Nonlinear Oscillatory Systems Research Articles

A fractional analysis of Noyes–Field model for the nonlinear Belousov–Zhabotinsky reaction

Nonlinear phenomena play an essential role in various field of natural sciences and engineering. In particular, the nonlinear chemical reactions are observed in various domains, as, for instance, in biological and chemical physics. For this reason, it is important to investigate the solution to this nonlinear phenomenon. This article investigates numerical solutions to a nonlinear oscillatory system called the Belousov–Zhabotinsky with Caputo fractional-time derivative. The simplified Noyes–Field fractional model reads $$\begin{aligned} {\mathcal {D}}_t^{\mu }p= & {} \xi _1p_{xx}+\beta \delta {w}+p-p^2-\delta {pw},\quad 0<\mu \le 1,\\ {\mathcal {D}}_t^{\mu }w= & {} \xi _2w_{xx}+ \gamma {w}-\lambda {pw}, \end{aligned}$$ where $$\xi _1$$ and $$\xi _2$$ are the diffusing constants for the concentration p and w respectively, $$\gamma $$ and $$\beta $$ are given constants, $$\lambda \ne 1$$ and $$\delta $$ are positive parameters. The two iterative techniques used in this work are the fractional reduced differential transform method and q-homotopy analysis transform method. The outcomes using these two methods reveal an efficient numerical solution with high accuracy and minimal computations. Furthermore, to better understand the effect of the fractional order, we present the solution profiles which demonstrate the behavior of the obtained results.

A novel two-degree-of-freedom model of nonlinear self-excited force for coupled flutter instability of bridge decks

This paper aims to propose a novel model for predicting the nonlinear heave-torsion coupled flutter instability of flat bridge decks. The nonlinear oscillatory system during coupled post-flutter instabilities was modeled as a weak perturbation of the classical linear flutter theory. A novel nonlinear self-excited force model was then proposed by introducing extra nonlinear terms in classical linear model to consider the effects of nonlinear aerodynamic damping, amplitude-dependent vibration mode and the coupling of aerostatic deformation. An efficient algorithm for parameters identification and solving of the nonlinear coupled oscillator was also developed. The effectiveness of the proposed analytical framework was validated through an elastically-supported sectional model test in estimating the self-sustained vibrations during post-critical states of a typical closed-box bridge deck.

On approach for obtaining approximate solution to highly nonlinear oscillatory system with singularity

This paper deals with further investigations of recently introduced so-called low-frequency pendulum mechanism, which represents an extended form of classical pendulum. Exact equation of motion, which is in Eksergian’s form, is a singular and highly nonlinear second order differential equation. It is transformed by suitable choice of a new “coordinates” into classical form of nonlinear conservative oscillator containing only inertial and restoring force terms. Also, due to the singularity of coefficient of governing equation that shows hyperbolic growth, Laurent series expansion was used. Using these, we derived a nonsingular nonlinear differential equation, for which there exists an exact solution in the form of a Jacobi elliptic function. By using this exact solution, and after returning to the original coordinate, both explicit expression for approximate natural period and solution of motion of mechanism were obtained. Comparison between approximate solution and solution is obtained by numerical integration of exact equation shows noticeable agreement. Analysis of impact of mechanism parameters on period is given.

Discrete Vortices in Systems of Coupled Nonlinear Oscillators: Numerical Results for an Electric Model

Vortex coherent structures on arrays of nonlinear oscillators joined by weak links into topologically nontrivial two-dimensional discrete manifolds have been theoretically studied. A circuit of nonlinear electric oscillators coupled by relatively weak capacitances has been considered as a possible physical implementation of such objects. Numerical experiments have shown that a time-monochromatic external force applied to several oscillators leads to the formation of long-lived and nontrivially interacting vortices in the system against a quasistationary background in a wide range of parameters. The dynamics of vortices depends on the method of "coupling" of the opposite sides of a rectangular array by links, which determines the topology of the resulting manifold (torus, Klein bottle, projective plane, M\"obius strip, ring, or disk).

Application of variable- and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators

Many physical processes in nature exhibit complex dynamics that result from a combination of multiscale, nonlinear, non-local, and memory effects. Recent experimental measurements conducted in a variety of physical domains have shown that, at the macroscale level, these effects typically result in significant deviations from the behavior predicted by classical models. Notably, the underlying dynamics was often shown to be of non-integer order and possibly better captured by fractional-order models. Fractional operators are intrinsically multiscale; thus, they provide a natural approach to account for non-local and memory effects. In this study, we present the possible application of variable-order (VO) and distributed-order (DO) fractional operators to a few classes of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. More specifically, we present a methodology to define VO and DO fractional operators that are capable of capturing various physical transitions characteristic of contact dynamics, nonlinear reversible systems, hysteretic systems, and nonlinear damped oscillator systems. Despite using simplified lumped parameters models to illustrate the application of VO and DO operators to mechanics, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex systems at the continuum scale. Further, for a selected problem involving distributed nonlinear damping, we provide approximate analytical solutions that are helpful to better understand the underlying dynamics and to quantify the accuracy of our numerical models.

