½πœ”πœŠπ‘‘ βˆšπœ‹ οΏ½ 1 2 2 ±𝑛𝑓 2 (16) where πœ‹ 2 < πœ”πœŠπ‘‘ ≀ 2πœ‹ and 0 ≀ πœ”πœŠπ‘‘ < πœ‹ 2 , .i.e., πœ”πœŠπ‘‘ = 𝛼 β‰  πœ‹ 2 Eq(16) represents shape of profile for aperiodic stable attractors with Lyapunov exponents, ±𝑛𝑓. The periodic unstable attractors at 𝛼 = πœ”πœŠπ‘‘ = πœ‹ 2 will be produced following the FT. Eq (16) obeys WD and WS in (𝑑,πœ”) plane. All attractors whether periodic or aperiodic are, in fact, the manifestation of Lyaponov exponents. 3. Conclusion The shape of energy profile for aperiodic stable attractors with Lyaponov exponents ±𝑛𝑓 is determined by using fractional Fourier transform. The formula aperiodic stable attractors eq (16) is established. Withπœ”πœŠπ‘‘ = πœ‹ 2 , it is found that the enegy profile exists for periodic unstable attractors at different values of Lyaponov exponents, 𝑛𝑓.Critical analysis about chaos is revisited with emphasis on time series modeling. References [1] J. Jimenez, J.A. Moreno, and G.J. Ruggieri, β€œForecasting on chaotic time series: A local optimal linear reconstruction method,” Phy.Rev.A., vol. 45, no. 6, pp. 3553–3558, 1992. [2] J.H Lefebvre, D.A. Goodings, M.V. Kannath, and E.L. Fallen, β€œPredictability of normal heart rhythms and deterministic chaos,” Chaos, vol. 3, pp. 267–277, 1993. [3] D.M. Rubin, β€œUse 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, β€œStatiRstical decision theory and [4Bayesian analysis,” Springer Verlag ,Berlin, 1982. [5] M. Casdagli and J. Roy, β€œChaos and deterministic versus stochastic nonlinear modeling,” Stat. Soc, B., vol. 54, pp. 303–328, 1991. 287 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 15, No 1, pp 282-289 [6] Y.Z. Jafri, β€œIs chaotic time series deterministic?” Sci International (LHR), vol. 8, no.1, pp. 13–14, 1996. [7] H. Casdagli, β€œA dynamical system approach to modeling input-output system in non-linear modeling and forecasting,” SFI (Scientific Forum International) studies in the Series of Complexity Process, vol. 12, pp. 265– 281, 1992. [8] S. Neil Rasband, β€œChaotic dynamics of non-linear systems,” Wiley, 1997 [9] J.D. Farmer, β€œChaotic attractors of infinite dimensional dynamic systems,” Physica D: Nonlinear Phenomena vol, 4. no.3 pp. 366-393 ,1982 [10] E. Canessa and A. Calmetta, β€œPhysics of biological random process,” Phy Rev.E.50, 1994. [11] P.V. Baylay and N.L. Virgin, β€œPractical considerations in the control of chaos,” Physical Review E 50, no. 1, pp. 604, 1994. [12] L.X. Wang, β€œAdaptive fuzzy system and control, design and stability analysis,” Prentice Hall, Englewoodcliffs, 1994. [13] E. Lorenz, β€œDeterministic no periodic flow,” Atmospheric Science, vol. 20, pp. 130, 1963. [14] O.E. Rossler, β€œAn equation for continuous chaos,” Physical Review Letters, vol. 57A, pp. 397–398, 1976. [15] Y.Z Jafri, L. Kamal, and S. Massod Raza, β€œChaotic 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, β€œRule 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, β€œStochastic 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, β€œTime 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, β€œChaos and deterministic versus stochastic nonlinear modeling” Jr. Royal Stat.Soc.B, vol. B54, pp. 303–328, 1991. [20] B. Kosko, β€œFuzzy engineering,” 1997. [21] Y.Z Jafri, β€œResents 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. 288 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 15, No 1, pp 282-289 [22] L. Kamal and Y.Z. Jafri, β€œInformation theory in stochastic process,” Science International Lahore, vol. 9, no. 3, pp. 329–330, 1997. [23] L.E. Reided, β€œThe transition to chaos,” no. 3, 1992. [24] L. Kamal and Y.Z. Jafri, β€œDisorder in a stochastic processes,” vol. 7, no. 4, pp. 465–467, 1995. [25] R.N. Bracewell, β€œThe Fourier transform,” Scientific American, vol. 260, no. 6, pp. 86–95, 2000. [26] L.B.Almeida, β€œThe 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 β€œHow 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. 289