scholarly journals Nonlinear Dynamics of a Periodically Driven Duffing Resonator Coupled to a Van der Pol Oscillator

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
X. Wei ◽  
M. F. Randrianandrasana ◽  
M. Ward ◽  
D. Lowe
Keyword(s):  
Strong Coupling ◽  
Internal Resonance ◽  
Weak Coupling ◽  
Response Pattern ◽  
Coupled System ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Periodically Driven ◽  
Coupling Case ◽  
The Stability

We explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of 1 : 1 internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.

FUNCTIONAL INTEGRALS AND PHASE STRUCTURES IN SINE-GORDON–THIRRING MODEL

2012 ◽  
Vol 26 (27) ◽  
pp. 1250178 ◽  
Author(s):  
JUN YAN
Keyword(s):  
Strong Coupling ◽  
Finite Temperature ◽  
Thirring Model ◽  
Attractive Potential ◽  
Functional Integrals ◽  
One Dimensional ◽  
Phase Structures ◽  
Coexistence Phase ◽  
Coupling Case ◽  
The Stability

The phase structures of one-dimensional quantum sine-Gordon–Thirring model with N-impurities coupling are studied in this paper. The effective actions at finite temperature are derived by means of the perturbation and non-perturbation functional integrals method. The stability of coexistence phase is analyzed respectively in the weak and strong coupling case. It is shown that the coexistence phase is not stable when fermions have an attractive potential g < 0, and the stable coexistence phase can form when fermions have an exclude potential g > 0.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε &gt; 0 and k &gt; 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

scholarly journals COUPLED GROWING NETWORKS

2003 ◽  
Vol 06 (04) ◽  
pp. 507-514 ◽  
Author(s):  
DAFANG ZHENG ◽  
GÜLER ERGÜN
Keyword(s):  
Strong Coupling ◽  
Weak Coupling ◽  
Citation Network ◽  
Power Laws ◽  
Coupling Case ◽  
Coupled Networks ◽  
Growing Networks ◽  
Sexual Contacts ◽  
Cross Links ◽  
The Web

We introduce and solve a model which considers two coupled networks growing simultaneously. The dynamics of the networks is governed by the new arrival of network elements (nodes) making preferential attachments to pre-existing nodes in both networks. The model segregates the links in the networks as intra-links, cross-links and mix-links. The corresponding degree distributions of these links are found to be power-laws with exponents having coupled parameters for intra- and cross-links. In the weak coupling case, the model reduces to a simple citation network. As for the strong coupling, it mimics the mechanism of the web of human sexual contacts.

On the Delayed van der Pol Oscillator with Time-Varying Feedback Gain

2014 ◽  
Vol 706 ◽  
pp. 149-158 ◽  
Author(s):  
Mustapha Hamdi ◽  
Mohamed Belhaq
Keyword(s):  
Numerical Simulation ◽  
Type System ◽  
Stationary Solutions ◽  
Van Der Pol Oscillator ◽  
Feedback Gain ◽  
Time Varying ◽  
Van Der Pol ◽  
Existence Region ◽  
The Stability ◽  
Time Delayed Feedback

This work studies the effect of time delayed feedback on stationary solutions in a van derPol type system. We consider the case where the feedback gain is harmonically modulated with a resonantfrequency. Perturbation analysis is conducted to obtain the modulation equations near primaryresonance, the stability analysis for stationary solutions is performed and bifurcation diagram is determined.It is shown that the modulated feedback gain position can influence significantly the steadystates behavior of the delayed van der Pol oscillator. In particular, for appropriate values of the modulateddelay parameters, the existence region of the limit cycle (LC) can be increased or quenched.Moreover, new regions of quasiperiodic vibration may emerge for certain values of the modulatedgain. Numerical simulation was conducted to validate the analytical predictions.

Period-m Motions in a Periodically Forced van der Pol Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Fourier Series ◽  
Periodic Motion ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Stable Period ◽  
Stability And Bifurcation Analysis ◽  
Series Expression ◽  
The Stability

Period-m motions in a periodically forced, van der Pol oscillator are investigated through the Fourier series expression, and the stability and bifurcation analysis of such periodic motions are carried out. To verify the approximate solutions of period-m motions, numerical illustrations are given. Period-m motions are separated by quasi-periodic motion or chaos, and the stable period-m motions are in independent periodic motion windows.

Analytical Solutions of Period-1 Motions in a Periodically Forced van der Pol Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Van Der Pol Oscillator ◽  
Comprehensive Understanding ◽  
Van Der Pol ◽  
Stability And Bifurcation ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balanced method. The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis, and numerical illustrations of periodic-1 solutions are given to verify the approximate motion. This investigation provides more accurate solutions of period-1 motions in the van der pol oscillator for a better and comprehensive understanding of motions in such an oscillator.

