Comparative dynamics of chains of coupled van der Pol equations and coupled systems of van der Pol equations

Theoretical and Mathematical Physics - We consider chains of van der Pol equations closed into a ring and chains of systems of two first-order van der Pol equations. We assume that the couplings...     

Dynamics of three coupled van der Pol oscillators with application to circadian rhythms

In this work we study a system of three van der Pol oscillators. Two of the oscillators are identical, and are not directly coupled to each other, but rather are coupled via the third oscillator. We investigate the existence of the in-phase mode in which the two identical oscillators have the same behavior. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the computer algebra system MACSYMA and the numerical bifurcation software AUTO.Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator. Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator.     

Stability and bifurcation analysis in the delay-coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol equations with a finite delay are investigated. We show that there exist the stability switches and a sequence of Hopf bifurcations occur at the zero equilibrium when the delay varies. Using the theory of normal form and the center manifold theorem, the explicit expression for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.     

Limit Cycle-Strange Attractor Competition

To understand the competition between what are known as limit cycle and strange attractor dynamics, the classical oscillators that display such features were coupled and studied with and without external forcing. Numerical simulations show that, when the Duffing equation (the strange attractor prototype) forces the van der Pol oscillator (the limit cycle prototype), the limit cycle is destroyed. However, when the van der Pol oscillator is coupled to the Duffing equation as linear forcing, the two traditionally stable steady states are destabilized and a quasi-periodic orbit is born. In turn, this limit cycle is eventually destroyed because the coupling strength is increased and eventually gives way to strange attractor or chaotic dynamics. When two van der Pol oscillators are coupled in the absence of external periodic forcing, the system approaches a stable, nonzero steady state when the coupling strengths are both unity; trajectories approach a limit cycle if coupling strengths are equal and less than 1. Solutions grow unbounded if the coupling strengths are equal and greater than 1. Quasi-periodic solutions give way to chaos as the coupling strength increases and one oscillator is strongly coupled to the other. Finally, increasing the nonlinearity in both the oscillators is stabilizing whereas increasing the nonlinearity in a single oscillator results in subcritical instability.     

Oscillatory region and asymptotic solution of fractional van der Pol oscillator via residue harmonic balance technique

In this paper, a powerfully analytical technique is proposed for predicting and generating the steady state solution of the fractional differential system based on the method of harmonic balance. The zeroth-order approximation using just one Fourier term is applied to predict the parametric function for the boundary between oscillatory and non-oscillatory regions of the fractional van der Pol oscillator. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear algebraic equations to improve the accuracy of the solutions successively. The highly accurate solutions to the angular frequency and limit cycle of fractional van der Pol oscillator are obtained and compared. The results reveal that the technique described in this paper is very effective and simple for obtaining asymptotic solution of nonlinear system having fractional order derivative.     

Stochastic optimal control of first-passage failure for coupled Duffing–van der Pol system under Gaussian white noise excitations

In this paper, the stochastic averaging method of quasi-non-integrable-Hamiltonian systems is applied to Duffing–van der Pol system to obtain partially averaged Ito stochastic differential equations. On the basis of the stochastic dynamical programming principle and the partially averaged Ito equation, dynamical programming equations for the reliability function and the mean first-passage time of controlled system are established. Then a non-linear stochastic optimal control strategy for coupled Duffing–van der Pol system subject to Gaussian white noise excitation is taken for investigating feedback minimization of first-passage failure. By averaging the terms involving control forces and replacing control forces by the optimal ones, the fully averaged Ito equation is derived. Thus, the feedback minimization for first-passage failure of controlled system can be obtained by solving the final dynamical programming equations. Numerical results for first-passage reliability function and mean first-passage time of the controlled and uncontrolled systems are compared in illustrative figures to show effectiveness and efficiency of the proposed method.     

Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay

A system of three coupled van der Pol oscillators with delay is considered. Hopf bifurcations at the zero equilibrium as the delay increases are exhibited. The existence and stability of multiple periodic solutions are established using a symmetric Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799-4838).     

A nonconvex dissipative system and its applications (I)

In order to study the uniformly translating solution of some non-linear evolution equations such as the complex Ginzburg–Landau equation, this paper presents a qualitative analysis to a Duffing–van der Pol non-linear oscillator. Monotonic property of the bounded exact solution is established based on the construction of a convex domain. Under certain parametric choices, one first integral to the Duffing–van der Pol non-linear system is obtained by using the Lie symmetry analysis, which constitutes one of the bases for further work of obtaining uniformly translating solutions of the complex Ginzburg–Landau equation. Dedicated to Professor G. Strang on the occasion of his 70th birthday     

A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.     

