Duffing Van Der Pol Oscillator Research Articles

New aspects of integrability of force-free Duffing–van der Pol oscillator and related nonlinear systems

In this letter we show that the force-free Duffing-van der Pol oscillator is completely integrable for a specific parametric choice. We derive a general solution for this parametric choice. Further, we describe a procedure to construct the transformation which removes the time dependent part from the first integral and provide the general solution by quadrature. The procedure is shown to have a wider applicability through additional examples. We also show that through our method one can deduce linearizing transformations in a simple and straightforward way and illustrate it with a specific example.

Duration of the process of complete synchronization of two coupled identical chaotic systems

We consider the time required for complete synchronization of two identical one-way coupled vander Pol-Duffing oscillators occurring in the regime of dynamic chaos. The influence of the initial phase differ-ence between oscillators on the duration of the process of complete synchronization has been studied. At a fixedphase of chaotic oscillations of the self-excited drive oscillator, the period of time (past the coupling onset) during which the complete synchronization regime is established depends on the phase of the self-excited responseoscillator

Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to combined harmonic and white noise excitations. According to the proposed method, an n+ α+ β-dimensional averaged Fokker–Planck–Kolmogorov (FPK) equation governing the transition probability density of n action variables or independent integrals of motion, α combinations of angle variables and β combinations of angle variables and excitation phase angles can be constructed when the associated Hamiltonian system has α internal resonant relations and the system and harmonic excitations have β external resonant relations. The averaged FPK equation is solved by using the combination of the finite difference method and the successive over relaxation method. Two coupled Duffing–van der Pol oscillators under combined harmonic and white noise excitations is taken as an example to illustrate the application of the proposed procedure and the stochastic jump and its bifurcation as the system parameters change are examined.

Stochastic bifurcation in Duffing–Van der Pol oscillators

Global analysis of bifurcation in the Duffing–van der Pol oscillators under both additive and multiplicative random excitations is explored in detail by the generalized cell mapping method using digraph. System parameters are chosen in the range that two co-exist attractors and a saddles. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. It is found that under certain conditions stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.

Resonant oscillation and homoclinic bifurcation in a Φ6-Van der Pol oscillator

In this paper, we study the dynamics of a system consisting of extended Duffing–Van der Pol oscillator. Resonance and nonresonance oscillations are analyzed using the multiple scale method. From the Melnikov theorem, we obtain the condition for the existence of homoclinic bifurcation both in the case where the Φ 6 potential is a bounded or unbounded double hump. A convenient demonstration of the accuracy of the method is obtained from the fractal basin boundary.

Strange attractors and chaos control in a Duffing–Van der Pol oscillator with two external periodic forces

An anharmonic Duffing–Van der Pol oscillator with two external forces is considered. By applying numerical results, strange attractors are presented and the chaotic behaviour is investigated. The problem of directing a chaotic state of the system to a periodic orbit is studied. By assuming that the exact model of the system is not known and that the position is the only state available for measurements, the controller comprises a linearizing-like feedback and an estimator. Simulations are provided to illustrate the performance of the controller.

Synchronization law for a van der Pol array.

We explore the transition to in-phase synchronization in globally coupled oscillator arrays, and compare results for van der Pol arrays with Josephson junction arrays. Our approach yields in each case an analytically tractable iterative map; the resulting stability formulas are simple because the expansion procedure identifies natural parameter groups. A third example, an array of Duffing-van der Pol oscillators, is found to be of the same fundamental type as the van der Pol arrays, but the Josephson arrays are fundamentally different owing to the absence of self-resonant interactions.

Identification of the state equation in complex non-linear systems

Building on the basic idea behind the Restoring Force Method for the non-parametric identification of non-linear systems, a general procedure is presented for the direct identification of the state equation of complex non-linear systems. No information about the system mass is required, and only the applied excitation(s) and resulting acceleration are needed to implement the procedure. Arbitrary non-linear phenomena spanning the range from polynomial non-linearities to the noisy Duffing–van der Pol oscillator (involving product-type non-linearities and multiple excitations) or hysteretic behavior such as the Bouc–Wen model can be handled without difficulty. In the case of polynomial-type non-linearities, the approach yields virtually exact results for sufficiently rich excitations. For other types of non-linearities, the approach yields the optimum (in least-squares sense) representation in non-parametric form of the dominant interaction forces induced by the motion of the system. Several examples involving synthetic data corresponding to a variety of highly non-linear phenomena are presented to demonstrate the utility as well as the range of validity of the proposed approach.