Improved numerical solutions of nonlinear oscillatory systems

SUMMARYWe consider numerical solutions of nonlinear oscillatory systems where closed‐form solutions do not exist. Such systems occur in buckling of columns, electrical oscillations of circuits containing inductance with an iron core, and vibration of mechanical systems with nonlinear restoring forces. We have improved the accuracy, stability, and speed of the generalized Newmark scheme of Zienkiewicz and Taylor using one‐step multiple‐value algorithms. Previously, a Laplace transform‐based method was developed to determine initial conditions of higher order derivatives of predictor algorithms together with the corrector algorithms cast in form of higher order Newton‐Raphson schemes, which are considered as consistent tangent operators that preserve quadratic rate of asymptotic convergence characteristics. For convenience, algorithms are applied to the solution of the van der Pol and Duffing's equations to show the concepts, but the procedures can also be applied to other systems such as the Bouc's, Coulomb's, and Mathieu equations. Comparisons are carried out regarding speed, accuracy, and stability, of results from the Runge‐Kutta and one‐step multiple‐value methods with remainder (truncation error) terms, previously not considered. Results compare favorably. A system of two‐degrees of freedom is also covered to illustrate how to extend the methods to deal with multiple‐degree of freedom systems.

AK-ARBIS: An improved AK-MCS based on the adaptive radial-based importance sampling for small failure probability

The pivotal problem in reliability analysis is how to use a smaller number of model evaluations to get more accurate failure probabilities. To achieve this aim, an iterative method based on the Monte Carlo simulation and the adaptive Kriging (AK) model (abbreviated as AK-MCS) has been proposed in 2011 by Echard et al. But for small failure probability, the number of the candidate points is extremely large for convergent solution. These points need to be evaluated by the current Kriging model to select the best next sample for updating the Kriging model in AK-MCS method, and the large candidate points will make the adaptive updating process of Kriging model much more time-consuming. Therefore, to improve the applicability of the AK-MCS method for small failure probability, the adaptive radial-based importance sampling (ARBIS) is employed to reduce the number of candidate points in the AK-MCS method, and an ARBIS combined with AK model method (abbreviated as AK-ARBIS) is proposed. The idea of the ARBIS is adaptively to find the optimal β-sphere, i.e., the largest sphere of the safe domain, and then samples inside the optimal β-sphere is directly recognized as safety and do not need to call the true limit state function to judge their states (safe or failed). During the adaptive process of finding the optimal β-sphere, the Kriging model is ceaselessly updated layer after layer based on the U learning scheme in each sampling pool which only contains the samples between the current spherical rings. The updating process of Kriging model stops until the optimal β-sphere is adaptively found and the convergent condition is satisfied. By finding the optimal β-sphere, the total number of candidate samples is reduced which only includes the samples outside the optimal β-sphere. Besides, the whole candidate sampling pool is partitioned into several sub-candidate sampling pool sequentially. The proposed method not only inherits the advantage of the AK-MCS but also reduces the reliability analysis time of the AK-MCS from two aspects. One is the size reduction of the candidate sampling pool, the other is the reduction of the actual limit state function evaluations because the sampling points locating inside the adaptively searched optimal β-sphere do not need to participate in the training process. By analyzing a highly nonlinear numerical case, a non-linear oscillator system, a simplified wing box structural model, an aero-engine turbine disk and a planar ten-bar structure, the effectiveness and the accuracy of the proposed AK-ARBIS method for estimating the small failure probability are verified.

THE PREDICTOR-CORRECTOR METHOD FOR MODELLING OF NONLINEAR OSCILLATORS

In the work physically reasonable algorithm of numerical modeling of nonlinear oscillatory and self-oscillatory systems are offered. The algorithm is based on discrete in time model of the linear oscillator. Nonlinearity is considered by the introduction to the oscillator of additional communications by the structural analysis of an initial system. For approximation of a temporary derivative in nonlinear communications it is offered to use the scheme of the prediction and correction. In spite of the fact that theoretically the algorithm has the second order of accuracy, within the numerical experiment with Van der Pol oscillator it shows better results, than a standard method of the second order the Heuns method.

