用平方映射理解tanh
Tanh的表达式
对tanh求导
对其进行反向传导就是 d*（1+tanhx）（1-tanhx）
而平方映射为μx(1-x)
d对应μ
1+tanhx 对应 x 只是多了一个1
Tanhx（n+1）=d*（1+tanhx（n））（1-tanhx（n））
先写出结论
| d | |
| d<0.86 | 趋近于一个大于零的解 |
| <0.86d<1.19 | 周期解 |
| 1.19<d | 无规则混沌解 |
当d<0.86的时候有一个解就是X=d（1-x^2）这个方程的解图像形如
这时当d=0.86时的图像，图像最终收敛于0.575333034
也就是说当残差小于0.86是对tanh进行反向传导时只要运行次数足够多，比如对0.86这种情况只要迭代次数大于401次就会得到一个确定的值。这样的网络很显然是稳定的。
当d>0.86是图像会分岔，出现倍频周期解
这是d=0.87时的图像 有一大一小两个解
这是d=1.1 时 可以看到两大两小4个解
这是d=1.18可以看到两大两小4个解更加明显，也就是说tanh函数在反向传导时如果d>0.87即便是同样的输入、迭代同样的次数，两次得到的网络也可能不同，或者对同样的输入随着迭代次数的增加输出也不见得稳定。
当d>1.2是图像变的没有规律
比如当d=1.4时
可以想见如果tanh反向传导的残差大于1.4这个网络的性能是很难稳定的，而tanh的输出是-1到1,1.4的残差是可能的。当d更大时输出变化范围更大，更不稳定
对比sigmoid函数
| d | dx（1-x） |
| d<1 | 趋近于0 |
| 1<d<2.8 | 趋近于一个大于零的解 |
| 2.8<d<3.5 | 周期波动 |
| d>3.5 | 混沌 |
具体迭代数据
| * | d(1+f)(1-f) | ||||
| d | 0.1 | 0.8 | -0.86 | 0.87 | 1 |
| f | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
| 1 | 0.099 | 0.792 | -0.8514 | 0.8613 | 0.99 |
| 2 | 0.0990199 | 0.2981888 | -0.23660151 | 0.22460121 | 0.0199 |
| 3 | 0.099019506 | 0.728866752 | -0.81185696 | 0.826112238 | 0.99960399 |
| 4 | 0.099019514 | 0.375002607 | -0.29316391 | 0.276258556 | 0.000791863 |
| 5 | 0.099019514 | 0.687498436 | -0.78608723 | 0.803602653 | 0.999999373 |
| 6 | 0.099019514 | 0.42187672 | -0.3285775 | 0.308173815 | 1.25409E-06 |
| 7 | 0.099019514 | 0.657616026 | -0.76715167 | 0.787375143 | 1 |
| 8 | 0.099019514 | 0.45403293 | -0.35387135 | 0.330635135 | 3.14548E-12 |
| 9 | 0.099019514 | 0.635083279 | -0.75230656 | 0.774891955 | 1 |
| 10 | 0.099019514 | 0.477335383 | -0.37326997 | 0.347601939 | 0 |
| 11 | 0.099019514 | 0.617720746 | -0.7401758 | 0.764880416 | 1 |
| 12 | 0.099019514 | 0.494736864 | -0.38884022 | 0.361013416 | 0 |
| 13 | 0.099019514 | 0.604188348 | -0.72997082 | 0.756612303 | 1 |
| 14 | 0.099019514 | 0.507965152 | -0.40174263 | 0.371957906 | 0 |
| 15 | 0.099019514 | 0.593577124 | -0.72119846 | 0.749633165 | 1 |
| 16 | 0.099019514 | 0.518132959 | -0.4126906 | 0.381103603 | 0 |
| 17 | 0.099019514 | 0.58523059 | -0.71353037 | 0.743641238 | 1 |
| 18 | 0.099019514 | 0.526004126 | -0.422152 | 0.388888007 | 0 |
| 19 | 0.099019514 | 0.578655728 | -0.70673741 | 0.738426523 | 1 |
| 20 | 0.099019514 | 0.532126039 | -0.43044912 | 0.395611855 | 0 |
| 21 | 0.099019514 | 0.573473503 | -0.70065366 | 0.733837396 | 1 |
| 22 | 0.099019514 | 0.536902513 | -0.43781263 | 0.401489928 | 0 |
| 23 | 0.099019514 | 0.569388553 | -0.69515529 | 0.729761079 | 1 |
| 24 | 0.099019514 | 0.54063734 | -0.44441285 | 0.406680428 | 0 |
| 25 | 0.099019514 | 0.566169013 | -0.69014761 | 0.726111596 | 1 |
| d(1+f)(1-f) | ||||||
| d | 1.1 | 1.18 | 1.19 | 1.2 | 1.6 | 1.9 |
| f | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 5 |
| 1 | 1.089 | 1.1682 | 1.1781 | 1.188 | 1.584 | -45.6 |
| 2 | -0.2045131 | -0.43033566 | -0.46162434 | -0.4936128 | -2.4144896 | -3948.884 |
