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Figure 1.
Stable and Unstable Regions of the $ \delta $-$ \varepsilon $ plane
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Figure 2.
Curves corresponding to expansions of larger periods: curves with $ 2\pi $-periodic solutions (black solid and black dashed), curves with $ 4\pi $-periodic solutions (orange solid and orange dashed), curves with $ 8\pi $-periodic solutions (red dashed) and curves with $ 16\pi $-periodic solutions (blue dashed)
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Figure 3.
The width of the $ 5 $th stable band
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Figure 4.
The width of first $ 5 $ stable bands
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Figure 5.
The width of $ 6 $th and $ 7 $th stable bands
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Figure 6.
The triangle area corresponding to $ R_{w_1},R_{w_2},R_{w_3}, $ and $ R_{w_4} $. The green area is the stable region, and shaded area is the unstable region
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Figure 7.
Probabilities $ P_i $ and the fitting curve
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Figure 8.
Normalized solutions corresponding to $ p(\sin t,0) $ (solid black), $ p(\sin t,5) $ (red), $ p(\sin t,10) $ (orange), $ p(\sin t,20) $ (green), $ p(\sin t,40) $ (blue), $ p(\sin t,80) $ (purple)
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Figure 9.
Normalized solutions corresponding to $ p(\sin 2t,0) $ (solid black), $ p(\sin 2t,5) $ (red), $ p(\sin 2t,10) $ (orange), $ p(\sin 2t,20) $ (green), $ p(\sin 2t,40) $ (blue), $ p(\sin 2t,80) $ (purple)
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Figure 10.
Normalized solutions corresponding to $ p(\sin 3t,0) $ (solid black), $ p(\sin 3t,5) $ (red), $ p(\sin 3t,10) $ (orange), $ p(\sin 3t,20) $ (green), $ p(\sin 3t,40) $ (blue), $ p(\sin 3t,80) $ (purple)
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Figure 11.
t-position of maximal points, with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $
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Figure 12.
t-position of the second maximal points, with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $
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Figure 13.
t-position of the third maximal points on curve $ p(\sin 3t, \varepsilon) $, with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $, $ a = 9.567,b = 23.07,c = 22.86,d = 40.46 $
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Figure 14.
u-position of the second maximal points on curve $ p(\sin 2t, \varepsilon) $, with fitting curve $ t = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+d \varepsilon+e} $. $ a = 0.8393,b = -0.5638,c = 5.566,d = -0.1973,e = 5.571 $
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Figure 15.
u-position of the second and third maximal points corresponding to curve $ p(\sin 3t, \varepsilon) $, with fitting curve $ u = \frac{a \varepsilon^3+b \varepsilon^2+c \varepsilon+d}{ \varepsilon^3+e \varepsilon^2+f \varepsilon+g} $
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Figure 16.
Normalized solutions corresponding to $ p(\cos 0t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
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Figure 17.
Normalized solutions corresponding to $ p(\cos t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10 $ in the left graph, and $ \varepsilon = 10,20,40,80,160 $ in the right graph
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Figure 18.
Normalized solutions corresponding to $ p(\cos 2t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,6 $ in the left, and $ \varepsilon = 6,7,8,9,10,20,30,40,60,80,100,160 $ in the right
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Figure 19.
The $ u $ coordinate of minimum points of solutions for points $ p(\cos 0, \varepsilon) $, with $ \varepsilon = 0,1,\cdots, 200 $. Fitting curve $ u = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $, with $ a = -0.02171,b = 0.2895,c = 0.2289,d = 0.2895 $
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Figure 20.
The $ t $ coordinate of minimum points of solutions for points $ p(\cos t, \varepsilon) $, with $ \varepsilon = 2,\cdots, 200 $. Fitting curve $ y = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $, with $ a = 241,9,b = 1284,c = 310.8,d = 177.1 $
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Figure 21.
The $ u $ coordinate of minimum points of solutions for points $ p(\cos t, \varepsilon) $, with $ \varepsilon = 0,1,2,\cdots, 200 $. Fitting curve $ u = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+d \varepsilon+e} $, with $ a = 0.8687,b = -1.631,c = 0.8498,d = -1.72 $ and $ e = 0.8497 $
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Figure 22.
