Duffing-type Equations Research Articles

On a periodic problem for super-linear second-order ODEs

Abstract The present paper concerns the periodic problem u ″ = p ( t ) u − q ( t , u ) u + f ( t ) ; u ( 0 ) = u ( ω ) , u ′ ( 0 ) = u ′ ( ω ) , $$\begin{array}{} \displaystyle u''=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\, u'(0)=u'(\omega), \end{array}$$ where p, f : [0, ω] → ℝ are Lebesgue integrable functions and q : [0, ω] × ℝ → ℝ is a Carathéodory function. We assume that the anti-maximum principle holds for the corresponding linear problem and provide sufficient conditions guaranteeing the existence and uniqueness of a positive solution to the given non-linear problem. The general results obtained are applied to the non-autonomous Duffing type equation with a super-linear power non-linearity.

Nonlinear Dynamic Stability Analysis of Ground Effect Vehicles in Waves Using Poincaré–Lindstedt Perturbation Method

In this study, we present an analytical tool that can be used to predict the nonlinear dynamic response of ground effect vehicles (GEVs) advancing through sinusoidal head-sea waves. GEVs exhibit a unique instability phenomenon known as porpoising, which is an oscillatory motion along the heave and pitch axes that can cause serious structural damage. The heaving and pitching equations of motion are presented in the form of coupled, forced, and nonlinear Duffing-type equations with cubic nonlinearity. The analytical model developed in this study leverages the Poincaré–Lindstedt perturbation method to express the amplitude and frequency of motion in terms of all physical parameters. The accuracy and reliability of the proposed model were validated through computational fluid dynamics (CFD) simulations based on incompressible unsteady Reynolds-averaged Navier–Stokes (RANS) equations. The results show a strong agreement between the analytical tool and the CFD simulations, with minor discrepancies due to assumptions inherent in the simulations, particularly the assumption that seawater only passes beneath the hull, resulting in increased buoyancy forces and reduced damping. This study offers a novel and practical method for predicting the dynamic stability of GEVs under realistic sea conditions, potentially enhancing safety and operational efficiency by mitigating the risks associated with porpoising.

Lie algebra classification, conservation laws and invariant solutions for the kind generalization of the Duffing-type equation

This paper makes significant contributions to the study of a generalized form of the Duffing-type equation. We derive the generating operators of the optimal system associated with this equation, enabling us to characterize an implicit solution. Additionally, we present a complete classification of group symmetries and obtain the Lagrangian for the equation. Our results include the classification of the Lie algebra and the optimal system, providing a thorough understanding of the equation’s underlying structure. These contributions serve to enhance the current body of knowledge on the Duffing-type equation and provide useful insights for future research in this area.

Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients

Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients

On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives

In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes.

Nonlinear vibrations of doubly curved micropanels reinforced by graphene nanoplates in length direction under non-uniform thermal loading

Structures made of composites are quite common in different industries entailing automotive, marine, and aircraft industries, thanks to their favorable lightweight and strength. This study investigated the nonlinear vibrations of the various shapes of size-dependent graphene nanoplatelets reinforced micropanels under non-uniform thermal loading. In diverse ways in the axial direction, the material characteristics of graphene nanoplates (GPLs) alter. Taylor’s series expansion terms in curvature coordinates are used to illustrate the displacement field. Nonlinear von Karman theory and Hamiltonian principle work together to derive the nonlinear form of the pertinent equations and boundary conditions. Strain gradient theory with consideration for inherent length scale characteristics is investigated to account for size effects. By discretizing the spatial domain equations, deriving Duffing-type equations, and using Kronecker and Hadamard products, the governing equations and numerous nonlinear boundary edges are resolved. Using the techniques and answers outlined above, the results section is produced. There are four sections for this research. The findings are cross-checked with those from other studies that have been published in the first section. The effects of geometric and physical factors are shown in Parts 2, 3, and 4 on the nonlinear to linear frequency curve, the nonlinear frequency curve, and the sensitivity of the nonlinear frequency-to-size effect (SNFSE).

