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On the critical values of parametric resonance in Meissner’s equation by the method of difference equations. (English) Zbl 1197.70013

Summary: The second-order linear differential equation

x '' +a 2 (t)x=0,a(t)=π+ε,if2nTt<2nT+T 1 ,π-ε,if2nT+T 1 t<2nT+T 1 +T 2 ,(n=0,1,2,...),

is investigated, where T 1 >0, T 2 >0 (T:=(T 1 +T 2 )/2) and ε[0,π). We say that a parametric resonance occurs in this equation, if for every ε>0 sufficiently small there are T 1 (ε),T 2 (ε) such that the equation has solutions with amplitudes tending to , as t. The period 2T * of the parametric excitation is called a critical value of the parametric resonance if T * =T 1 (ε)+T 2 (ε) with some T 1 ,T 2 for all sufficiently small ε>0. We give a new simple geometric proof for the fact that the critical values are the natural numbers. We apply our method also to find the most effective control destabilizing the equilibrium x=0,x ' =0, and to give a sufficient condition for the parametric resonance in the asymmetric case T 1 T 2 .

MSC:
70J40Parametric resonances (linear vibrations)
39A60Applications of difference equations