Applied Mathematics
pISSN: 21631409 eISSN: 21631425
2019; 9(2): 2548
doi:10.5923/j.am.20190902.01
Salah H. Abid, Saad S. Mahmood, Yaseen A. Oraibi
Department of Mathematics, College of Education, ALMustansiriyah University, Baghdad, Iraq
Correspondence to: Salah H. Abid, Department of Mathematics, College of Education, ALMustansiriyah University, Baghdad, Iraq.
Email: 
Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
The aim of this paper is to design a feed forward artificial neural network (Ann) to estimate two dimensional Tinkerbell dynamical map by selecting an appropriate network, transfer function and node weights. The proposed network side by side with using Fast Fourier Transform (FFT) as transfer function is used. For different cases of the system, chaotic and noisy, the experimental results of proposed algorithm will compared empirically, by means of the mean square error (MSE) with the results of the same network but with traditional transfer functions, Logsig and Tansig. The performance of proposed algorithm is best from others in all cases from Both sides, speed and accuracy.
Keywords: FFT, Logsig, Tansig, Feed Forward neural network, Transfer function, Tinkerbell map, Logistic noise, Normal noise
Cite this paper: Salah H. Abid, Saad S. Mahmood, Yaseen A. Oraibi, Estimation of Tinkerbell Dynamical Map by Using Neural Network with FFT as Transfer Function, Applied Mathematics, Vol. 9 No. 2, 2019, pp. 2548. doi: 10.5923/j.am.20190902.01.
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Figure 1. Tinkerbell attractor for a = 0.9 and b = 0.6103, c=2, d=0.5 
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Figure 2. Flowchart for training algorithm with BFGS 

Figure 3. Real and approximate Tinkerbell dynamical map with initial point x=0.1 using FFT transfer function 
Figure 4. Real and approximate Tinkerbell dynamical map with initial point x=0.4 using FFT transfer function 
Figure 5. Real and approximate Tinkerbell dynamical map with initial point x=0.7 using FFT transfer function 
Figure 6. Real and approximate Tinkerbell dynamical map with initial point x=0.9 using FFT transfer function 
Figure 7. Real and approximate Tinkerbell dynamical map with initial point y=0.1 using FFT transfer function 
Figure 8. Real and approximate Tinkerbell dynamical map with initial point y=0.4 using FFT transfer function 
Figure 9. Real and approximate Tinkerbell dynamical map with initial point y=0.7 using FFT transfer function 
Figure 10. Real and approximate Tinkerbell dynamical map with initial point y=0.9 using FFT transfer function 
Figure 11. Real and approximate Tinkerbell dynamical map with initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 12. Real and approximate Tinkerbell dynamical map with initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 13. Real and approximate Tinkerbell dynamical map with initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 14. Real and approximate Tinkerbell dynamical map with initial points x=0.9 & y=0.9 using FFT transfer function 
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Figure 15. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point x=0.1 using FFT transfer function 
Figure 16. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point x=0.4 using FFT transfer function 
Figure 17. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point x=0.7 using FFT transfer function 
Figure 18. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point x=0.9 using FFT transfer function 
Figure 19. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point y=0.1 using FFT transfer function 
Figure 20. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point y=0.4 using FFT transfer function 
Figure 21. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point y=0.7 using FFT transfer function 
Figure 22. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial point y=0.9 using FFT transfer function 
Figure 23. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 24. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 25. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 26. Real and approximate Tinkerbell dynamical map with normal noise (v=0.05) and initial points x=0.9 & y=0.9 using FFT transfer function 

Figure 27. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point x=0.1 using FFT transfer function 
Figure 28. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point x=0.4 using FFT transfer function 
Figure 29. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point x=0.7 using FFT transfer function 
Figure 30. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point x=0.9 using FFT transfer function 
Figure 31. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point y=0.1 using FFT transfer function 
Figure 32. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point y=0.4 using FFT transfer function 
Figure 33. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point y=0.7 using FFT transfer function 
Figure 34. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial point y=0.9 using FFT transfer function 
Figure 35. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 36. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 37. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 38. Real and approximate Tinkerbell dynamical map with normal noise (v=0.5) and initial points x=0.9 & y=0.9 using FFT transfer function 

