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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Disappearance of Resonance Tongues

Journal of Applied Nonlinear Dynamics 4(1) (2015) 1--9 | DOI:10.5890/JAND.2015.03.001

Rocio E. Ruelas$^{1}$; Richard H. Rand$^{2}$

$^{1}$ Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA

$^{2}$ Department of Mathematics, Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA

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We investigate a phenomenon observed in systems of the form dx/dt = a1 (t)x + a2(t)y, dy/dt = a3(t)x + a4(t)y, where ai(t) = Pi + εQicos2t, where Pi, Qi and ε are given constants, and where it is assumed that when ε=0 this system exhibits a pair of linearly independent solutions of period 2π. Since the driver cos2t has period π, we have the ingredients for a 2:1 subharmonic resonance which typically results in a tongue of instability involving unbounded solutions when ε>0. We present conditions on the coefficients Pi, Qi such that the expected instability does not occur, i.e., the tongue of instability has disappeared.


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