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Research Papers

# Time Delay Control for Two van der Pol Oscillators

[+] Author and Article Information
Attilio Maccari

Via Alfredo Casella 3, 00013 Mentana, Rome, Italy

J. Comput. Nonlinear Dynam 6(1), 011016 (Oct 07, 2010) (7 pages) doi:10.1115/1.4002390 History: Received October 20, 2009; Revised March 21, 2010; Published October 07, 2010; Online October 07, 2010

## Abstract

A method for time delay vibration control of the principal and fundamental resonances of two nonlinearly coupled van der Pol oscillators is investigated Using the asymptotic perturbation method, four slow-flow equations on the amplitude and phase of the oscillators are obtained. Their fixed points correspond to a two-period quasi-periodic phase-locked motion for the original system. In the system without control, stable periodic solutions (if any) exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of these solutions do not correspond to the technical requirements. On the contrary, it is demonstrated that, if vibration control terms are added, stable two-period quasi-periodic solutions with arbitrarily chosen amplitudes can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.

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## Figures

Figure 1

(a) Stability chart in the plane (α,ρ) for the nonlinear model systems 11,12,13,14 with the parameter values C=0.06, D=1.0, E=0.78, F=0.01, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 24,25,26,27. The phase varies from 0 to 2π and the response from 0 to 0.4. White (black) regions stand for stable (unstable) solutions. Note that A and B are different. (b) The prevision of our method for the stability chart in the plane (X0,X•0=V0) for the van der Pol system 1 with the parameter values C=0.06, D=1.0, E=0.78, F=0.01, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 24,25,26,27. X0 and V0 vary from −0.8 to 0.8. White (black) regions stand for stable (unstable) solutions. (c) The result of numerical integration by the Runge–Kutta–Fehlberg method for the stability chart in the plane (X0,Ẋ0=V0) for the van der Pol system 1 with the parameter values C=0.06, D=1.0, E=0.78, F=0.01, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 24,25,26,27. X0 and V0 vary from −0.8 to 0.8. White (black) regions stand for stable (unstable) solutions.

Figure 2

Stability chart in the plane (α,ρ) for the nonlinear model system 11,12,13,14 with the parameter values C=0.06, D=1.0, E=0.78, F=0.1, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 24,25,26,27. The phase varies from 0 to 2π and the response from 0 to 0.4. White (black) regions stand for stable (unstable) solutions.

Figure 3

Stability chart in the plane (α,ρ) for the nonlinear model system 11,12,13,14 with the parameter values C=0.06, D=2.3, E=0.78, F=0.01, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 24,25,26,27. The phase varies from 0 to 2π and the response from 0 to 0.4. White (black) regions stand for stable (unstable) solutions.

Figure 4

Stability chart in the plane (α,ρ) for the nonlinear model system 11,12,13,14 with the parameter values C=0.2, D=1.0, E=0.78, F=0.01, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 24,25,26,27. The phase varies from 0 to 2π and the response from 0 to 0.4. White (black) regions stand for stable (unstable) solutions.

Figure 5

Stability chart in the plane (α,ρ) for the nonlinear model system 29,30,31,32 with the parameter values C=0.06, D=1.0, E=0.78, F=0.01, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 26,27,36,37. The phase varies from 0 to π and the response from 0 to 0.4. White (black) regions stand for stable (unstable) solutions.

Figure 6

Stability chart in the plane (α,ρ) for the nonlinear model system 29,30,31,32 with the parameter values C=0.06, D=1.0, E=0.78, F=0.1, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 26,27,36,37. The phase varies from 0 to π and the response from 0 to 0.4. White (black) regions stand for stable (unstable) solutions.

Figure 7

Stability chart in the plane (α,ρ) for the nonlinear model system 29,30,31,32 with the parameter values C=0.06, D=2.3, E=0.78, F=0.01, σ=−0.04, Ω=1, and T=π/4. Note that A and B are not constant because for a specific value in the plane (α,ρ), A and B are calculated from Eqs. 26,27,36,37. The phase varies from 0 to π and the response from 0 to 0.4. White (black) regions stand for stable (unstable) solutions.

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