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mathieu_mathieuexp

Evaluates the characteristic exponent.

Calling Sequence

[nu,c] = mathieu_mathieuexp(a,q,ndet)

Parameters

a, q

parameters in Mathieu Equation:

ndet

a positive integer number: it is the matrix dimension used in the algorithm. Precision increases by increasing ndet. Typical value is ndet=12.

nu

characteristic exponent of mathieu equation

c

coefficients of the expansion of the periodic factor

Description

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.

Examples

// 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');

See Also

Authors

R.Coisson and G. Vernizzi, Parma University

X. K. Yang

N. O. Strelkov, NRU MPEI

2011 - DIGITEO - Michael Baudin

Bibliography

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).


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