Nonlinear continuum thermo-mechanics of the composite thick panel used in concrete ceilings via general expression of the polynomial series theory

Proper concrete vibration is vital to the final quality and durability of concrete structures. There is a lack of objective methods to assess conformance to the requirements of vibration behavior of concrete systems used in various engineering structures. So, in the current work for the first time, nonlinear vibrations of a composite thick panel made of concrete materials are presented. In the current work, the GPLs are used to reinforce the thick concrete panel in the length direction. In the current work, according to the higher-order shear deformation shell theory the effects of the von Kármán strain-displacement kinematic nonlinearity are presented in the constitutive laws of the shell. The nonlinear governing equations for various nonlinear boundary edges are solved via discretization of equations on the space domain, derivation of Duffing-type equations, and Hadamard and Kronecker Products. The results are validated by comparing the current results with deep neural networks (DNN) and open-source results in the literature. For DNN, it is introduced a supervised neural network based on physical information to predict the vibrational behavior of the current system. In this context, data-driven solutions and data-driven discovery are presented to solve the problem of determining nonlinear frequency. Consequently, new results are investigated to show the effects of the Pasternak modulus, Winkler modulus, and the GPLs’ weight fraction on the nonlinear vibrations of concrete panels.

Effect of thermal pre/post-buckling regimes on vibration and instability of graphene-reinforced composite beams

The aim of this research is to analyze the natural frequencies and instability of laminated nanocomposite beams in thermally pre/post-buckled configurations. The thick beam consists of ten composite layers, each reinforced with graphene platelets (GPLs). It is also assumed that the beam is in contact with a generalized linear/nonlinear elastic medium. Different patterns are used to simulate selected distribution profiles of material properties through the beam thickness. Reddy’s third-order shear deformation theory within the framework of the von-Kármán assumption is implemented to derive the kinematic relation. Hamilton’s principle is used to derive the equations of motion for the uniformly heated graphene-reinforced composite beam. In the absence of an external pressure load, the Duffing-type equation is obtained and solved for the nonlinear vibration problem of the beam. The analytical solutions for the vibration and stability problems of the thermally buckled beam are obtained using the perturbation-based method. This study presents new insights into the vibration and stability problems of laminated nanocomposite beams. The influence of important parameters of the nanocomposite beam such as the weight fraction of nanofillers, the distribution pattern of GPLs, slenderness ratio, temperature variation and foundation components on the nonlinear behavior of the beam is numerically presented and discussed.

Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient

We consider the equation $u_{xx}^{}-u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $|u(x)|<\xi$, can be put in one-to-one correspondence with bi-infinite sequences of numbers $n\in \{-N,\,\ldots,\,N\}$ (called ``codes'' of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a ``great part'' of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

We do not know Van der Pol-type equations with nonlinear restoring force having explicitly an exact periodic solution. We present, for the first time, nonpolynomial Van der Pol oscillator equations that do not satisfy the classical existence theorems. We exhibit their exact harmonic and isochronous solutions and prove the existence of limit cycles by using averaging theory. We also present first integrals and exact solutions of polynomial Van der Pol-Duffing equations to show that they do not have any limit cycle. Additionally, we prove that the damped Duffing-type equations are equivalent to the conservative Duffing equations exhibiting nonoscillatory solutions.

A topological approach to the problem of chaotic tides

We study the existence of subharmonic solutions and solutions with complicated dynamics in a Duffing-type equation which arises from the study of forced tides. This is a model which involves an almost enclosed basin, connected to the sea by a narrow strait. Such a basin turns out to be a nonlinear Helmholtz resonator. The proofs are based on developments of the theory of topological horseshoes in connection with some geometric features associated to linked twist maps.

Nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermal loading based on a novel HSDT

This paper studies the nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermal loading based on a novel higher-order shear deformation theory (HSDT). Nonlocal elasticity theory is applied to consider the small-scale effect. The nonlinear equations of motion are derived using Hamilton’s principle. First, a Galerkin-based numerical method is utilized to decrease the nonlinear governing equation into a set of Duffing-type time-dependent differential equations. Then, the analytical solutions are obtained based on the method of multiple scales (MMS) and perturbation technique. All of the mechanical properties of the beam are temperature-dependent. The impacts of the several variables are examined on the nonlinear frequency ratio of the nanobeams. The results illustrate that when the maximum deflection is smaller/greater than 0.2, its impact on the nonlinear frequency ratio will decrease/increase.

Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators

Some novel exact solutions and approximations to the damped Duffing–Mathieu‐type oscillator with cubic nonlinearity are obtained. This work is divided into two parts: in the first part, some exact solutions to both damped and undamped Mathieu oscillators are obtained. These solutions are expressed in terms of the Mathieu functions of the first kind. In the second part, the equation of motion to the damped Duffing–Mathieu equation (dDME) is solved using some effective and highly accurate approaches. In the first approach, the nonintegrable dDME with cubic nonlinearity is reduced to the integrable dDME with linear term having undermined optimal parameter (maybe called reduced method). Using a suitable technique, we can determine the value of the optimal parameter and then an analytical approximation is obtained in terms of the Mathieu functions. In the second approach, the ansatz method is employed for deriving an analytical approximation in terms of trigonometric functions. In the third approach, the homotopy perturbation technique with the extended Krylov–Bogoliubov–Mitropolskii (HKBM) method is applied to find an analytical approximation to the dDME. Furthermore, the dDME is solved numerically using the Runge–Kutta (RK) numerical method. The comparison between the analytical and numerical approximations is carried out. All obtained approximations can help a large number of researchers interested in studying the nonlinear oscillations and waves in plasma physics and many other fields because many evolution equations related to the nonlinear waves and oscillations in a plasma can be reduced to the family of Mathieu‐type equation, Duffing‐type equation, etc.

A power-form method for dynamic systems: investigating the steady-state response of strongly nonlinear oscillators by their equivalent Duffing-type equation

A power-form method for dynamic systems: investigating the steady-state response of strongly nonlinear oscillators by their equivalent Duffing-type equation

Duffing-type Equations: Singular Points of Amplitude Profiles and Bifurcations

We study the Duffing equation and its generalizations with polynomial nonlinearities. Recently, we have demonstrated that metamorphoses of the amplitude response curves, computed by asymptotic methods in implicit form as $F\left( \Omega ,\ A\right) =0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F\left(\Omega ,\ A\right) =0$. In the present work we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of form $\Omega _{\pm }=f_{\pm }\left( A\right) $, obtained within other asymptotic approaches.

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

In this paper we prove the existence of multiple periodic (harmonic and subharmonic) solutions for a class of planar Hamiltonian systems which includes the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincare-Birkhoff fixed point theorem as well as some refinements on the side of the theory of topological horseshoes. A Duffing-type equation and an exponential nonlinearity case are studied as applications.

Chaos in a Periodically Perturbed Second-Order Equation with Signum Nonlinearity

We study the periodically perturbed Duffing-type equation [Formula: see text] and its damped counterpart [Formula: see text] The main feature of our model is the presence of a “signum term” in [Formula: see text] We prove the existence of infinitely many subharmonic solutions as well as the presence of chaotic dynamics for some [Formula: see text]-periodic forcing terms.

Nonlinear boundary value problems for nondegenerate differential-algebraic systems

The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

Lyapunov Equivalent Representation Form of Forced, Damped, Nonlinear, Two Degree-of-Freedom Systems

The aim of this paper focuses on finding equivalent representation forms of forced, damped, two degree-of-freedom, nonlinear systems in the sense of Lyapunov by using a nonlinear transformation approach that provides decoupled, forced, damped, nonlinear equations of the Duffing type, under the assumption that the driving frequency and the external forces are equal in both systems. The values of Lyapunov characteristic exponents (LCEs), Lyapunov largest exponents (LLE), and time-amplitude and frequency-amplitude curves computed from numerical integration solutions, indicate that the decoupled Duffing-type equations are equivalent, in the sense of Lyapunov, to the original dynamic system, since both set of motion equations tend to have the same qualitative and quantitative behaviors.

Nonlinear primary resonance of micro/nano-beams made of nanoporous biomaterials incorporating nonlocality and strain gradient size dependency

Abstract A wide range of biological applications such as drug delivery, biosensors and hemodialysis can be provided by nanoporous biomaterials due to their uniform pore size as well as considerable pore density. In the current study, the size dependency in the nonlinear primary resonance of micro/nano-beams made of nanoporous biomaterials is anticipated. To accomplish this end, a refined truncated cube is introduced to model the lattice structure of nanoporous biomaterial. Accordingly, analytical expressions for the mechanical properties of material are derived as functions of pore size. After that, based upon a nonlocal strain gradient beam model, the size-dependent nonlinear Duffing type equation of motion is constructed. The Galerkin technique together with the multiple time-scales method is employed to obtain the nonlocal strain gradient frequency-response and amplitude-response related to the nonlinear primary resonance of a micro/nano-beam made of the nanoporous biomaterial with different pore sizes. It is indicated that the nonlocality causes to decrease the response amplitudes associated with the both bifurcation points of the jump phenomenon, while the strain gradient size dependency causes to increase them. Also, it is found that increasing the pore size leads to enhance the nonlinearity, so the maximum deflection of response occurs at higher excitation frequency.