Time Delay Control for Two van der Pol Oscillators

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Attilio Maccari
Keyword(s):  
Time Delay ◽  
Vibration Control ◽  
Periodic Solutions ◽  
Original System ◽  
Excitation Amplitude ◽  
Van Der Pol ◽  
Flow Equations ◽  
Technical Requirements ◽  
Stable Periodic ◽  
Van Der Pol Oscillators

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.

Vibration Control for Parametrically Excited Coupled Nonlinear Oscillators

2008 ◽  
Vol 3 (3) ◽  
Author(s):  
Attilio Maccari
Keyword(s):  
Time Delay ◽  
Vibration Control ◽  
Periodic Solutions ◽  
Parametric Excitation ◽  
Nonlinear Oscillators ◽  
Excitation Amplitude ◽  
State Feedback Control ◽  
Coupled Equations ◽  
Technical Requirements ◽  
Coupled Nonlinear Oscillators

A complex nonlinear system under state feedback control with a time delay corresponding to two coupled nonlinear oscillators with a parametric excitation is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. In the system without control, phase-locked solutions with period equal to the parametric excitation period are possible only if the oscillator amplitudes are equal, but they depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. It is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude and phase can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.

scholarly journals The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control

2012 ◽  
Vol 19 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Xinye Li ◽  
Huabiao Zhang ◽  
Lijuan He
Keyword(s):  
Time Delay ◽  
Feedback Control ◽  
Parametric Resonance ◽  
Relaxation Oscillation ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Flow Equations ◽  
Principal Parametric Resonance ◽  
Van Der Pol Oscillators ◽  
And Control

The principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The slow-flow equations are obtained by the averaging method and simplified by truncating the first term of Taylor expansions for those terms with time delay. It is found that nontrivial solutions corresponding to periodic motions exist only for one oscillator if no feedback control is applied although the two oscillators are nonlinearly coupled. Based on Levenberg-Marquardt method, the effects of excitation and control parameters on the amplitude of periodic solutions of the system are graphically given. It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback. However, the amplitudes of the periodic vibrations are independent of the sign of feedback gains. In addition, the influence of time delay on the response of the system is periodic. In terms of numerical simulations, it is shown that both of the two oscillators can also have quasi-periodic motions, periodic motions about a new equilibrium position and other complex motions such as relaxation oscillation when feedback control is considered.

scholarly journals Modelling chronic hepatitis B using the Marchuk-Petrov model

2021 ◽  
Vol 2099 (1) ◽  
pp. 012036
Author(s):  
M Yu Khristichenko ◽  
Yu M Nechepurenko ◽  
D S Grebennikov ◽  
G A Bocharov
Keyword(s):  
Time Delay ◽  
Hepatitis B ◽  
Immune Responses ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Stable Periodic ◽  
Delay Differential ◽  
Parameter Values

Abstract Systems of time-delay differential equations are widely used to study the dynamics of infectious diseases and immune responses. The Marchuk-Petrov model is one of them. Stable non-trivial steady states and stable periodic solutions to this model can be interpreted as chronic viral diseases. In this work we briefly describe our technology developed for computing steady and periodic solutions of time-delay systems and present and discuss the results of computing periodic solutions for the Marchuk-Petrov model with parameter values corresponding to the hepatitis B infection.

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

2016 ◽  
Vol 26 (08) ◽  
pp. 1650141 ◽  
Author(s):  
Adrian C. Murza ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Coupled Oscillators ◽  
Invariant Manifolds ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Oscillatory Systems ◽  
Weakly Coupled ◽  
Form Theory ◽  
Van Der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

scholarly journals Periodic Solutions of a System of Delay Differential Equations for a Small Delay

2002 ◽  
Vol 7 (2) ◽  
pp. 295
Author(s):  
Adu A.M. Wasike ◽  
Wandera Ogana
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Original System ◽  
System Of Equations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Stable Periodic ◽  
Delay Differential

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system.  This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.   

Synchronization or cluster synchronization in coupled Van der Pol oscillators networks with different topological types

2021 ◽  
Author(s):  
Shuai Wang ◽  
Yong Li
Keyword(s):  
Periodic Solutions ◽  
Phase Synchronization ◽  
Van Der Pol Oscillator ◽  
Cluster Synchronization ◽  
Critical Conditions ◽  
Complete Synchronization ◽  
Van Der Pol ◽  
Special Groups ◽  
Different Types ◽  
Van Der Pol Oscillators

Abstract In this paper, we try to discuss the mechanism of synchronization or cluster synchronization in the coupled Van der Pol oscillator networks with different topology types by using the theory of rotating periodic solutions. The synchronous solutions here are transformed into rotating periodic solutions of some dynamical systems. By analyzing the bifurcation of rotating periodic solutions, the critical conditions of synchronous solutions are given in three different networks. We use the rotating periodic matrix in the rotating periodic theory to judge various types of synchronization phenomena, such as complete synchronization, anti-phase synchronization, periodic synchronization, or cluster synchronization. All rotating periodic matrices which satisfy the exchange invariance of multiple oscillators form special groups in these networks. By using the conjugate classes of these groups, we obtain various possible synchronization solutions in the three networks. In particular, we find symmetry has different effects on synchronization in different networks. The network with better symmetry has more elements in the corresponding group, which may have more types of synchronous solutions. However, different types of symmetry may get the same type of synchronous solutions or different types of synchronous solutions, depending on whether their corresponding rotating periodic matrices are similar.

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