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In this paper, we first develop a systematic procedure of state feedback control, based on a Lur'e-type system, to analyze the synchronization of two chaotic systems in the presence of random white noise. With the aid of the modified independent component analysis (ICA), the real chaotic signal can be extracted from a noisy source where the chaotic signal has been contaminated by random white noise. Hence, a new scheme has been proposed in this paper to combine the modified ICA design and the state feedback control method for achieving chaos synchronization. The synchronization time can be arbitrarily designed to guarantee stability, even if the system's output is corrupted by measuring noise. A Duffing system example is provided to show the effectiveness of the proposed scheme. The new scheme is first used for control systems with measurement noise which can replace the conventional Kalman filter.

This paper implements He's max–min method to solve accurately strong nonlinear oscillators. Maximal and minimal solution thresholds of a nonlinear problem can be easily found, and an approximate solution of the nonlinear equation can be easily deduced using He Chengtian's interpolation, which has a millennia history. Some typical examples are employed to illustrate its validity, effectiveness, and flexibility.

The nonlinear capacitor that obeys of a cubic polynomial voltage–charge relation (usually a power series in charge) is introduced. The quantum theory for a mesoscopic electric circuit with charge discreteness is investigated, and the Hamiltonian of a quantum mesoscopic electrical circuit comprised by a linear inductor, a linear resistor and a nonlinear capacitor under the influence of a time-dependent external source is expressed. Using the numerical solution approaches, a good analytic approximate solution for the quantum cubic Duffing equation is found. Based on this, the persistent current is obtained antically. The energy spectrum of such nonlinear electrical circuit has been found. The dependency of the persistent current and spectral property equations to linear and nonlinear parameters is discussed by the numerical simulations method, and the quantum dynamical behavior of these parameters is studied.

In this paper, the variational iterative method (VIM) with the Laplace transform is utilized to solve the nonlinear problems of a simple pendulum and mass spring oscillator, which corresponds to the Duffing equation. Finding the Lagrange multiplier (LM) is a significant phase in the VIM, and variational theory is frequently employed for this purpose. This paper demonstrates how the Laplace transform can be utilized to locate the LM in a more efficient manner. The frequency obtained by Laplace-based VIM is the same as that defined in the already existing methods in the literature in order to ensure the clarity of the results. Numerous analytical techniques can be used to solve the Duffing equation, but we are the first to do it using a Laplace-based VIM and a distinctive LM. The fundamental results of my paper are that LM is also the same in the Elzaki transformation. In the vast majority of instances, Laplace-based VIM only requires one iteration to arrive at an answer with high precision and linearization, discretization or intensive computational work is required for this purpose. Comparing analytical results of VIM by Laplace transform to the built-in Simulink command in MATLAB which gives us the surety about the method’s applicability for solving nonlinear problems. Future work on the basic pendulum may examine the effects of nonlinearities and damping on its motion and the application of advanced control mechanisms to regulate its behavior. Future research on mass spring oscillators could examine the system’s response to random or harmonic input. The mass spring oscillator could also be used in vibration isolation to minimize vibrations from one building to another.

Duffing-type electrical oscillator is a second-order nonlinear electric circuit driven by a sinusoidal voltage source. The nonlinear element is a nonlinear inductor. We have studied the dynamics of two resistively coupled oscillators of this type in two cases. The first, when the oscillators are identical having chaotic dynamics, and the second, when the oscillators are in different dynamic states (periodic and chaotic, respectively). In the first case, chaotic synchronization is observed, while in the second case control of the chaotic behavior is achieved.

This paper is Part II of an earlier paper dealing with the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, and an external periodical excitation of period τ = 2π/ω (amplitude E). In the absence of this excitation, this equation of Duffing type does not give rise to self-oscillations. Part I was essentially devoted to analyze the harmonics behavior of period τ solutions, more precisely the behavior of rank-p harmonics according to the points of the parameter plane (ω,E). The present Part II deals with period kτ solutions related to a cascade of closed fold bifurcation curves related to fractional harmonics p/k, k = 3, p = 3,4,…. With respect to the organization of bifurcation curves associated with rank-p harmonics of the basic period τ, this study shows that the situation is a lot more complex for the sequence of bifurcation curves related to rank-p/3 harmonics.