Global existence of solutions for a class of nonlinear differential equations

By making use of a special Lyapunov-type function and applying the comparison method due to Conti, we prove global existence of solutions for a general class of nonlinear second-order differential equations that includes, in particular, van der Pol, Rayleigh, and Liénard equations, widely encountered in applications. Relevant examples are discussed.     

Nonautonomous dynamics of coupled van der Pol oscillators in the regime of amplitude death

A pulse driven system of two coupled van der Pol oscillators in the regime of amplitude death is studied. The existence of islands of quasiperiodic regimes on the parameter plane of period and amplitude of the external force is shown in numerical and electronic experiments. A number of different types of oscillations in this system are illustrated.     

Existence of limit cycles for coupled van der Pol equations

In this paper, we consider the existence of limit cycles of coupled van der Pol equations by using S¹-degree theory due to Dylawerski et al. (see Ann. Polon. Math. 62 (1991) 243).     

Poincaré bifurcation of a three-dimensional system

Consider a three-dimensional system having an invariant surface. By using bifurcation techniques and analyzing the solutions of bifurcation equations, we study the spatial bifurcation phenomena near a family of periodic orbits and a center in the invariant surface respectively. New formula of Melnikov function is derived and sufficient conditions for the existence of periodic orbits are obtained. An application of our results to a modified van der Pol–Duffing electronic circuit is given.     

Nonintegrability of Parametrically Forced Nonlinear Oscillators

We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by second-order homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using the Kovacic algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable, i. e., they can be solved by quadratures. The obtained results are very general and reveal their rational nonintegrability for the wide class of parametrically forced nonlinear oscillators. We also give two examples for the van der Pol and Duffing oscillators to demonstrate our results.     

Chaos in a generalized van der Pol system and in its fractional order system

In this paper, chaos of a generalized van der Pol system with fractional orders is studied. Both nonautonomous and autonomous systems are considered in detail. Chaos in the nonautonomous generalized van der Pol system excited by a sinusoidal time function with fractional orders is studied. Next, chaos in the autonomous generalized van der Pol system with fractional orders is considered. By numerical analyses, such as phase portraits, Poincaré maps and bifurcation diagrams, periodic, and chaotic motions are observed. Finally, it is found that chaos exists in the fractional order system with the order both less than and more than the number of the states of the integer order generalized van der Pol system.     

Period-doubling bifurcation in an extended van der Pol system with bounded random parameter

An extended van der Pol system with bounded random parameter subjected to harmonic excitation is investigated by Chebyshev polynomial approximation. Firstly the stochastic extended van der Pol system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then we explored nonlinear dynamical behavior about period-doubling bifurcation in stochastic system. Numerical simulations show that similar to the conventional period-doubling phenomenon in deterministic extended van der Pol system, stochastic period-doubling bifurcation may also occur in the stochastic extended van der Pol system. Besides, different from the deterministic case, in addition to the conventional bifurcation parameters, i.e. the amplitude and frequency of harmonic excitation, in the stochastic case the intensity of random parameter should also be taken as a new bifurcation parameter.     

Chaos in a modified van der Pol system and in its fractional order systems

Chaos in a modified van der Pol system and in its fractional order systems is studied in this paper. It is found that chaos exists both in the system and in the fractional order systems with order from 1.8 down to 0.8 much less than the number of states of the system, two. By phase portraits, Poincaré maps and bifurcation diagrams, the chaotic behaviors of fractional order modified van der Pol systems are presented.     

Approximate limit cycles of coupled nonlinear oscillators with fractional derivatives

In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled nonlinear oscillators with fractional derivatives. Approximate limit cycles (LCs) of two systems of the coupled fractional van der Pol (VDP) oscillators and the fractional damped Duffing resonator driven by a fractional VDP oscillator are exampled for illustrating the validity and great potential of the HAM. The presented approach can provide approximate LCs very accurately and efficiently compared with some direct simulation results. This method can keep high accuracy and efficiency for both weakly and strongly nonlinear problems with any given fractional order. Furthermore, it is capable of tracking unstable LCs which cannot be generated by some time-marching numerical algorithm. Based on the obtained results, we analyze effect of different fractional orders, coupling coefficient, and nonlinear coefficient of the coupled equations on amplitudes and frequencies of the LCs.     

Global Algebraic Poincaré–Bendixson Annulus for the van der Pol System

Differential Equations - We construct analytically two algebraic closed curves forming a Poincaré–Bendixson annulus for the van der Pol system for all values of its parameter. The inner...     

A study of homotopy analysis method for limit cycle of van der Pol equation

The homotopy analysis method for limit cycle of the van der Pol equation is computationally extensive. Aiming at reducing the computational efforts, we construct the deformation equations using different initial conditions from those in the existing literature. The computational efforts are significantly reduced. And, the calculated amplitude and frequency of the limit cycle are in excellent agreement with the exact solutions.