Evaluates the characteristic exponent.
[nu,c] = mathieu_mathieuexp(a,q,ndet)
parameters in Mathieu Equation:
a positive integer number: it is the matrix dimension used in the algorithm. Precision increases by increasing ndet. Typical value is ndet=12.
characteristic exponent of mathieu equation
coefficients of the expansion of the periodic factor
mathieu_mathieuexp evaluates the characteristic exponent nu, corresponding to solutions of Mathieu Equation
where a and q are real variables.
The alghoritm consider two different cases: a=(2k)² or not (k integer). nu is such that its real part belongs to the interval [0,1). Of course, every other solutions are obtained by the formula ±nu+2⋅k, with k integer.
// Stability curves of Mathieu equation // (see fig. 3 of [1] for detailed view or change N to 180 manually) N=30; a=linspace(-2, 7, N); q=linspace(-4, 4, N); st=zeros(N,N); for j=1:length(a) for h=1:length(q) st(h, j)=mathieu_mathieuexp(a(j), q(h), 12); end end contour(q, a, imag(st), N); xtitle('Stability curves of Mathieu equation', 'q', 'a'); | ![]() | ![]() |
R.Coisson and G. Vernizzi, Parma University
X. K. Yang
N. O. Strelkov, NRU MPEI
2011 - DIGITEO - Michael Baudin
1. R. Coïsson, G. Vernizzi and X.K. Yang, "Mathieu functions and numerical solutions of the Mathieu equation", IEEE Proceedings of OSSC2009 (online at http://www.fis.unipr.it/~coisson/Mathieu.pdf).