This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Van der Pol’s Oscillator under Delayed Feedback Fatihcan M. Atay Ko¸c University Department of Mathematics Istinye 80860, Istanbul, Turkey E-mail: [email protected] Preprint . Final version in J. Sound and Vibration 218 (2):333-339, 1998 Abstract The effect of delayed feedback on oscillatory behaviour is investigated in the classical van der Pol’s oscillator. It is shown how the presence of delay can change the amplitude of limit cycle oscillations, or suppress them altogether. The result is compared to the conventional proportional-and- derivative feedback. The derivative-like effect of delay is also demonstrated in a modified equation where a delayed term provides the damping. 1 INTRODUCTION One of the classical equations of nonlinear dynamics was formulated by and bears the name of the Dutch physicist van der Pol . Originally it was a model for an electrical circuit with a triode valve, and was later extensively studied as a host of a rich class of dynamical behavior, including relaxation oscillations, quasiperiodicity, elementary bifurcations, and chaos [2, 3, 4]. Nevertheless, it is perhaps best known as a prototype system exhibiting limit cycle oscillations. This celebrated equation has a nonlinear damping term: ¨ x + ε ( x 2- 1)˙ x + x = f ( t ) , x ∈ R , ε > , (1) which is responsible for energy generation at low amplitudes and dissipation at high amplitudes. The unforced case ( f ≡ 0) is an equation of Li´ enard type, and thus can be shown to have a unique periodic solution which attracts all other orbits except the origin, which is an unstable equilibrium point . Limit cycle oscilla- tions with such strong stability properties are important in applications; hence, being able to modify their behavior through feedback is a question of interest. On the other hand, most practical implementations of feedback have inherent delays, 1 the presence of which results in an infinite-dimensional system, thus complicating the analysis. This study uses averaging methods to investigate the behavior of the limit cycle of (1) when the forcing f is a delayed feedback of the position x . Since the limit cycle disappears when ε is zero, it is convenient to scale the parameters by ε . Hence equation (1) will be considered with f ( t ) = εkx ( t- τ ) , (2) where τ is a positive quantity representing the delay and k is the feedback gain. It is shown in Section 2 that both the amplitude and frequency of the oscillations can be modified by changing the delay and the gain. In particular, it is possible to reduce the amplitude to zero, thereby preventing oscillations and stabilizing the origin. To gain more insight into the mechanism of delayed feedback, the results are compared to the conventional feedback of the state coordinates x ( t ) and ˙ x ( t ). While derivative feedback can change the amplitude of oscillations (but not the frequency), position feedback without delay affects only the frequency....
View Full Document
- Spring '11
- limit cycle, θ, van der Pol, limit cycle oscillations