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The purpose of this paper is to detect the periodic signal under strong noise background, and estimate its amplitude/phase.

Melnikov method is adopted as calculating the threshold value when chaos occurs, and the detected signal is taken as a system parameter. The system's output state is changed if the parameter has a slight change near the threshold. Meantime, the phase of system's output is recognized to judge whether the output state changes, and the signal parameter is estimated according to the necessary condition.

A small periodic signal in noise can be detected by Duffing oscillator via a transition from chaotic motion to periodic motion.

The paper shows how to calculate the amplitude/phase in low signal‐to‐noise ratios.

The Duffing system is sensitive to the weak periodic signal and has definite immunity to noise, so it is easy to construct a system composed of many oscillators that could process complex signals, even though the environmental noise is intense.

This paper presents a nonlinear method for detecting and extracting the weak signal.

An image contrast enhancement is one of the most important low‐level image pre‐processing tasks required by the vision‐based advanced driver assistance systems (ADAS). This paper seeks to address this important issue keeping the real time constraints in focus, which is especially vital for the ADAS.

The approach is based on a paradigm of nonlinear‐coupled oscillators in image processing. Each layer of the colored images is treated as an independent grayscale image and is processed separately by the paradigm. The pixels with the lowest and the highest gray levels are chosen and their difference is enhanced to span all the gray levels in an image over the entire gray level range, i.e. [0 1]. This operation enhances the contrast in each layer and the enhanced layers are finally combined to produce a color image of a much improved quality.

The approach performs robust contrast enhancement as compared to other approaches available in the relevant literature. Generally, other approaches do need a new setting of parameters for every new image to perform its task, i.e. contrast enhancement. These approaches are not useful for real‐time applications such as ADAS. Whereas, the proposed approach presented in this paper performs contrast enhancement for different images under the same setting of parameters, hence giving rise to the robustness in the system. The unique setting of parameters is derived through a bifurcation analysis explained in the paper.

The proposed approach is novel in different aspects. First, the proposed paradigm comprises of coupled differential equations, and therefore, offers a continuous model as opposed to other approaches in the relevant literature. This continuity in the model is an inherent feature of the proposed approach, which could be useful in realizing real‐time image processing with an analog implemented circuit of the approach. Furthermore, a novel framework combining coupled oscillatory paradigm and cellular neural network is also possible to achieve ultra‐fast solution in image contrast enhancement.

It is often observed in practice that the essential behavior of mathematical models involving many variables can be captured by a much smaller model involving only a few variables. Further, the simpler model very often displays oscillatory behavior of some sort, especially when critical problem parameters are varied in certain ranges. This paper attempts to supply arguments from the theory of dynamical systems for why oscillatory behavior is so frequently observed and to show how such behavior emerges as a natural consequence of focusing attention upon so‐called “essential” variables in the process of model simplification. The relationship of model simplification and oscillatory behavior is shown to be inextricably intertwined with the problems of bifurcation and catastrophe in that the oscillations emerge when critical system parameters, i.e. those retained in the simple model, pass through critical regions. The importance of the simplification, oscillation and bifurcation pattern is demonstrated here by consideration of several examples from the environmental, economic and urban areas.

The purpose of this paper is to present an analytical approximate technique to solve nonlinear conservative oscillator based on the He’s energy balance method (improved version recently presented by Khan and Mirzabeigy). The method is illustrated by solving double well Duffing oscillator.

The Duffing equation with a double-well potential (with a negative linear stiffness) is an important model of a mass particle moving in a symmetric double well potential. This form of the equation also appears in the transverse vibrations of a beam when the transverse and longitudinal deflections are coupled (Thompsen, 2003).

The approximate solutions obtained by the present technique have good agreement with the numerical solution and also provide better results than other existing methods.

The results are more accurate than those obtained by other existing methods. The relative errors obtained by the present paper are less than those obtained by other existing methods. Therefore, the present technique is very effective and convenient for solving nonlinear conservative oscillator.

