# American Institute of Mathematical Sciences

March  2020, 19(3): 1795-1845. doi: 10.3934/cpaa.2020073

## The mathieu differential equation and generalizations to infinite fractafolds

 1 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA 2 Department of Mathematics, Indiana University Bloomington, Bloomington, IN 47405, USA 3 Department of Mathematics, The University of Hong Kong, Hong Kong 4 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

* Corresponding author

Received  April 2019 Revised  August 2019 Published  November 2019

Fund Project: 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.

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.

Citation: Shiping Cao, Anthony Coniglio, Xueyan Niu, Richard H. Rand, Robert S. Strichartz. The mathieu differential equation and generalizations to infinite fractafolds. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1795-1845. doi: 10.3934/cpaa.2020073
##### References:
 [1] N. Asai, D. Cai, Y. Ikebe and Y. Miyazaki, The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Linear Algebra Appl., 241 (1996), 599-618.  doi: 10.1016/0024-3795(95)00699-0. [2] J. Avron and B. Simon, The asymptotics of the gap in the mathieu equation, Ann. Physics, 134 (1981), 76-84.  doi: 10.1016/0003-4916(81)90005-1. [3] O. Ben-Bassat, R. Strichartz and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.  doi: 10.1006/jfan.1999.3431. [4] K. Dalrymple, R. Strichartz and J. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.  doi: 10.1007/BF01261610. [5] K. Falconer, Fractal Geometry, John Wiley & Sons, Chichester, 2014. [6] M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.  doi: 10.1007/BF00249784. [7] A. Gil, J. Segura and N. Temme, Numerical Methods for Special Functions, vol. 99, Society for Industrial and Applied Mathematics, 2007. doi: 10.1137/1.9780898717822. [8] B. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 78 (1999), 431-458.  doi: 10.1112/S0024611599001744. [9] H. Hochstadt, On the width of the instability intervals of the mathieu equation, SIAM J. Math. Anal., 15 (1984), 105-107.  doi: 10.1137/0515005. [10] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055. [11] M. Ionescu, L. Rogers and R. Strichartz, Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.  doi: 10.4171/RMI/752. [12] J. Kigami, A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.  doi: 10.1007/BF03167882. [13] J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.  doi: 10.2307/2154402. [14] J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. [15] D. Levy and J. Keller, Instability intervals of Hill's equation, Comm. Pure Appl. Math., 16 (1963), 469-476.  doi: 10.1002/cpa.3160160406. [16] W. Loud, Stability regions for Hill's equation, J. Differential Equations, 19 (1975), 226-241.  doi: 10.1016/0022-0396(75)90003-0. [17] R. Rand, Mathieu's Equation, International Centre for Mechanical Sciences, 2016. [18] H. Ruan and R. Strichartz, Covering maps and periodic functions on higher dimensional Sierpinski gaskets, Canad. J. Math., 61 (2009), 1151-1181.  doi: 10.4153/CJM-2009-054-5. [19] J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, vol. 2, New York: Interscience Publishers, 1950. [20] R. Strichartz, Fractals in the large, Can. J. Math., 50 (1996), 638-657.  doi: 10.4153/CJM-1998-036-5. [21] R. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.  doi: 10.1090/S0002-9947-03-03171-4. [22] R. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006. [23] R. Strichartz, Periodic and almost periodic functions on infinite Sierpinski gaskets, Canad. J. Math., 61 (2009), 1182-1200.  doi: 10.4153/CJM-2009-055-9. [24] R. Strichartz and A. Teplyaev, Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297.  doi: 10.1007/s11854-012-0007-5. [25] A. Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.  doi: 10.1006/jfan.1998.3297. [26] M. Weinstein and J. Keller, Asymptotic behavior of stability regions for Hill's equation, J. Appl. Math., 47 (1987), 941-958.  doi: 10.1137/0147062.

