Van Der Pol-Duffing Equation Research Articles

Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of O(h4) for a small time stepsize h.

The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

Regularising transformations for the (n,,n+1)-Liénard equations

We discuss regularising transformations for the Liénard equations of the form y''=F(z,y)y'+G(z,y), where F and G are polynomials of degrees n and n+1 respectively using the geometric approach. As a particular case we find a transformation for the Duffing–van der Pol equation which leads to the regularisation.

Analytic approximate solutions of the cubic–quintic Duffing–van​ der Pol equation with two-external periodic forcing terms: Stability analysis

In the light of the potential applications in engineering, electronics, physics, chemistry, and biology, the current work applies several techniques to achieve analytic approximate and numerical solutions of the cubic–quintic Duffing–van der Pol equation. This equation represents a second-order ordinary differential equation with quintic nonlinearity and includes two external periodic forcing terms. A classical approximate solution involves the secular terms is obtained. Unfortunately, this traditional method does not enable us to ignore these secular terms. Additionally, along with the concept of the expanded frequency, a bounded approximate solution is achieved. The Homotopy perturbation method is utilized to obtain an approximate solution with an artificial frequency of the given system. Near the equilibrium points, in the case of the autonomous system, the linearized stability is accomplished. Furthermore, in the case of the non-autonomous system, by means of the multiple time scales, the stability analysis is effectuated, together with the resonance and the non-resonance cases. Numerical computations are performed to demonstrate, graphically, the perturbed solutions as well as the stability/instability regions. Various numerical solutions to initial–boundary value problems are deduced via a three-step finite difference scheme. These are plotted and discussed to show the chaotic nature of solutions.

Exact Solutions and Integrability of the Duffing – Van der Pol Equation

The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painleve test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.

Modeling nonlinear dissipative chemical dynamics by a forced modified Van der Pol-Duffing oscillator with asymmetric potential: Chaotic behaviors predictions

This paper addresses the issues of nonlinear chemical dynamics modeled by a modified Van der Pol-Duffing oscillator with asymmetric potential. The Melnikov method is utilized to analytically determine the domains boundaries where Melnikov’s chaos appears in chemical oscillations. Routes to chaos are investigated through bifurcations structures, Lyapunov exponent, phase portraits and Poincaré section. The effects of parameters in general and in particular the effect of the constraint parameter β which shows the difference between a nonlinear chemical dynamics order two differential equation and ordinary Van der Pol-Duffing equation are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.

Novel bursting patterns in a Van der pol-Duffing oscillator with slow varying external force

In this paper, we investigate the emergence of bursting dynamics with complex waveforms and their relation to periodic behavior in typical Van der pol-Duffing equation with fifth order polynomial stiffness nonlinearity, when the external force changes slowly with the variation of time. We exploit bifurcation characteristics of the fast subsystem using the slowly changing periodic excitation as a bifurcation parameter to show how the bursting oscillations are created in this model. We also identify that some regimes of bursting patterns are related to codimension two bifurcation type over a wide range of parameters. A subsequent two-parameter continuation reveals a transition in the bursting behavior from fold/fold hysteresis cycle to sup-Hopf/sup-Hopf or limit point cycle/sub-Hopf bursting type. Furthermore, the effects of external forcing item on bursting oscillations are investigated. For instance, the time interval between two adjacent spikes of bursting oscillations is dependent on the forcing frequency. Some numerical simulations are included to illustrate the validity of our study.

Differential properties of Van der Pol — Duffing mathematical model of cerebrovascular haemodynamics based on clinical measurements

The present paper discusses the method of identification (diseased/healthy) human cerebral vessels by using of mathematical model. Human cerebral circulation as a single tuned circuit, which consists of blood flow, elastic vessels and elastic brain gel tissue is under consideration. Non linear Van der Pol-Duffing equation is assumed as mathematical model of cerebrovascular circulation. Hypothesis of vascular pathology existence in some position of blood vessel, based on mathematical model properties for this position is formulated. Good reliability of hypothesis is proved statistically for 7 patients with arterial aneurysms.

Bifurcations and Chaos of Duffing–van der Pol Equation with Nonsymmetric Nonlinear Restoring and Two External Forcing Terms

Bifurcations and chaos of Duffing–van der Pol equation with nonsymmetric nonlinear restoring and two external forcing terms are investigated. The threshold values of the existence of chaotic motion are obtained under periodic perturbation. By the second-order averaging method, we prove the criteria of the existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 2, 3, 5, and cannot prove the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 4, 6, 7, …, where σ is not rational to ω1, but can show the occurrence of chaos in the original system by numerical simulation. Numerical simulation including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincaré maps, not only show the consistence with the theoretical analysis but also exhibit more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations lead to interleaving occurrence of chaotic behaviors and quasi-periodic orbits, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos occurring more than once, chaos suddenly disappearing to periodic orbits, strange nonchaotic attractor, nonattracting chaotic set and nice chaotic attractors.

Phenomenological model for torsional galloping of an elastic flat plate due to hydrodynamic loads

This study investigates the torsional galloping phenomenon, an instability type flow-induced oscillation, in an elastic structure due to hydrodynamic loads into the water current. The structure applied here is a rectangular flat plate with an elastic axis in its mid-chord length. The elasticity is provided by torsion spring. The flat plate has only one degree of freedom which is rotation in pure yaw about its axis. It is observed that as the current speed is higher than a critical velocity, the flat plate becomes unstable. The instability leads to torsional galloping occurrence, as a result of which the flat plate begins to yaw about the elastic axis. By testing two different chord lengths each with several torsion spring rates, the flat plate behavior is investigated and three different responses are recognized. Then, a phenomenological model is developed with the original kernel in the form of the van der Pol-Duffing equation. The model explains these three responses observed experimentally.

