Invariants and approximate solutions for certain non-linear oscillators by means of the field method

Certain strongly non-linear conservative oscillators are approached with the field method, which is combined with the convolution integral method. A complete set of their adiabatic invariants are derived, on the basis of which approximate solutions for motion can be obtained.     

Elliptic harmonic balance method for two degree-of-freedom self-excited oscillators

Elliptic harmonic balance (EHB) method as an analytical method is widely used for strongly non-linear oscillators with a single degree-of-freedom (DOF). The oscillation equations are expressed by elliptic functions, and then the expressions are expanded as harmonics of elliptic type while only the first harmonic is retained. To the best of our knowledge, however, it seems that the EHB method has not been found applications in two or multiple DOFs systems. One possible reason is that the EHB method may cause a difficult problem that the number of equations obtained by harmonic balancing is not equal to that of unknowns. To this end, in the present paper, the EHB method is therefore extended to study a class of strongly self-excited oscillators with two DOFs. Prior to harmonic balancing, an additional equation is established to tackle the aforementioned problem. Illustrative examples show that solutions of limit cycles obtained by the proposed method are in good agreement with the numerical solutions.     

An elliptic averaging method for harmonically excited oscillators with a purely non-linear non-negative real-power restoring force

Harmonically excited oscillators with a purely non-linear non-negative real-power restoring force are considered in this paper. The solution for motion is assumed in the form of a Jacobi cn elliptic function, the frequency and the parameter of which are obtained from the exact period of motion calculated from the energy conservation law. The parameter of the elliptic function is found to be negative for under-linear restoring forces and positive for over-linear restoring forces. By taking into account the time variation of the parameter of the elliptic function, a new elliptic averaging method is developed. The use of the equations derived is illustrated on the examples of oscillators with van der Pol damping and linear viscous damping with various integer and non-integer powers of the restoring force. New insights into dynamics of these oscillators are engendered. Numerical confirmations of analytical results are provided.     

Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators

A system of two non-linear differential equations is considered that simulates the dynamics of two completely identical weakly coupled oscillators both in the case of dissipative and active coupling. The system of normal modes is investigated. All the self-similar periodic solutions, including the asymmetric solutions describing the natural ascillations of oscillators with dissimilar amplitude's, are found analytically. The stability is investigated as well as the local bifurcations of the self-similar cycles when there is a change in stability. In particular, the possibility of the creation of two-dimensional invariant tori is pointed out. In the case of active coupling, it is shown that the basic version of the natural oscillations is a stable antiphase cycle that was observed in Huygens experiments.     

The relation between the mechanics of dissipative finite-dimensional systems with heredity and the mechanics of infinite-dimensional Hamiltonian systems

It is shown that the dynamics of a non-linear multi-dimensional oscillator interacting with a field of harmonic oscillators, continuously distributed with respect to frequency, is governed by a non-linear integro-differential equation. The investigation centres on the possibility of an inverse transition from a non-linear oscillator with heredity to be embeddable in a larger Hamiltonian system is the usual condition that the entropy production should not be negative. Existence and uniqueness theorems are proved and several a priori estimates are found for the solution. It is also proved that, subject to certain restrictions on the relaxation kernel, the solution converges to one of the critical points of the effective potential.     

Mathematical models of bipolar disorder

We use limit cycle oscillators to model bipolar II disorder, which is characterized by alternating hypomanic and depressive episodes and afflicts about 1% of the United States adult population. We consider two non-linear oscillator models of a single bipolar patient. In both frameworks, we begin with an untreated individual and examine the mathematical effects and resulting biological consequences of treatment. We also briefly consider the dynamics of interacting bipolar II individuals using weakly-coupled, weakly-damped harmonic oscillators. We discuss how the proposed models can be used as a framework for refined models that incorporate additional biological data. We conclude with a discussion of possible generalizations of our work, as there are several biologically-motivated extensions that can be readily incorporated into the series of models presented here.     

Quasiperiodicity,strange non-chaotic and chaotic attractors in a forced two degrees-of-freedom system

Strange non-chaotic, strange chaotic and quasiperiodic attractors are demonstrated to exist for a system of two non-linear coupled oscillators with almost periodic excitations. For same parameter values a transition from a strange non-chaotic to a quasiperiodic attractor is presented, whereas for other parameter values a shift from the strange chaotic attractor to a quasiperiodic one is found.     