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##### References:
 [1] N. Asai, D. Cai, Y. Ikebe and Y. Miyazaki, The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Linear Algebra Appl., 241 (1996), 599-618.  doi: 10.1016/0024-3795(95)00699-0. [2] J. Avron and B. Simon, The asymptotics of the gap in the mathieu equation, Ann. Physics, 134 (1981), 76-84.  doi: 10.1016/0003-4916(81)90005-1. [3] O. Ben-Bassat, R. Strichartz and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.  doi: 10.1006/jfan.1999.3431. [4] K. Dalrymple, R. Strichartz and J. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.  doi: 10.1007/BF01261610. [5] K. Falconer, Fractal Geometry, John Wiley & Sons, Chichester, 2014. [6] M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.  doi: 10.1007/BF00249784. [7] A. Gil, J. Segura and N. Temme, Numerical Methods for Special Functions, vol. 99, Society for Industrial and Applied Mathematics, 2007. doi: 10.1137/1.9780898717822. [8] B. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 78 (1999), 431-458.  doi: 10.1112/S0024611599001744. [9] H. Hochstadt, On the width of the instability intervals of the mathieu equation, SIAM J. Math. Anal., 15 (1984), 105-107.  doi: 10.1137/0515005. [10] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055. [11] M. Ionescu, L. Rogers and R. Strichartz, Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.  doi: 10.4171/RMI/752. [12] J. Kigami, A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.  doi: 10.1007/BF03167882. [13] J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.  doi: 10.2307/2154402. [14] J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. [15] D. Levy and J. Keller, Instability intervals of Hill's equation, Comm. Pure Appl. Math., 16 (1963), 469-476.  doi: 10.1002/cpa.3160160406. [16] W. Loud, Stability regions for Hill's equation, J. Differential Equations, 19 (1975), 226-241.  doi: 10.1016/0022-0396(75)90003-0. [17] R. Rand, Mathieu's Equation, International Centre for Mechanical Sciences, 2016. [18] H. Ruan and R. Strichartz, Covering maps and periodic functions on higher dimensional Sierpinski gaskets, Canad. J. Math., 61 (2009), 1151-1181.  doi: 10.4153/CJM-2009-054-5. [19] J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, vol. 2, New York: Interscience Publishers, 1950. [20] R. Strichartz, Fractals in the large, Can. J. Math., 50 (1996), 638-657.  doi: 10.4153/CJM-1998-036-5. [21] R. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.  doi: 10.1090/S0002-9947-03-03171-4. [22] R. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006. [23] R. Strichartz, Periodic and almost periodic functions on infinite Sierpinski gaskets, Canad. J. Math., 61 (2009), 1182-1200.  doi: 10.4153/CJM-2009-055-9. [24] R. Strichartz and A. Teplyaev, Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297.  doi: 10.1007/s11854-012-0007-5. [25] A. Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.  doi: 10.1006/jfan.1998.3297. [26] M. Weinstein and J. Keller, Asymptotic behavior of stability regions for Hill's equation, J. Appl. Math., 47 (1987), 941-958.  doi: 10.1137/0147062.
Stable and Unstable Regions of the $\delta$-$\varepsilon$ plane
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)
The width of the $5$th stable band
The width of first $5$ stable bands
The width of $6$th and $7$th stable bands
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
Probabilities $P_i$ and the fitting curve
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)
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)
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)
t-position of maximal points, with fitting curve $t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d}$
t-position of the second maximal points, with fitting curve $t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d}$
t-position of the third maximal points on curve $p(\sin 3t, \varepsilon)$, with fitting curve $t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d}$, $a = 9.567,b = 23.07,c = 22.86,d = 40.46$
u-position of the second maximal points on curve $p(\sin 2t, \varepsilon)$, with fitting curve $t = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+d \varepsilon+e}$. $a = 0.8393,b = -0.5638,c = 5.566,d = -0.1973,e = 5.571$
u-position of the second and third maximal points corresponding to curve $p(\sin 3t, \varepsilon)$, with fitting curve $u = \frac{a \varepsilon^3+b \varepsilon^2+c \varepsilon+d}{ \varepsilon^3+e \varepsilon^2+f \varepsilon+g}$
Normalized solutions corresponding to $p(\cos 0t, \varepsilon)$, with $\varepsilon = 0,1,2,3,4,5,10,20,40,80,160$
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
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
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 $a = -0.02171,b = 0.2895,c = 0.2289,d = 0.2895$
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 $a = 241,9,b = 1284,c = 310.8,d = 177.1$
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 $e = 0.8497$
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 $a = 0.2495,b = -4.991,c = -3.037,d = 6.722$
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 $e = -157.7$
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 $f = -6017$
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 $g = 13.44$
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 $a = -0.006875,b = 0.7009,c = -8.537,d = -2.457,e = 8.857$
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 $f = 1002$
Normalized solutions corresponding to $p(\sin \frac12t, \varepsilon)$, with $\varepsilon = 0,1,2,3,4,5,10,20,40,80,160$
Normalized solutions corresponding to $p(\sin\frac32t, \varepsilon)$, with $\varepsilon = 0,5,10,20,40,80,160$
Normalized solutions corresponding to $p(\cos \frac12t, \varepsilon)$, with $\varepsilon = 0,1,2,3,4,5,10,20,40,80,160$
Normalized solutions corresponding to $p(\cos \frac32t, \varepsilon)$, with $\varepsilon = 0,1,2,3,4,5,10,20,40,80,160$
Approximating graphs $\Gamma_0,\Gamma_1,\Gamma_2$
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)
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)
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)
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)
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)
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)
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)
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)
Initial values of the eigenfunctions with birth of generation $2$ and initial eigenvalues $5$ (left) and $6$(right)
Solutions on the first curve for Version 1, initial eigenvalue $5$, with $\varepsilon = 1000,2000,3000$
Solutions on the first curve for Version 1, initial eigenvalue $5$, with $\varepsilon = 10000,11000,12000$
Solutions on the first curve for Version 1, initial eigenvalue $5$, with $\varepsilon = 20000,25000,30000$
Solutions on the first curve for Version 1, initial eigenvalue $6$, with $\varepsilon = 1000,2000,3000$
Solutions on the first curve for Version 1, initial eigenvalue $6$, with, with $\varepsilon = 10000,11000,12000$
Solutions on the first curve for Version 1, initial eigenvalue $6$, with $\varepsilon = 20000,25000,30000$
The position of peaks for version 1, 5 series
The position of peaks for version 1, 6-series
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
 $i$ $P_i$ 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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