It is shown that, for analytic functions f, systems of the form and cannot produce chaos; and that systems of the form and are conservative. Eight simple chaotic systems of the form with quadratic and cubic polynomial f(z, z*) are given. Lyapunov spectra are calculated, and the systems' phase space trajectories are displayed. For each system, a Hamiltonian is given, if one exists.

The dynamic behavior of bidirectionally coupled Duffing-type circuits is numerically and experimentally investigated. Two related phenomena are explored such as anti-phase synchronization and inverse π-lag synchronization. Anti-phase synchronization is observed for the first time in coupled Duffing-type systems, while the inverse π-lag synchronization phenomenon in mutually coupled identical oscillators is a new type of synchronization. Inverse π-lag synchronization can be characterized by the vanishing of the sum of two relevant periodic signals. The first one is the signal of the first circuit x₁ and the second is the signal of the second circuit x₂ with a time lag τ_min. This time lag is equal to T/2, where T is the period of x₁ and x₂. The measurements and numerical computations suggest that these two synchronization phenomena are correlated with the symmetry under the transformation, x → -x, y → -y and t → t + T/2, which the Duffing equation exhibits.

The novel coalescence of the secondary responses for the coupled Duffing equations are observed in this study. Two secondary responses that do not bifurcate from the primary responses merge into one due to saddle-node bifurcation generation within a specific parameter range. The frequency responses of the coupled Duffing equations are calculated using the harmonic balance method while the periodic orbits are detected by the shooting method. The stability of the periodic orbits is determined on utilizing Floquet theory. The parametric continuation algorithm is used to obtain the bifurcation points and bifurcation lines for a Duffing system with two varying parameters. The analytical results demonstrate the novel phenomenon that occurs in the Duffing equations.

By using the reduction technique to impulsive differential equations [Akhmet & Turan, 2006], we rigorously prove the presence of chaos in dynamic equations on time scales (DETS). The results of the present study are based on the Li–Yorke definition of chaos. This is the first time in the literature that chaos is obtained for DETS. An illustrative example is presented by means of a Duffing equation on a time scale.

The simplest frequency formulation for nonlinear oscillators is introduced and proved, and a modification is suggested. A fractal vibration in a porous medium is introduced, and its low-frequency property is elucidated by the frequency formulation. It reveals that the inertia force in a fractal space is equivalent to a couple of the inertia force and the damping force for the traditional differential model.

We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant. We study the stability of the trivial solution of this Hill equation and we show that a criterion due to Burdina [Boundedness of solutions of a system of differential equations, Dokl. Akad. Nauk. SSSR92 (1953) 603–606] is very helpful for this analysis. In some cases, we are also able to determine exact solutions in terms of Jacobi elliptic functions. Overall, we obtain a fairly complete picture of the stability and instability regions. These results are then used to study the stability of nonlinear modes in some beam equations.

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar$é$ method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

In this paper, a highly accurate analytical approximation solution method to a class of mixed-parity Duffing equation is proposed. The system may be attributed to the free vibration of laminated anisotropic plates or the simplified problem of particle vibration. The system is first analyzed qualitatively and subsequently, the analytic approximate periodic solution within the parametric range is constructed. The first approximate solution with a suitable initial condition can be obtained by using the proposed method. Subsequently, higher precision approximate solutions are constructed by combining Newton’s method and harmonic balance method. These analytical solutions have excellent approximation accuracy as verified by numerical solutions derived from exact analytical expressions.

In this paper, the Adomian Decomposition Method (ADM) is used to study the Duffing equation. The series solution is constructed and compared with the solution obtained by the perturbation method.

In this article, an approximate solution for Duffing equations with cubic and quantic nonlinearities is obtained using the Differential Transform Method and Pade approximation technique. The concept of Differential Transform Method is briefly introduced and applied it to given problem to derive solution of nonlinear equation. The major concern of this paper is successfully use of Adomian polynomial to assess the nonlinearities. The results are compared with the numerical solution by fourth-order Runge–Kutta method and found with good agreement. Results are shown by graphs for whole range of time domains accurately.