The predictor-corrector method for modelling of self-oscillatory systems

In the work physically reasonable algorithm of numerical modeling of nonlinear oscillatory and self-oscillatory systems is offered. The algorithm is based on discrete in time model of the linear oscillator. Nonlinearity is considered by introduction to the oscillator of additional communications by the structural analysis of an initial system. For approximation of a temporary derivative in nonlinear communications it is offered to use the scheme of the prediction and correction. In spite of the fact that theoretically the algorithm has the second order of accuracy, within the numerical experiment with Van der Pol oscillator it shows the best results, than a standard method of the second order &ndash; the Heun&rsquo;s method.

Rapidly Convergent Solution of Nonlinear Oscillators with General Non-rational Restoring Force

A rapidly convergent solution for nonlinear oscillatory system with non-rational restoring force has been presented. The energy balance method has been extended and applied to solve said nonlinear oscillator. It has been verified that the second approximate solution is better than that obtained by the usual energy balance method and it is close to corresponding third approximate solution obtained by harmonic balance method.

Modified harmonic balance method for solving strongly nonlinear oscillators

In this paper, a modified harmonic balance method is applied to obtain higher-order approximate periodic solutions of strongly nonlinear oscillatory systems having a rational force. Approximate frequency and periodic solution are obtained. Frequency and solutions are analytically analyzed. Obtained results are compared with several authors and corresponding numerical results. Error analysis is carried out and found satisfactory results for the proposed method. The new technique also avoids the necessity of solving sets of equations with very complex nonlinearities numerically as in the classical harmonic balance method. Effectiveness of the method found from comparison with other articles. The method is illustrated by examples.

Modeling cochlear two-tone suppression using a system of nonlinear oscillators with feed-forward coupling

Mechanism of two-tone suppression is studied using a coupled-oscillator model of the cochlea with feed-forward coupling. Local amplification of sound signals is modeled by using Stuart-Landau oscillators near the Hopf bifurcation, and transmission of sound signals is described as feed-forward coupling between the oscillators. Effect of suppressor signals on the response to probe signals is analyzed by numerical simulations. It is found that the effect of suppression is qualitatively different depending on relative frequency between probe and suppressor signals. By analyzing a simplified two-oscillator model, we explain the mechanism of the suppression, where configuration of the oscillators plays an essential role.

Using the 0-1 Test for Chaos in Nonlinear Continuous Systems With Two Varying Parameters: Parallel Computations

Two-parameter diagrams obtained through the 0-1 test of chaos for nonlinear oscillatory continuous systems are presented in this paper. The diagrams are the results of a parallel approach to tackle enormous memory and computational time requirements due to the known oversampling problem associated with the use of the 0-1 test for chaos in continuous systems. Our rectangular diagrams with black-and-white shades of gray levels correspond to the numbers between 0 and 1 obtained as the result of the 0-1 test for chaos. A comparison between the two-parameter diagrams for the 0-1 test with the color bifurcation diagrams for oscillatory systems obtained from another method (period-n identification) is also considered. Illustrative examples are based on both the well-known Lorenz model and a model describing two equivalent electric arc circuits.

Estimation of oscillation velocities of oscillator network

The study of the collective behavior of multiscale dynamic processes is currently one of the most urgent problems of nonlinear dynamics. Such systems arise on modelling of many cyclical biological or physical processes. It is of fundamental importance for understanding the basic laws of synchronous dynamics of distributed active subsystems with oscillations, such as neural ensembles, biomechanical models of cardiac or locomotor activity, models of turbulent media, etc. Since the nonlinear oscillations that are observed in such systems have a stable limit cycle , which does not depend on the initial conditions, then a system of interconnected nonlinear oscillators is usually used as a model of multiscale processes. The equations of Lienar type are often used as the main dynamic model of each of these oscillators. In a number of practical control problems of such interconnected oscillators it is necessary to determine the oscillation velocities by known data. This problem is considered as observation problem for nonlinear dynamical system. A new method – a synthesis of invariant relations is used to design a nonlinear observer. The method allows us to represent unknowns as a function of known quantities. The scheme of the construction of invariant relations consists in the expansion of the original dynamical system by equations of some controlled subsystem (integrator). Control in the additional system is used for the synthesis of some relations that are invariant for the extended system and have the attraction property for all of its trajectories. Such relations are considered in observation problems as additional equations for unknown state vector of initial oscillators ensemble. To design the observer, first we introduce a observer for unique oscillator of Lienar type and prove its exponential convergence. This observer is then extended on several coupled Lienar type oscillators. The performance of the proposed method is investigated by numerical simulations.

Locally exact discretization method for nonlinear oscillation systems

Nonlinear oscillations are ubiquitous in various fields of engineering and applied sciences. In engineering, vibration isolators such as absorbers and dampers have been used to reduce damages of mo...