| 3 | 1.053991831 | 0.961477236 | 0.936414537 | 0.907615684 | -7.72761605 | -29627999.3 |
| 4 | -0.12198866 | 0.089162599 | 0.146522099 | 0.211480523 | -93.9456796 | -1.6679E+15 |
| 5 | 1.083630644 | 1.170619037 | 1.164452217 | 1.146331186 | -14119.6651 | -5.2853E+30 |
| 6 | -0.19168091 | -0.43701174 | -0.42357927 | -0.37689023 | -318983908 | -5.3075E+61 |
| 7 | 1.059584272 | 0.954644476 | 0.976490918 | 1.02954451 | -1.628E+17 | -5.352E+123 |
| 8 | -0.13499071 | 0.10461163 | 0.055293929 | -0.07195428 | -4.2407E+34 | -5.443E+247 |
| 9 | 1.079955259 | 1.16708656 | 1.186361672 | 1.193787098 | -2.8773E+69 | #NUM! |
| 10 | -0.1829337 | -0.42726743 | -0.48487028 | -0.51015316 | -1.325E+139 | #NUM! |
| 11 | 1.063188789 | 0.964582205 | 0.910231966 | 0.887692499 | -2.808E+278 | #NUM! |
| 12 | -0.14340744 | 0.08210578 | 0.204058544 | 0.254402432 | #NUM! | #NUM! |
| 13 | 1.077377736 | 1.172045196 | 1.140448532 | 1.122335283 | #NUM! | #NUM! |
| 14 | -0.17681707 | -0.44095413 | -0.3577412 | -0.31156379 | #NUM! | #NUM! |
| 15 | 1.065609298 | 0.950560156 | 1.037705272 | 1.083513609 | #NUM! | #NUM! |
| 16 | -0.14907549 | 0.113793761 | -0.09143036 | -0.20880209 | #NUM! | #NUM! |
| 17 | 1.075554147 | 1.164720156 | 1.180052183 | 1.147682025 | #NUM! | #NUM! |
| 18 | -0.1724984 | -0.42075619 | -0.46710255 | -0.38060884 | #NUM! | #NUM! |
| 19 | 1.067268734 | 0.97109779 | 0.930360092 | 1.026164296 | #NUM! | #NUM! |
| 20 | -0.15296881 | 0.067223518 | 0.159971818 | -0.06361579 | #NUM! | #NUM! |
| 21 | 1.074260599 | 1.174667578 | 1.159546731 | 1.195143637 | #NUM! | #NUM! |
| 22 | -0.16943942 | -0.44821583 | -0.41001286 | -0.51404197 | #NUM! | #NUM! |
| 23 | 1.068419312 | 0.942941037 | 0.989948452 | 0.882913018 | #NUM! | #NUM! |
| 24 | -0.15567181 | 0.130817396 | 0.023802453 | 0.264557524 | #NUM! | #NUM! |
| 25 | 1.073342917 | 1.159806435 | 1.189325797 | 1.11601118 | #NUM! | #NUM! |
| * | d*f*(1-f) | |||||
| d | 0.1 | 0.2 | 0.9 | 1 | 1.1 | 1.2 |
| f | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
| 1 | 0.009 | 0.018 | 0.081 | 0.09 | 0.099 | 0.108 |
| 2 | 0.000908 | 0.003665 | 0.066995 | 0.0819 | 0.098119 | 0.115603 |
| 3 | 9.09E-05 | 0.000736 | 0.056256 | 0.075192 | 0.097341 | 0.122687 |
| 4 | 9.09E-06 | 0.000147 | 0.047782 | 0.069538 | 0.096652 | 0.129162 |
| 5 | 9.09E-07 | 2.95E-05 | 0.040949 | 0.064703 | 0.096041 | 0.134975 |
| 6 | 9.09E-08 | 5.89E-06 | 0.035345 | 0.060516 | 0.095499 | 0.140108 |
| 7 | 9.09E-09 | 1.18E-06 | 0.030686 | 0.056854 | 0.095017 | 0.144573 |
| 8 | 9.09E-10 | 2.36E-07 | 0.02677 | 0.053622 | 0.094588 | 0.148406 |
| 9 | 9.09E-11 | 4.71E-08 | 0.023448 | 0.050746 | 0.094205 | 0.151658 |
| 10 | 9.09E-12 | 9.42E-09 | 0.020608 | 0.048171 | 0.093863 | 0.15439 |
| 11 | 9.09E-13 | 1.88E-09 | 0.018165 | 0.045851 | 0.093558 | 0.156664 |
