# 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 & Applied Analysis, 2020, 19 (3) : 1795-1845. doi: 10.3934/cpaa.2020073
##### References:

show all references

##### References:
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
 [1] Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013 [2] G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509 [3] Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445 [4] David W. Pravica, Michael J. Spurr. Unique summing of formal power series solutions to advanced and delayed differential equations. Conference Publications, 2005, 2005 (Special) : 730-737. doi: 10.3934/proc.2005.2005.730 [5] Oktay Veliev. Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 397-424. doi: 10.3934/cpaa.2019020 [6] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 [7] Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118 [8] Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic & Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036 [9] Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 [10] Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061 [11] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [12] P. Alonso Ruiz, Y. Chen, H. Gu, R. S. Strichartz, Z. Zhou. Analysis on hybrid fractals. Communications on Pure & Applied Analysis, 2020, 19 (1) : 47-84. doi: 10.3934/cpaa.2020004 [13] Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic & Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020 [14] Robert F. Bailey, John N. Bray. Decoding the Mathieu group M12. Advances in Mathematics of Communications, 2007, 1 (4) : 477-487. doi: 10.3934/amc.2007.1.477 [15] M. A. Christou, C. I. Christov. Fourier-Galerkin method for localized solutions of the Sixth-Order Generalized Boussinesq Equation. Conference Publications, 2001, 2001 (Special) : 121-130. doi: 10.3934/proc.2001.2001.121 [16] Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109 [17] Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : ⅰ-ⅳ. doi: 10.3934/dcdss.201702i [18] Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221 [19] Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129 [20] Changfeng Gui, Zhenbu Zhang. Spike solutions to a nonlocal differential equation. Communications on Pure & Applied Analysis, 2006, 5 (1) : 85-95. doi: 10.3934/cpaa.2006.5.85

2018 Impact Factor: 0.925