BIFURCATION STRUCTURE OF THE DRIVEN VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (06) ◽  
pp. 1529-1555 ◽  
Author(s):  
R. METTIN ◽  
U. PARLITZ ◽  
W. LAUTERBORN
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Van Der Pol Oscillator ◽  
Chaotic Attractors ◽  
Transitional Region ◽  
Driving Frequency ◽  
Van Der Pol ◽  
Cycle Frequency ◽  
Dimensional Parameter ◽  
Periodically Driven

The bifurcation set in the three-dimensional parameter space of the periodically driven van der Pol oscillator has been investigated by continuation of local bifurcation curves. The two regions in which the driving frequency ω is greater or less than the limit cycle frequency ω0 of the nondriven oscillator are considered separately. For the case ω > ω0, the subharmonic region, the extent and location of the largest Arnol'd tongues are shown, as well as the period-doubling cascades and chaotic attractors that appear within most of them. Special attention is paid to the pattern of the bifurcation curves in the transitional region between low and large dampings that is difficult to approach analytically. In the case ω < ω0, the ultraharmonic region, a recurrent pattern of the bifurcation curves is found for small values of the damping d. At medium damping the structure of the bifurcation curves becomes involved. Period-doubling sequences and chaotic attractors occur.

Memristor Based van der Pol Oscillation Circuit

2014 ◽  
Vol 24 (12) ◽  
pp. 1450154 ◽  
Author(s):  
Yimin Lu ◽  
Xianfeng Huang ◽  
Shaobin He ◽  
Dongdong Wang ◽  
Bo Zhang
Keyword(s):  
Analog Circuit ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Circuit Element ◽  
Nonlinear Properties ◽  
Oscillation Circuit ◽  
Excited Oscillation ◽  
Exciting Source ◽  
The Stability ◽  
Passive Circuit

The memristor is referred to as the fourth fundamental passive circuit element of which inherent nonlinear properties offer to construct the chaos circuits. In this paper, a flux-controlled memristor circuit is developed, and then a van der Pol oscillator is implemented based on this new memristor circuit. The stability of the circuit, the occurring conditions of Hopf bifurcation and limit circle of the self-excited oscillation are analyzed; meanwhile, under the condition of the circuit with an external exciting source, the circuit exhibits a complicated nonlinear dynamic behavior, and chaos occurs within a certain parameter set. The memristor based van der Pol oscillator, furthermore, has been created by an analog circuit utilizing active elements, and there is a good agreement between the circuit responses and numerical simulations of the van der Pol equation. In the consequence, a new approach has been proposed to generate chaos within a nonautonomous circuit system.

Identification of van der Pol oscillator parameters

2021 ◽  
Vol 34 ◽  
pp. 11-18
Author(s):  
Iryna Baraniukova ◽  
Volodymyr Shcherbak
Keyword(s):  
Stability Property ◽  
Original System ◽  
Oscillatory Behavior ◽  
Parameters Estimation ◽  
Van Der Pol Oscillator ◽  
Extended System ◽  
Van Der Pol ◽  
Simulation Results ◽  
The Stability

An invariant relations method for parameters estimation is used for the wellknown van der Pol oscillator. The approach is based on dynamical extension of original system and synthesis of appropriate invariant relations, from which the unknowns can be expressed as a functions of the known quantities on the trajectories of extended system. The stability property is formally checked taking into account the oscillatory behavior of the system. The simulation results confirm efficiency of the proposed scheme of nonlinear identificator design.

scholarly journals Coupling the Neural and Physical Dynamics in Rhythmic Movements

1996 ◽  
Vol 8 (3) ◽  
pp. 567-581 ◽  
Author(s):  
Nicholas G. Hatsopoulos
Keyword(s):  
Oscillation Frequency ◽  
Sensory Feedback ◽  
Motor Behavior ◽  
Angular Displacement ◽  
Oscillation Amplitude ◽  
Coupled System ◽  
Van Der Pol Oscillator ◽  
Rhythmic Movements ◽  
Van Der Pol ◽  
Pendulum System

A pair of coupled oscillators simulating a central pattern generator (CPG) interacting with a pendular limb were numerically integrated. The CPG was represented as a van der Pol oscillator and the pendular limb was modeled as a linearized, hybrid spring-pendulum system. The CPG oscillator drove the pendular limb while the pendular limb modulated the frequency of the CPG. Three results were observed. First, sensory feedback influenced the oscillation frequency of the coupled system. The oscillation frequency was lower in the absence of sensory feedback. Moreover, if the muscle gain was decreased, thereby decreasing the oscillation amplitude of the pendular limb and indirectly lowering the effect of sensory feedback, the oscillation frequency decreased monotonically. This is consistent with experimental data (Williamson and Roberts 1986). Second, the CPG output usually led the angular displacement of the pendular limb by a phase of 90° regardless of the length of the limb. Third, the frequency of the coupled system tuned itself to the resonant frequency of the pendular limb. Also, the frequency of the coupled system was highly resistant to changes in the endogenous frequency of the CPG. The results of these simulations support the view that motor behavior emerges from the interaction of the neural dynamics of the nervous system and the physical dynamics of the periphery.

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