The $ u $ coordinate of minimum points of solutions for points $ p(\cos t, \varepsilon) $, with $ \varepsilon = 0,1,2,\cdots, 200 $. Fitting curve $ u = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $, with $ a = 0.2495,b = -4.991,c = -3.037,d = 6.722 $
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Figure 23.
The $ t $ coordinate of maximum points of solutions for points $ p(\cos 2t, \varepsilon) $, with $ \varepsilon = 5,6,\cdots, 200 $. Fitting curve $ t = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+d \varepsilon+e} $, with $ a = 0.5572,b = 88.5,c = -3.009,d = 55.35 $ and $ e = -157.7 $
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Figure 24.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $, with $ \varepsilon = 5,6,\cdots, 200 $. Fitting curve $ t = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^3+d \varepsilon^2+e \varepsilon+f} $, with $ a = 269.3,b = 4745,c = -9715,d = 641.3,e = -2040 $ and $ f = -6017 $
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Figure 25.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $, with $ \varepsilon = 1,2,\cdots, 200 $. Fitting curve $ t = \frac{a \varepsilon^3+b \varepsilon^2+c \varepsilon+d}{ \varepsilon^3+e \varepsilon^2+f \varepsilon+g} $, with $ a = 0.7931,b = -4.642,c = 6.596,d = 13.45,e = --5.508,f = 9.829 $ and $ g = 13.44 $
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Figure 26.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $, with $ \varepsilon = 1,2,\cdots, 200 $. Fitting curve $ y = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+e \varepsilon+f} $, with $ a = -0.006875,b = 0.7009,c = -8.537,d = -2.457,e = 8.857 $
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Figure 27.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $, with $ \varepsilon = 5,6,\cdots, 200 $. Fitting curve $ y = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^3+d \varepsilon^2+e \varepsilon+f} $, with $ a = 0.2268,b = -25.1,c = 587.7,d = 4.775,e = -156 $ and $ f = 1002 $
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Figure 28.
Normalized solutions corresponding to $ p(\sin \frac12t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
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Figure 29.
Normalized solutions corresponding to $ p(\sin\frac32t, \varepsilon) $, with $ \varepsilon = 0,5,10,20,40,80,160 $
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Figure 30.
Normalized solutions corresponding to $ p(\cos \frac12t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
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Figure 31.
Normalized solutions corresponding to $ p(\cos \frac32t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
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Figure 32.
The Sierpinski Gasket
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Figure 33.
Approximating graphs $ \Gamma_0,\Gamma_1,\Gamma_2 $
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Figure 34.
Transition curves for version 1, 5-series. The left picture shows transition curves of a single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 35.
Transition curves for version 1, 6-series. The left picture shows transition curves of a single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 36.
Transition curves for version 2, 5-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 37.
Transition curves for version 2, 6-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 38.
Transition curves for version 3, 5-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 39.
Transition curves for version 3, 6-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 40.
Transition curves for version 4, 5-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 41.
Transition curves for version 4, 6-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $. The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
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Figure 42.
Initial values of the eigenfunctions with birth of generation $ 2 $ and initial eigenvalues $ 5 $ (left) and $ 6 $(right)
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Figure 43.
Solutions on the first curve for Version 1, initial eigenvalue $ 5 $, with $ \varepsilon = 1000,2000,3000 $
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Figure 44.
Solutions on the first curve for Version 1, initial eigenvalue $ 5 $, with $ \varepsilon = 10000,11000,12000 $
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Figure 45.
Solutions on the first curve for Version 1, initial eigenvalue $ 5 $, with $ \varepsilon = 20000,25000,30000 $
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Figure 46.
Solutions on the first curve for Version 1, initial eigenvalue $ 6 $, with $ \varepsilon = 1000,2000,3000 $
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Figure 47.
Solutions on the first curve for Version 1, initial eigenvalue $ 6 $, with, with $ \varepsilon = 10000,11000,12000 $
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Figure 48.
Solutions on the first curve for Version 1, initial eigenvalue $ 6 $, with $ \varepsilon = 20000,25000,30000 $
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Figure 49.
The position of peaks for version 1, 5 series
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Figure 50.
The position of peaks for version 1, 6-series
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