Extended Kalman filters using explicit and derivative-free local linearizations

We propose three variants of the extended Kalman filter (EKF) especially suited for parameter estimations in mechanical oscillators under Gaussian white noises. These filters are based on three versions of explicit and derivative-free local linearizations (DLL) of the non-linear drift terms in the governing stochastic differential equations (SDE-s). Besides a basic linearization of the non-linear drift functions via one-term replacements, linearizations using replacements through explicit Euler and Newmark expansions are also attempted in order to ensure higher closeness of true solutions with the linearized ones. Thus, unlike the conventional EKF, the proposed filters do not need computing derivatives (tangent matrices) at any stage. The measurements are synthetically generated by corrupting with noise the numerical solutions of the SDE-s through implicit versions of these linearizations. In order to demonstrate the effectiveness and accuracy of the proposed methods vis-à-vis the conventional EKF, numerical illustrations are provided for a few single degree-of-freedom (DOF) oscillators and a three-DOF shear frame with constant parameters.     

Analysis of bifurcations for non-linear stochastic non-smooth vibro-impact system via top Lyapunov exponent

Bifurcations are discussed by the criterion of top Lyapunov exponent. Based on the local map and Kaminski’s algorithms, a general formulation of the top Lyapunov exponents is proposed for non-linear vibro-impact oscillators with Gaussian white noise perturbation. The analytical results are verified by phase portraits and bifurcation diagrams for a classical stochastic Duffing vibro-impact oscillator. Both results are consistent.     

A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions

Most available integration techniques for stochastically driven engineering dynamical systems are based on stochastic Taylor expansions of the response variables and thus require numerical modelling of multiple stochastic integrals (MSI-s). Since the latter is an extremely involved numerical task and becomes inaccurate for higher level MSI-s, these methods fail to achieve an accuracy beyond a limited order. Recently, the first author has proposed a locally transversal linearization (LTL) technique that completely avoids the use of Taylor-like expansions in the construction of the integration map [Proc. Roy. Soc. Ser. A, 457 (2001) 539; Int. J. Numer. Methods Eng. 61 (2004) 764]. A crucial step in the implementation of the LTL method is to arrive at a conditionally linearized solution which is then made to transversally intersect the non-linear solution manifold in the associated phase space. The present paper is the first part of an investigation consisting of two parts to considerably simplify and numerically expedite the generation of the conditionally linearized solution without affecting the local and global error orders of the original LTL method. In particular, the derivation of the conditionally linear form of the stochastic differential equations is done in such a way that the corresponding fundamental solution matrix (and consequently its inverse) remains unchanged during the entire integration process. In this part of the work, only strong (path wise) stochastic solutions are constructed. Through formal error estimates, it is verified that the present version of the LTL method has the same error orders as its older counterpart. Further, a host of numerical examples on stochastically driven non-linear oscillators are presented to illustrate its superior computational speed and ease of implementation. In a companion paper related to this work, a weak form of the alternative LTL approach will be derived.     

The optimality of the velocity-gradient method in the problem of controlling the escape from a potential well

The problem of controlling the escape of a particle from a potential well for a nonlinear system with friction is considered. The velocity-gradient method [Polushin IG, Fradkov AL, Hill, D. Passivity and passivation in non-linear systems. Avtomatika i Telemekhanika 2000;3:3–37] is proved to be optimal in the sense that if it does not guarantee escape from the well, then this is also impossible with any other control law. Nonlinear Duffing and Helmholtz oscillators with one degree of freedom and negative stiffness are considered. For each of them a curve is constructed separating the parameter plane of the problem into two parts: one where escape is feasible and one where it is not. An estimate is obtained for the inclination angle of the tangent to that curve near the origin.     

Passages through resonance in weakly nonlinear systems

A method of determining asymptotic expansions for weakly couplednonlinearly perturbed systems of harmonic oscillators with slowlyvarying frequencies is presented. In an example with two oscillators,each one experiences a separate resonance passage that producesa first-order amplitude change. Simultaneously, second-orderadjustments occur to both oscillators. The determination isachieved by carrying the calculations to third order.     