| 12 | 9.09E-14 | 3.77E-10 | 0.016052 | 0.043749 | 0.093286 | 0.158545 |
| 13 | 9.09E-15 | 7.54E-11 | 0.014215 | 0.041835 | 0.093042 | 0.16009 |
| 14 | 9.09E-16 | 1.51E-11 | 0.012611 | 0.040084 | 0.092824 | 0.161353 |
| 15 | 9.09E-17 | 3.02E-12 | 0.011207 | 0.038478 | 0.092628 | 0.162382 |
| 16 | 9.09E-18 | 6.03E-13 | 0.009973 | 0.036997 | 0.092453 | 0.163217 |
| 17 | 9.09E-19 | 1.21E-13 | 0.008887 | 0.035628 | 0.092296 | 0.163893 |
| 18 | 9.09E-20 | 2.41E-14 | 0.007927 | 0.034359 | 0.092155 | 0.164438 |
| 19 | 9.09E-21 | 4.83E-15 | 0.007078 | 0.033178 | 0.092029 | 0.164878 |
| 20 | 9.09E-22 | 9.65E-16 | 0.006325 | 0.032078 | 0.091915 | 0.165232 |
| 21 | 9.09E-23 | 1.93E-16 | 0.005656 | 0.031049 | 0.091814 | 0.165516 |
| 22 | 9.09E-24 | 3.86E-17 | 0.005062 | 0.030085 | 0.091722 | 0.165745 |
| 23 | 9.09E-25 | 7.72E-18 | 0.004533 | 0.02918 | 0.09164 | 0.165928 |
| 24 | 9.09E-26 | 1.54E-18 | 0.004061 | 0.028328 | 0.091567 | 0.166075 |
| 25 | 9.09E-27 | 3.09E-19 | 0.00364 | 0.027526 | 0.0915 | 0.166193 |
| * | d*f*(1-f) | |||||
| d | 2 | 3 | 3.5 | 3.6 | 3.8 | 4 |
| f | 0.1 | 0.9 | 0.1 | 0.1 | 0.1 | 0.1 |
| 1 | 0.18 | 0.27 | 0.315 | 0.324 | 0.342 | 0.36 |
| 2 | 0.2952 | 0.5913 | 0.755213 | 0.788486 | 0.855137 | 0.9216 |
| 3 | 0.416114 | 0.724993 | 0.647033 | 0.600392 | 0.470736 | 0.289014 |
| 4 | 0.485926 | 0.598135 | 0.799335 | 0.863717 | 0.946746 | 0.821939 |
| 5 | 0.499604 | 0.721109 | 0.561396 | 0.423756 | 0.191589 | 0.585421 |
| 6 | 0.5 | 0.603333 | 0.861807 | 0.879072 | 0.588555 | 0.970813 |
| 7 | 0.5 | 0.717967 | 0.416835 | 0.382695 | 0.9202 | 0.113339 |
| 8 | 0.5 | 0.607471 | 0.850793 | 0.850462 | 0.27904 | 0.401974 |
| 9 | 0.5 | 0.71535 | 0.444306 | 0.457835 | 0.764472 | 0.961563 |
| 10 | 0.5 | 0.610873 | 0.864144 | 0.893599 | 0.684208 | 0.147837 |
| 11 | 0.5 | 0.713121 | 0.410898 | 0.342286 | 0.821056 | 0.503924 |
| 12 | 0.5 | 0.613738 | 0.847213 | 0.810455 | 0.558307 | 0.999938 |
| 13 | 0.5 | 0.711191 | 0.453051 | 0.553025 | 0.937081 | 0.000246 |
| 14 | 0.5 | 0.616195 | 0.867285 | 0.889878 | 0.224049 | 0.000985 |
| 15 | 0.5 | 0.709496 | 0.402856 | 0.352782 | 0.660634 | 0.003936 |
| 16 | 0.5 | 0.618334 | 0.84197 | 0.821977 | 0.851947 | 0.015682 |
| 17 | 0.5 | 0.707991 | 0.465697 | 0.526792 | 0.479305 | 0.061745 |
| 18 | 0.5 | 0.620219 | 0.870882 | 0.897416 | 0.948373 | 0.23173 |
| 19 | 0.5 | 0.706642 | 0.393564 | 0.331418 | 0.186056 | 0.712124 |
| 20 | 0.5 | 0.621897 | 0.83535 | 0.797689 | 0.575468 | 0.820014 |
| 21 | 0.5 | 0.705423 | 0.481392 | 0.580973 | 0.928357 | 0.590364 |
| 22 | 0.5 | 0.623404 | 0.873788 | 0.876396 | 0.252738 | 0.967337 |
| 23 | 0.5 | 0.704315 | 0.385989 | 0.389974 | 0.717674 | 0.126384 |
| 24 | 0.5 | 0.624767 | 0.829505 | 0.856419 | 0.769948 | 0.441645 |
| 25 | 0.5 | 0.7033 | 0.494993 | 0.442675 | 0.673086 | 0.986379 |