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.     

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.     

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

Dynamics of three coupled limit cycle oscillators with application to artificial intelligence

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can make decisions by identifying a given initial condition with its associated steady state.     

Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method

In this paper, the iteration perturbation method proposed by He [J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, 2006; J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26 (2005) 827–833] is used to generate periodic solutions of van der Pol oscillator with a forcing term, forcing oscillator with quadratic type damping and van der Pol oscillator with excitation term. The comparison of the obtained results verifies its convenience and effectiveness.     

Strange attractors and synchronization dynamics of coupled Van der Pol–Duffing oscillators

We consider in this paper the synchronization dynamics of coupled chaotic Van der Pol–Duffing systems. We first find that with the judicious choose of the set of initial conditions, the model exhibits two strange chaotic attractors. The problem of synchronizing chaos both on the same and different chaotic orbits of two coupled Van der Pol–Duffing systems is investigated. The stability boundaries of the synchronization process between two coupled driven Van der Pol model are derived and the effects of the amplitude of the periodic perturbation of the coupling parameter on these boundaries are analyzed. The results are provided on the stability map in the (q, K) plane.     

Anharmonic oscillator systems

The anharmonic oscillator is solved quickly, easily, and elegantly by Adomian's methods for solution of nonlinear stochastic differential equations emphasizing its applicability to nonlinear deterministic equations as well as stochastic equations. No difficulty is encountered in treating the case of the forced anharmonic oscillator or the stochastic case or of any nonlinear oscillating system such as the Duffing or Van der Pol oscillators, for example, with coefficients, as well as forcing functions, which are stochastic processes, since statistical separability is inherent in the Adomian method.     

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.     

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.     

Global analysis of boundary and interior crises in an elastic impact oscillator

The crisis phenomena of a Duffing–Van der Pol oscillator with a one-side elastic constraint are studied by the composite cell coordinate system method in this paper. By computing the global properties such as attractors, basins of attraction and saddles, the vivid evolutionary process of two kinds of crises: boundary crisis and interior crisis are shown. The boundary crisis is resulted by the collision of a chaotic attractor and a periodic saddle on the basin boundary. It is observed that there are two types of interior crises. One is caused by the collision of a chaotic attractor and a chaotic saddle within the interior of basin of attraction. The other one occurs because a period attractor collides with a chaotic saddle within the interior of basin of attraction. The saddles of system play an important role in the crisis process. The results show that this method is an efficient tool to perform the global analysis of elastic impact oscillators.     

Nonlinear Optimal Control Problems and the Method of Multiple Scales

The method of multiple scales is used to obtain an approximatesolution to the problem of the optimal control of a second-orderdifferential equation containing a small nonlinearity. Applicationsare made to the Duffing and van der Pol equations.     

Effect of fast harmonic excitation on frequency-locking in a van der Pol–Mathieu–Duffing oscillator

We study the effect of high-frequency harmonic excitation on the entrainment area of the main resonance in a van der Pol–Mathieu–Duffing oscillator. An averaging technique is used to derive a self- and parametrically driven equation governing the slow dynamic of the oscillator. The multiple scales method is then performed on the slow dynamic near the main resonance to obtain a reduced autonomous slow flow equations governing the modulation of amplitude and phase of the slow dynamic. These equations are used to determine the steady state response, bifurcation and frequency–response curves. A second multiple scales expansion is used for each of the dependent variables of the slow flow to obtain slow slow flow modulation equations. Analysis of non-trivial equilibrium of this slow slow flow provides approximation of the slow flow limit cycle corresponding to quasi-periodic motion of the slow dynamic of the original system. Results show that fast harmonic excitation can change the nonlinear characteristic spring behavior and affect significantly the entrainment region. Numerical simulations are used to confirm the analytical results.     

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.     

Duffing–van der Pol oscillator type dynamics in Murali–Lakshmanan–Chua (MLC) circuit

We have constructed a simple second-order dissipative nonautonomous circuit exhibiting ordered and chaotic behaviour. This circuit is the well known Murali–Lakshmanan–Chua(MLC) circuit but with diode based nonlinear element. For chosen circuit parameters this circuit admits familiar MLC type attractor and also Duffing–van der Pol circuit type chaotic attractors. It is interesting to note that depending upon the circuit parameters the circuit shows both period doubling route to chaos and quasiperiodic route to chaos. In our study we have constructed two-parameter bifurcation diagrams in the forcing amplitude–frequency plane, one parameter bifurcation diagrams, Lyapunov exponents, 0–1 test and phase portrait. The performance of the circuit is investigated by means of laboratory experiments, numerical integration of appropriate mathematical model and explicit analytic studies.     

Relationship between the homotopy analysis method and harmonic balance method

This paper presents a study of the relationship between the homotopy analysis method (HAM) and harmonic balance (HB) method. The HAM is employed to obtain periodic solutions of conservative oscillators and limit cycles of self-excited systems, respectively. Different from the usual procedures in the existing literature, the HAM is modified by retaining a given number of harmonics in higher-order approximations. It is proved that as long as the solution given by the modified HAM is convergent, it converges to one HB solution. The Duffing equation, the van der Pol equation and the flutter equation of a two-dimensional airfoil are taken as illustrations to validate the attained results.     

Dynamics in coupled oscillators with recurrent/random forcing, a PDE approach

The current paper is devoted to the study of coupled oscillators with recurrent/random forcing. Special attention is given to the solutions having the same recurrence/randomness as that of the forcing (recurrent/random solutions for short). By embedding coupled oscillators into coupled parabolic equations, it establishes a general theorem on the existence of recurrent/random solutions. It also finds conditions under which such solutions are unique. When the recurrent forcing is actually quasi-periodic or almost periodic, recurrent solutions are refereed to as quasi-periodic or almost periodic solutions in a weak sense and they are quasi-periodic or almost periodic in the classical sense under the uniqueness conditions. In addition, applications of the general theory to coupled Duffing type oscillators and Josephson junctions are considered and the results obtained extend several existing ones for quasi-periodic Duffing oscillators.     

A preliminary study into emergent behaviours in a lattice of interacting nonlinear resonators and oscillators

Future sensor arrays will be composed of interacting nonlinear components with complex behaviours with no known analytic solutions. This paper provides a preliminary insight into the expected behaviour through numerical and analytical analysis. Specifically, the complex behaviour of a periodically driven nonlinear Duffing resonator coupled elastically to a van der Pol oscillator is investigated as a building block in a 2D lattice of such units with local connectivity. An analytic treatment of the 2-device unit is provided through a two-time-scales approach and the stability of the complex dynamic motion is analysed. The pattern formation characteristics of a 2D lattice composed of these units coupled together through nearest neighbour interactions is analysed numerically for parameters appropriate to a physical realisation through MEMS devices. The emergent patterns of global and cluster synchronisation are investigated with respect to system parameters and lattice size.     

An analysis of phase noise and Fokker-Planck equations

A local moving orthonormal transformation has been introduced to rigorously study phase noise in stochastic differential equations (SDEs) arising from nonlinear oscillators. A general theory of phase and amplitude noise equations and its corresponding Fokker-Planck equations are derived to characterize the dynamics of phase and amplitude error. As an example, a van der Pol oscillator is considered by using the general theory.