Resonance behaviour and jump phenomenon in a two coupled Duffing–van der Pol oscillators

We consider a two coupled Duffing–van der Pol oscillators with a nonlinear coupling. The method of multiple scales is used to obtain a set of autonomous equations for the amplitudes and phases of the response of the system. The stability boundaries of fixed points of the approximate equations are obtained using Routh–Hurwitz criterion. The stability boundaries are drawn in various parameters space. The influence of nonlinear damping, detuning, nonlinearity and the coupling strength on the response dynamics is studied. Jump phenomenon is found to occur for a range of values of the parameters of the system. The results obtained from the approximate equations are verified by numerically solving the original system and good agreement is obtained.

Global analysis of stochastic bifurcation in a Duffing-van der Pol system

Stochastic bifurcation of the Duffing-van der Pol oscillators under both additive and multiplicative random excitations is studied in detail by the generalized cell mapping method using digraph.As an alternative definition,stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value.It is found that under certain conditions stochastic bifurcation mostly occurs when a stochastic attractor collides with a stochastic saddle.Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation.By this global analysis ,the mechanism of development,occurrence,and evolution of a stochastic bifurcation can be explored clearly and vividly.

Amplitude envelope synchronization in coupled chaotic oscillators

A peculiar type of synchronization has been found when two Van der Pol-Duffing oscillators, evolving in different chaotic attractors, are coupled. As the coupling increases, the frequencies of the two oscillators remain different, while a synchronized modulation of the amplitudes of a signal of each system develops, and a null Lyapunov exponent of the uncoupled systems becomes negative and gradually larger in absolute value. This phenomenon is characterized by an appropriate correlation function between the returns of the signals, and interpreted in terms of the mutual excitation of new frequencies in the oscillators power spectra. This form of synchronization also occurs in other systems, but it shows up mixed with or screened by other forms of synchronization, as illustrated in this paper by means of the examples of the dynamic behavior observed for three other different models of chaotic oscillators.

Principal response of van der pol-duffing oscillator under combined deterministic and random parametric excitation

The principal resonance of Van der Pol-Duffing oscillator to combined deterministic and random parametric excitations is investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied. Jumps were shown to occur under some conditions. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analysis were verified by numerical results.

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise.

An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator

The Lyapunov exponents of the linearization x = - x + 2betax + rho zeta t x of a noisy Duffing-van der Pol oscillator are key quantities in the investigation of the stochastic Hopf bifurcation of this system. Considering the white noise case we derive a simple equation exhibiting them explicitly as functions of the fourth moment of the invariant measure of an associated diffusion with drift given by a potential function and additive noise, and, consequently, in terms of hypergeometric functions. This representation leads to diff erent kinds of complete and explicit asymptotic expansions, as well as a rather complete account of global properties of the Lyapunov exponents as functions of beta and rho.

Statistical dynamics at critical bifurcations in Duffing-van der Pol oscillator

We study the characteristic features of certain statistical quantities near critical bifurcations such as onset of chaos, sudden widening and band-merging of chaotic attractor and intermittency in a periodically driven Duffing-van der Pol oscillator. At the onset of chaos the variance of local expansion rate is found to exhibit a self-similar pattern. For all chaotic attractors the variance Σn(q) of fluctuations of coarse-grained local expansion rates of nearby orbits has a single peak. However, multiple peaks are found just before and just after the critical bifurcations. On the other hand, Σn (q) associated with the coarse-grained state variable is zero far from the bifurcations. The height of the peak of Σn(q) is found to increase as the control parameter approached the bifurcation point. It is maximum at the bifurcation point. Power-law variation of maximal Lyapunov exponent and the mean value of the state variablex is observed near sudden widening and intermittency bifurcations while linear variation is seen near band-merging bifurcation. The standard deviation of local Lyapunov exponent λ(X,L) and the local mean valuex(L) of the coordinatex calculated after everyL time steps are found to approach zero in the limitL → ∞ asL-Β. Β is sensitive to the values of control parameters. Further weak and strong chaos are characterized using the probability distribution of ak-step difference quantity δxk = xi+kxi.