Some aspects of investigation of limit cycles of FitzHugh-Nagumo oscillator with degree memory

In this paper, oscillograms and phase trajectories are constructed using numerical simulation to study the limiting cycles of a nonlinear FitzHugh-Nagumo oscillatory system with power memory. The simulation results showed that in the absence of power memory (α=2, β=1) or the classical dynamic FitzHugh-Nagumo system, there is a single stable limit cycle, i.e. the Lienard theorem is fulfilled. In the case of viscous friction (α=2, 0< β<1), there is a family of stable limit cycles of different shapes. In other cases, the limit cycle destroyed according to two scenarios: Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of maximum Lyapunov exponents for a purpose of identifying chaotic oscillatory regimes for the considered hereditary dynamic system (HDS).

Description of the front automobile collision in a model of the system of nonlinear oscillators

The article reviewed the automobile collision process on the basis of a mathematical model of nonlinear interacting oscillators. A mathematical model for research of the front collision automobile extends to the limit of aggregates is detailed. The system was used (four masses are connected by three "springs") for mathematical modeling, where mass 1 corresponds to bumper, mass 2 corresponds to the frame with engine, mass 3, 4 to the steering mechanism with a column. Mass values were taken for the Iveco Evrocargo 75E18 car. Two types of numerical experiments were performed: at a fixed collision speed parameters changed model dependence of force on displacement; for fixed parameters of the model dependence of force on displacement changed the collision velocity. Qualitative analysis of the dynamic properties of the system is presented in the graphs of phase trajectories. It was found that the general form of the phase trajectories depends on the initial conditions, the most non-linear effects will manifest for the oscillator 1, which is shock excited. On account of nonlinearity of the return force passes "dropping" of energy, which is wasted on the deformation or destruction of the material.

Periodic solution of the cubic nonlinear Klein–Gordon equation and the stability criteria via the He-multiple-scales method

The current work demonstrated a new technique to improve the accuracy and computational efficiency of the nonlinear partial differential equation based on the homotopy perturbation method (HPM). In this proposal, two different homotopy perturbation expansions, the outer expansion and the inner one, are introduced based on two different homotopy parameters. The multiple-scale homotopy technique (He-multiple-scalas method) is applied as an outer perturbation for the nonlinear Klein–Gordon equation. A highly accurate periodic temporal solution has been derived from three orders of perturbation. The amplitude equation, which is imposed as a uniform condition, is of the fourth-order cubic–quintic nonlinear Schrodinger equation. The standard HPM with another homotopy parameter has been used as an inner perturbation to obtain a spatial solution of the nonlinear Schrodinger equation. The cubic–quintic Landau equation is obtained in the inner perturbation technique. Finally, the approximate solution is derived from the temporal and spatial solutions. Further, two different tools are used to obtain the same stability conditions. One of them is a new tool based on the HPM, by constructing the nonlinear frequency. The method adopted here is important and powerful for solving partial differential nonlinear oscillator systems arising in nonlinear science and engineering.

Application of AG method and its improvement to nonlinear damped oscillators

In this paper, a new and innovative semi analytical technique, namely Akbari-Ganji’s method (AGM), is employed for solving three nonlinear damped oscillatory systems. Applying this method to nonlinear problems is very simple because in solving process only a trial solution, the main differential equation and its derivatives are required. The analytical solutions obtained by the AGM are utilized to study the impact of amplitude on nonlinear frequency and damping ratio. It is found that the AGM leads to acceptable results for the problems considered in this paper. Also, in order to obtain a more accurate solution, instead of using a trial solution with higher-order terms which may result in complicated and time consuming mathematical calculations, the solution obtained by AGM is improved via variational iteration method (VIM). The usefulness and effectiveness of the approach is demonstrated through comparison of the obtained results with those achieved by the numerical method. Hence, the AGM can be applied to nonlinear problems consisting of significant nonlinear damping terms and, if necessary, can be easily improved.

Molecules and the Eigenstate Thermalization Hypothesis.

We review a theory that predicts the onset of thermalization in a quantum mechanical coupled non-linear oscillator system, which models the vibrational degrees of freedom of a molecule. A system of N non-linear oscillators perturbed by cubic anharmonic interactions exhibits a many-body localization (MBL) transition in the vibrational state space (VSS) of the molecule. This transition can occur at rather high energy in a sizable molecule because the density of states coupled by cubic anharmonic terms scales as N3, in marked contrast to the total density of states, which scales as exp(aN), where a is a constant. The emergence of a MBL transition in the VSS is seen by analysis of a random matrix ensemble that captures the locality of coupling in the VSS, referred to as local random matrix theory (LRMT). Upon introducing higher order anharmonicity, the location of the MBL transition of even a sizable molecule, such as an organic molecule with tens of atoms, still lies at an energy that may exceed the energy to surmount a barrier to reaction, such as a barrier to conformational change. Illustrative calculations are provided, and some recent work on the influence of thermalization on thermal conduction in molecular junctions is also discussed.