Analysis of non-linear stochastic oscillations by the averaging method

The solution of a quasiconservative non-linear oscillatory system is considered, the right-hand sides of which are proportional to a small parameter. Fundamental relations for solving the problem are obtained by changing to “slow” variables and a combination of the stochastic averaging method and the theory of Markov processes. An efficient numerical algorithm is developed based on the fast Fourier transform that enables the output parameter distribution density and the amplitudes of the oscillations to be obtained. Application of the theory to solve the Duffing–van der Pol equation for an additive and multiplicative stochastic action is considered.

The periodic oscillations in the ENSO recharge—discharge oscillator model

A class of recharge—discharge oscillator model for the El Niño/Southern Oscillation (ENSO) is considered. A stable limit cycle is obtained by transforming the ENSO model into the van der Pol-Duffing equation. We proved that there exists periodic oscillations in the ENSO recharge—discharge oscillator model.

BIFURCATIONS OF PERIODIC SOLUTIONS AND CHAOS IN DUFFING-VAN DER POL EQUATION WITH ONE EXTERNAL FORCING

The Duffing-Van der Pol equation withfifth nonlinear-restoring force and one external forcing term isinvestigated in detail: the existence and bifurcations of harmonicand second-order subharmonic, and third-order subharmonic,third-order superharmonic and $m$-order subharmonic under smallperturbations are obtained by using second-order averaging methodand subharmonic Melnikov function; the threshold values of existenceof chaotic motion are obtained by using Melnikov method. Thenumerical simulation results including the influences of periodicand quasi-periodic and all parameters exhibit more new complexdynamical behaviors. We show that the reverse period-doublingbifurcation to chaos, period-doubling bifurcation to chaos,quasi-periodic orbits route to chaos, onset of chaos, and chaossuddenly disappearing, and chaos suddenly converting to periodorbits, different chaotic regions with a great abundance of periodicwindows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, andso on), and more wide period-one window, and varied chaoticattractors including small size and maximum Lyapunov exponentapproximate to zero but positive, and the symmetry-breaking ofperiodic orbits. In particular, the system can leave chaotic regionto periodic motion by adjusting the parameters $p, \beta, \gamma, f$and $\omega$, which can be considered as a control strategy.

An Optimal Analytic Approximate Solution for the Limit Cycle of Duffing–van der Pol Equation

The present paper is concerned with the accurate analytic solution of the limit cycle of the Duffing–van der Pol equation. Instead of the traditional Taylor series or asymptotic methods, the homotopy analysis technique is employed, which does not require a small perturbation parameter or a large asymptotic parameter. It is known that such a method is extremely powerful in gaining the exact solution of the physical problem in terms of purely trigonometric functions, yet the computational cost of the method is considerably high. We propose here an approach that not only greatly reduces the computational efforts but also presents an easy to implement task of application of the homotopy analysis method to the Duffing–van der Pol equation. The explicit analytical expressions obtained using the proposed approach generates the displacement, amplitude, and frequency of the limit cycle that compare excellently with the numerically computed ones.

Uniformly valid solution of limit cycle of the Duffing–van der Pol equation

The limit cycle of the Duffing–van der Pol equation x ¨ + x + ε ( x 2 - 1 ) x ˙ + δ x 3 = 0 is studied. By considering the product of the frequency ω of the limit cycle and the coefficient ε as an independent parameter μ = ε ω , an equivalent equation is obtained and then solved by Liao’s homotopy analysis method. The frequency ω is deduced as a function of μ and δ . This function provides us with an algebraic equation for ω , according to which we have an analytical approximation for the frequency. Numerical examples show that the attained approximation is very accurate. More importantly, the results are uniformly valid for all positive values of ε .

Period-doubling cascades and strange attractors in the triple-well ϕ6-Van der Pol oscillator

Duffing–Van der Pol equation with the fifth nonlinear-restoring force is investigated. The bifurcation structure and chaotic motion under the periodic perturbation are obtained by numerical simulations. Numerical simulations, including bifurcation diagrams, Lyapunov exponents, phase portraits and Poincaré maps, exhibit some new complex dynamical behaviors of the system. Different routes to chaos, such as period doubling and quasi-periodic routes, and various kinds of strange attractors are also demonstrated.

A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems

A symbolic algorithm is developed for the automatic generation of harmonic balance equations for multitone input for a class of nonlinear differential systems with polynomial nonlinearities. Generalized expressions are derived for the construction of balance equations for a defined multitone signal form. Procedures are described for determining combinations for a given output frequency from the desired set obtained from box truncated spectra and their permutations to automate symbolic algorithm. An application of method is demonstrated using the well-known Duffing–Van der Pol equation. Then the obtained analytical results are compared with numerical simulations to show the accuracy of the approach. The computation times for both the numerical solutions of equations versus the number of frequency components and the symbolic generation of the equations versus the power of nonlinearity are also investigated.

Controlled chaotic autooscillatory system

A mathematical model of the chaotic system based on a three-dimensional modification of the van der Pol-Duffing equation is considered. Solutions possible in the presence of a chaotizing feedback algorithm are analyzed by numerical methods. It is shown that the chaotization of oscillations can be controlled by an autocommutator, and the behavior of a modified system for various parameters of the additional feedback is analyzed.

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω1, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.

The construction of homo- and heteroclinic orbits in non-linear systems

Pade and quasi-Pade approximants are used to construct homo- and heteroclinic orbits of non-linear systems. By using the convergence condition for Pade approximants and the conditions at infinity the problem can be solved with sufficiently high accuracy. Actual computations are carried out for the non-autonomous Duffing equation, the equations of vibrations of a parametrically driven mathematical pendulum, and the van der Pol-Duffing equation with non-linear elastic characteristic.