Strongly resonant bifurcations of nonlinearly coupled van der Pol-Duffing Oscillator

In this paper, the strongly resonant bifurcations of a nonlinearly coupled Van der Pol-Duffing Oscillator by the classical multi-scale method are studied. It is shown that there exist periodic motions of a single oscillator, frequency-locking and quasi-periodic motions of two oscillators when the parameters vary. Meanwhile, some numerical results are given to test the theoretical ones.

COMPLEX NORMAL FORM FOR STRONGLY NON-LINEAR VIBRATION SYSTEMS EXEMPLIFIED BY DUFFING–VAN DER POL EQUATION

We extend the normal form method to study the asymptotic solutions of strongly non-linear oscillatorsu+ω2u=f](u,u), wheref(u,u) contains only linear and cubic non-linear terms. The novel contribution is the ansatzu=ξ+ξ,u=iω1(ξ−ξ) where ω1is to be determined, allowing for the change of the fundamental frequency during the course of vibration, rather than usingu=ξ+ξ,u=iω(ξ−ξ) as suggested by Nayfeh. With the present method, not only the stability of the periodic solutions but also the asymptotic expressions for the periodic solutions can be obtained easily. The results obtained by the method presented coincide very well with the results obtained by numerical integration for the Duffing–van der Pol oscillator withf(u,u)=μ(1−u2)u−βu3. When ω=μ=β=1, Nayfeh's method givers qualitatively different results from the numerical integration while our method works well even when ω=1, μ=β=3, since Nayfeh's method is based on weak non-linearities and ω=1, μ=β=3 is beyond the valid range of assumption.

Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: Numerical and analytical studies

The behavior of a driven double-well Duffing--van der Pol oscillator for a specific parametric choice $(|\ensuremath{\alpha}|=\ensuremath{\beta})$ is studied. The existence of different attractors in the system parameters $f$-$\ensuremath{\omega}$ domain is examined and a detailed account of various steady states for fixed damping is presented. The transition from quasiperiodic to periodic motion through chaotic oscillations is reported. The intervening chaotic regime is further shown to possess islands of phase-locked states and periodic windows (including period-doubling regions), boundary crisis, three classes of intermittencies, and transient chaos. We also observe the existence of local-global bifurcation of intermittent catastrophe type and global bifurcation of blue-sky catastrophe type during the transition from quasiperiodic to periodic solutions. Using a perturbative periodic solution, an investigation of the various forms of instabilities allows one to predict Neimark instability in the $f$-$\ensuremath{\omega}$ plane and eventually results in the approximate predictive criteria for the chaotic region.

TOWARD AN UNDERSTANDING OF STOCHASTIC HOPF BIFURCATION

In this paper, asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through the study of a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied. In the first part of this paper, we present an introduction to the theory of random dynamical systems (in particular, their generation, their invariant measures, the multiplicative ergodic theorem, and Lyapunov exponents). We then present the two concepts of stochastic bifurcation theory: Phenomenological (based on the Fokker-Planck equation), and dynamical (based on Lyapunov exponents). The method of stochastic averaging of the nonlinear system yields a set of equations which, together with its variational equation, can be explicitly solved and hence its bifurcation behavior completely analyzed. We augment this analysis by asymptotic expansions of the Lyapunov exponents of the variational equation at zero. Finally, the stochastic normal form of the noisy Duffing-van der Pol oscillator is derived, and its bifurcation behavior is analyzed numerically. The result is that the (truncated) normal form retains the essential bifurcation characteristics of the full equation.