The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.

The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.

Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.

The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

The purpose of this paper is to propose an iterative Legendre technique to deal with a continuous optimal control problem (OCP).

For the system in the considered problem, the control variable is a function of the state variables and their derivatives. State variables in the problem are approximated by Legendre expansions as functions of time t. A constant matrix is given to express the derivatives of state variables. Therefore, control variables can be described as functions of time t. After that, the OCP is converted to an unconstrained optimization problem whose decision variables are the unknown coefficients in the Legendre expansions.

The convergence of the proposed algorithm is proved. Experimental results, which contain the controlled Duffing oscillator problem demonstrate that the proposed technique is faster than existing methods.

Experimental results, which contained the controlled Duffing oscillator problem demonstrate that the proposed technique can be faster while securing exactness.

The purpose of this paper is to propose a new algorithm for detection of chaos in oscillatory circuits. The algorithm is based on the wavelet transform.

The proposed detection is developed by using a specific measure obtained by averaging wavelet coefficients. This measure exhibits various values for chaotic and periodic states.

The proposed algorithm is applied to signals from autonomous systems such as the Chua’s oscillatory circuit, the Lorenz chaotic system and non-autonomous systems such as the Duffing oscillator. In addition, the detection is applied to sequences obtained from the logistic map. The results are compared to those obtained with a detrended fluctuation analysis and a time-frequency signal analysis based on detectors of chaotic states.

In this paper, a new algorithm is proposed for the detection of chaos from a single time series. The proposed technique is robust to the noise influence, having smaller calculation complexity with respect to the state-of-the-art techniques. It is suitable for real-time detection with delay that is about half of the window width.

This study aims to investigate the dynamic motion of an ultrasonic machining system comprising two Duffing oscillators, each with a single degree of freedom. After derivation of the differential equations of the system using the Lagrange equations and dimensionless time, numerical analysis was used to observe changes in the system caused by differences in excitation frequency.

To suppress this effect and improve performance, proportional differential (PD) control was used. The integral absolute error was used as the fitness function, and particle swarm optimization was used to find the best value for the gain constant of the PD controller.

The results showed that with specific changes of excitation frequency, the dynamic motion of the system became nonlinear and chaotic behavior resulted. This made the system unstable and affected performance.

A range of methods, including fuzzy control, was used to analyze the results, and exhaustive laboratory work was carried out. Means of control were found that were effective in suppressing the chaotic behavior, and differences in response to control were investigated and verified. The findings of this study can be used as a basis for system parameter settings or control circuit design.

The variational principle views a complex problem in an energy way, it gives good physical understanding of an iteration method, and the variational-based numerical methods always have a conservation scheme with a fast convergent rate. The purpose of this paper is to establish a variational principle for a fractal nano/microelectromechanical (N/MEMS) system.

This paper begins with an approximate variational principle in literature for the studied problem, and a genuine variational principle is obtained by the semi-inverse method.

The semi-inverse method is a good mathematical tool to the search for a genuine fractal variational formulation for the N/MEMS system.

The established variational principle can be used for both analytical and numerical analyses of the N/MEMS systems, and it can be extended to some more complex cases.

The variational principle can be used for variational-based finite element methods and energy-based analytical methods.

The new and genuine variational principle is obtained. This paper discovers the missing piece of the puzzle for the establishment of a variational principle from governing equations for a complex problem by the semi-inverse method. The new variational theory opens a new direction in fractal MEMS systems.

An iteration technique has been developed based on the Mickens iteration method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, coupled mass–spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced.

In this paper, the truncated terms of the Fourier series have been used and utilized in every step of the iteration.

The obtained results are valid for whole ranges of vibration amplitude of the oscillations. The approximated results are compared with existing and corresponding numerical (considered to be exact) results which show excellent agreement. The error analysis has been carried out and shown acceptable results for the proposed iteration technique.

Effectiveness of the proposed iteration technique is found in comparison with other existing methods. The method is demonstrated by examples.