Figure 39. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point x=0.1 using FFT transfer function 
Figure 40. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point x=0.4 using FFT transfer function 
Figure 41. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point x=0.7 using FFT transfer function 
Figure 42. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point x=0.9 using FFT transfer function 
Figure 43. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point y=0.1 using FFT transfer function 
Figure 44. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point y=0.4 using FFT transfer function 
Figure 45. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point y=0.7 using FFT transfer function 
Figure 46. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial point y=0.9 using FFT transfer function 
Figure 47. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 48. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 49. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 50. Real and approximate Tinkerbell dynamical map with normal noise (v=40) and initial points x=0.9 & y=0.9 using FFT transfer function 

Figure 51. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point x=0.1 using FFT transfer function 
Figure 52. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point x=0.4 using FFT transfer function 
Figure 53. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point x=0.7 using FFT transfer function 
Figure 54. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point x=0.9 using FFT transfer function 
Figure 55. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point y=0.1 using FFT transfer function 
Figure 56. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point y=0.4 using FFT transfer function 
Figure 57. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point y=0.7 using FFT transfer function 
Figure 58. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial point y=0.9 using FFT transfer function 
Figure 59. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 60. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 61. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 62. Real and approximate Tinkerbell dynamical map with normal noise (v=60) and initial points x=0.9 & y=0.9 using FFT transfer function 

Figure 63. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point x=0.1 using FFT transfer function 
Figure 64. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point x=0.4 using FFT transfer function 
Figure 65. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point x=0.7 using FFT transfer function 
Figure 66. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point x=0.9 using FFT transfer function 
Figure 67. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point y=0.1 using FFT transfer function 
Figure 68. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point y=0.4 using FFT transfer function 
Figure 69. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point y=0.7 using FFT transfer function 
Figure 70. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial point y=0.9 using FFT transfer function 
Figure 71. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 72. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 73. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 74. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.05) and initial points x=0.9 & y=0.9 using FFT transfer function 

Figure 75. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point x=0.1 using FFT transfer function 
Figure 76. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point x=0.4 using FFT transfer function 
Figure 77. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point x=0.7 using FFT transfer function 
Figure 78. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point x=0.9 using FFT transfer function 
Figure 79. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point y=0.1 using FFT transfer function 
Figure 80. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point y=0.4 using FFT transfer function 
Figure 81. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point y=0.7 using FFT transfer function 
Figure 82. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial point y=0.9 using FFT transfer function 
Figure 83. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 84. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 85. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 86. Real and approximate Tinkerbell dynamical map with logistic noise (v=0.5) and initial points x=0.9 & y=0.9 using FFT transfer function 

Figure 87. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point x=0.1 using FFT transfer function 
Figure 88. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point x=0.4 using FFT transfer function 
Figure 89. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point x=0.7 using FFT transfer function 
Figure 90. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point x=0.9 using FFT transfer function 
Figure 91. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point y=0.1 using FFT transfer function 
Figure 92. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point y=0.4 using FFT transfer function 
Figure 93. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point y=0.7 using FFT transfer function 
Figure 94. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial point y=0.9 using FFT transfer function 
Figure 95. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 96. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 97. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 98. Real and approximate Tinkerbell dynamical map with logistic noise (v=40) and initial points x=0.9 & y=0.9 using FFT transfer function 

Figure 99. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point x=0.1 using FFT transfer function 
Figure 100. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point x=0.4 using FFT transfer function 
Figure 101. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point x=0.7 using FFT transfer function 
Figure 102. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point x=0.9 using FFT transfer function 
Figure 103. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point y=0.1 using FFT transfer function 
Figure 104. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point y=0.4 using FFT transfer function 
Figure 105. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point y=0.7 using FFT transfer function 
Figure 106. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial point y=0.9 using FFT transfer function 
Figure 107. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial points x=0.1 & y=0.1 using FFT transfer function 
Figure 108. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial points x=0.4 & y=0.4 using FFT transfer function 
Figure 109. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial points x=0.7 & y=0.7 using FFT transfer function 
Figure 110. Real and approximate Tinkerbell dynamical map with logistic noise (v=60) and initial points x=0.9 & y=0.9 using FFT transfer function 