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The mathieu differential equation and generalizations to infinite fractafolds

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    * Corresponding author 

This research was hosted by the Cornell University Department of Mathematics through its 2018 Summer Program for Undergraduate Research. Anthony Coniglio's participation in this research was partly supported by Indiana University Bloomington. Xueyan Niu's participation in this research was partly supported by the Overseas Research Fellowship (ORF) of Faculty of Science, The University of Hong Kong

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  • One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by converting the equation into an infinite system of linear equations for the Fourier coefficients. We study the asymptotic behavior of these Fourier coefficients and discuss the ways in which to numerically approximate solutions. We present both theoretical and numerical results pertaining to the stability of the Mathieu differential equation and the properties of solutions. Further, based on the idea of using Fourier series, we provide a method in which the Mathieu differential equation can be generalized to be defined on the infinite Sierpinski gasket. We discuss the stability of solutions to this fractal differential equation and describe further results concerning properties and behavior of these solutions.

    Mathematics Subject Classification: Primary: 34D23; Secondary: 28A80.

    Citation:

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  • Figure 1.  Stable and Unstable Regions of the $ \delta $-$ \varepsilon $ plane

    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)

    Figure 3.  The width of the $ 5 $th stable band

    Figure 4.  The width of first $ 5 $ stable bands

    Figure 5.  The width of $ 6 $th and $ 7 $th stable bands

    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

    Figure 7.  Probabilities $ P_i $ and the fitting curve

    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)

    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)

    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)

    Figure 11.  t-position of maximal points, with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $

    Figure 12.  t-position of the second maximal points, with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $

    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 $

    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 $

    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} $

    Figure 16.  Normalized solutions corresponding to $ p(\cos 0t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $

    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

    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

    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 $

    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 $

    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 $

    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 $

    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 $

    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 $

    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 $

    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 $

    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 $

    Figure 28.  Normalized solutions corresponding to $ p(\sin \frac12t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $

    Figure 29.  Normalized solutions corresponding to $ p(\sin\frac32t, \varepsilon) $, with $ \varepsilon = 0,5,10,20,40,80,160 $

    Figure 30.  Normalized solutions corresponding to $ p(\cos \frac12t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $

    Figure 31.  Normalized solutions corresponding to $ p(\cos \frac32t, \varepsilon) $, with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $

    Figure 32.  The Sierpinski Gasket

    Figure 33.  Approximating graphs $ \Gamma_0,\Gamma_1,\Gamma_2 $

    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)

    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)

    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)

    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)

    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)

    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)

    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)

    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)

    Figure 42.  Initial values of the eigenfunctions with birth of generation $ 2 $ and initial eigenvalues $ 5 $ (left) and $ 6 $(right)

    Figure 43.  Solutions on the first curve for Version 1, initial eigenvalue $ 5 $, with $ \varepsilon = 1000,2000,3000 $

    Figure 44.  Solutions on the first curve for Version 1, initial eigenvalue $ 5 $, with $ \varepsilon = 10000,11000,12000 $

    Figure 45.  Solutions on the first curve for Version 1, initial eigenvalue $ 5 $, with $ \varepsilon = 20000,25000,30000 $

    Figure 46.  Solutions on the first curve for Version 1, initial eigenvalue $ 6 $, with $ \varepsilon = 1000,2000,3000 $

    Figure 47.  Solutions on the first curve for Version 1, initial eigenvalue $ 6 $, with, with $ \varepsilon = 10000,11000,12000 $

    Figure 48.  Solutions on the first curve for Version 1, initial eigenvalue $ 6 $, with $ \varepsilon = 20000,25000,30000 $

    Figure 49.  The position of peaks for version 1, 5 series

    Figure 50.  The position of peaks for version 1, 6-series

    Table 1.  The probability $ P_i $'s

    $ i $ $ P_i $
    1 0.625056436387445
    2 0.784428425813594
    3 0.845139663995868
    4 0.878143787704672
    5 0.899154589232086
    6 0.913800056779179
    7 0.924632580597566
    8 0.932988858616457
    9 0.939640860147421
    10 0.945067300763657
    11 0.949581603650156
    12 0.953397960289362
    13 0.956667925443240
    14 0.959502140424685
    15 0.961982968671620
    16 0.964174006346465
    17 0.966133547589692
    18 0.967239011961280
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