Time dependence of the solutions of van der Pol type equation with periodic forcing term

1991 ◽  
Vol 7 (1) ◽  
pp. 75-90
Author(s):  
Zheng Zhiming
Keyword(s):  
Time Dependence ◽  
Type Equation ◽  
Periodic Forcing ◽  
Van Der Pol ◽  
Forcing Term

scholarly journals Forced quasi-periodic oscillations in strongly dissipative systems of any finite dimension

2019 ◽  
Vol 21 (07) ◽  
pp. 1850064 ◽  
Author(s):  
Guido Gentile ◽  
Alessandro Mazzoccoli ◽  
Faenia Vaia
Keyword(s):  
Potential Energy ◽  
Finite Dimension ◽  
Dissipative Systems ◽  
Diophantine Condition ◽  
Frequency Vector ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Term ◽  
Degeneracy Condition ◽  
The One

We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence of dissipation, so that the forcing term is — up to the sign — the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a non-degeneracy condition on the forcing term, essentially that the time-average of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular, we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exists provided the dissipation is strong enough. This extends results previously available in the literature in the one-dimensional case.

scholarly journals Numerical simulation of a wind excited quasi-periodic regime of a stochastic van der Pol-type equation

2020 ◽  
Vol 313 ◽  
pp. 00044
Author(s):  
Cyril Fischer ◽  
Jiří Náprstek
Keyword(s):  
Numerical Simulation ◽  
Experimental Investigation ◽  
Type Equation ◽  
Random Excitation ◽  
Periodic Regime ◽  
Van Der Pol ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Configuration Parameter ◽  
Slender Structure

The contribution regards a mathematical single-degree-of-freedom model of a slender structure vibrating in an air flow. Based on an experimental investigation, movement of such structures can be expressed by van der PolDuffing-type equations. Several particular configuration parameter settings for a white and non-white Gaussian random excitation together with deterministic harmonic forcing are considered and numerically analysed. The results support recently published analytic formulas.

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