Vibration control of ultrasonic cutting via dynamic absorber

Ultrasonic machining (USM) is one of the most effective non-conventional techniques. Its application especially to hard-to-machine material (HTM) is growing rapidly. The main operation condition of USM is at resonance where an exciter derives a tuned blade or a tool. In this paper, the coupling of two non-linear oscillators of the main system and absorber representing ultrasonic cutting process are investigated. This leads to a two-degree-of-freedom Duffing’s oscillator in which such non-linear effects can be neutralized under certain dynamic conditions. The aim of this work is the control of the system behavior at principal parametric resonance condition where the system damage is probable. An approximate solution is derived up to the second order for the coupled system. A threshold value of linear damping has been obtained, where the system vibration can be reduced dramatically. The stability of the system is investigated applying both phase-plane and frequency response techniques. The effects of the different parameters of the absorber on system behavior are studied numerically. Comparison with the available published work is reported.     

Effect of viscosity on the conditions for the formation of centrifugal solitons in the translational-rotational flow of a liquid

The viscosity of the medium is taken into consideration in deriving an evolution equation describing the propagation of non-linear centrifugal waves along the free surface of a translational-rotational liquid flow. The result is the Burgers-Korteweg-de Vries (BKdV) equation, for which a steady solution is described in the form of a shock wave with soliton oscillators near the front. Estimates are presented for the effect of viscosity on the wave-front structure and the conditions of formation previously predicted by the author /1/ for centrifugal solitons, which play an important role in various atmosphere processes^**.     

Dynamics of a ring of three coupled relaxation oscillators

The dynamics of a ring of three identical relaxation oscillators is shown to exhibit a variety of periodic motions, including clockwise and counter-clockwise wave-like modes, and a synchronous mode in which all three oscillators are in phase. The model involves individual oscillators which exhibit sudden jumps, modeling the relaxation oscillations of van der Pol oscillators. Methods include (i) numerical integration, (ii) a semi-analytical method involving solving transcendental equations numerically, and (iii) perturbation methods. A variety of bifurcations of the periodic motions are identified. This work is motivated by application to the design of a decision-making machine which can sort initial conditions according to their steady state.     

Frequency dependent iteration method for forced nonlinear oscillators

A new iteration method for nonlinear vibrations has been developed by decomposing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.     

Synchronous states in time-delay coupled periodic oscillators: A stability criterion

There are several works showing that nonzero time delay between nodes in an oscillator network can be responsible for several kinds of behavior as synchronization and chaos. Here, by using the Lyapunov linearizing method, in a system of two coupled oscillators derived as a particular case of the full connected network, it is shown that the time delay parameter has two sets of values: one that destabilizes the whole system and other that implies stability. Besides, there is a set of time delay values responsible for chaotic behaviors, even in a simple coupled oscillators system.     

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.     

非线性自治振动系统同宿解的广义双曲函数摄动法

提出广义的双曲函数摄动法,用于求解强非线性自治振子的同宿解,克服一般摄动步骤中派生方程须存在显式精确同宿解的限制．以广义双曲函数作为摄动步骤的基本函数,拓展了基于双曲函数的摄动法的适用范围．对同时含2,3次和含4次强非线性项的系统进行求解分析,验证了方法的有效性和精度．     

An Ito–Taylor weak 3.0 method for stochastic dynamics of nonlinear systems

A new framework for development of order 3.0 weak Taylor scheme towards stochastic modeling and dynamics of coupled nonlinear systems is presented. The proposed method is derived by including third order multiple stochastic integral terms of Ito–Taylor expansion and developing them for a wide class of stochastic nonlinear systems. For computing the system responses of linear and a wide class of nonlinear structural systems, the use of lower order integration schemes is sufficient. But for highly non-linear stochastically driven systems like base isolated hysteretic systems and degrading stochastic systems the evaluation of higher order terms is necessary. Additionally, the use of higher order integration schemes for stochastic dynamics of higher dimensional nonlinear systems remains a challenge due to the arising mathematical complexities with the increase in the number of DOFs (degrees-of-freedom) which really necessitates the development of the proposed algorithm. The proposed algorithm is verified using a representative class of coupled nonlinear system in presence and absence of nonlinear degradation and hysteretic oscillators. The efficiency of the proposed numerical scheme over classical integration schemes is demonstrated through a practical engineering problem. Finally, an automated extension of the proposed algorithm is presented by generalizing it for a system of N-DOFs.