1/3 PURE SUB-HARMONIC SOLUTION AND FRACTAL CHARACTERISTIC OF TRANSIENT PROCESS FOR DUFFING'S EQUATION

The 1/3 sub-harmonic solution for the Duffing＇s with damping equation was investigated by using the methods of harmonic balance and numerical integration. The assumed solution is introduced, and the domain of sub-harmonic frequencies was found. The asymptotical stability of the subharmonic resonances and the sensitivity of the amplitude responses to the variation of damping coefficient were examined. Then, the subharmonic resonances were analyzed by using the techniques from the general fractal theory. The analysis indicates that the sensitive dimensions of the system time-field responses show sensitivity to the conditions of changed initial perturbation, changed damping coefficient or the amplitude of excitation, thus the sensitive dimension can clearly describe the characteristic of the transient process of the subharmonic resonances.

2π-periodic Solution for a Kind High Order Delay Duffing Equation

In this paper, the high order delay Duffing equation ax^（2n） ＋bx＋g（x（t-r）） = p（t） are considered,using the theory of coincidence degree, the sufficient condition for its there being at least a 2π-periodic solution is obtained.

Approximate Solutions of Primary Resonance for Forced Duffing Equation by Means of the Homotopy Analysis Method

Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approximate methods.Classical perturbation methods such as LP method,KBM method,multi-scale method and the averaging method on weakly nonlinear vibration system is effective,while the strongly nonlinear system is difficult to apply.Approximate solutions of primary resonance for forced Duffing equation is investigated by means of homotopy analysis method （HAM）.Different from other approximate computational method,the HAM is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.The HAM provides a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter and the auxiliary function.Therefore,HAM not only may solve the weakly non-linear problems but also may be suitable for the strong non-linear problem.Through the approximate solution of forced Duffing equation with cubic non-linearity,the HAM and fourth order Runge-Kutta method of numerical solution were compared,the results show that the HAM not only can solve the steady state solution,but also can calculate the unsteady state solution,and has the good computational accuracy.

Exact Solutions of the Cubic Duffing Equation by Leaf Functions under Free Vibration

Exact solutions of the cubic Duffing equation with the initial conditions are presented.These exact solutions are expressed in terms of leaf functions and trigonometric functions.The leaf function r=sleafn(t)or r=cleafn(t)satisfies the ordinary differential equation dx2/dt2=-nr2n-1.The second-order differential of the leaf function is equal to-n times the function raised to the(2n-1)power of the leaf function.By using the leaf functions,the exact solutions of the cubic Duffing equation can be derived under several conditions.These solutions are constructed using the integral functions of leaf functions sleaf2(t)and cleaf2(t)for the phase of a trigonometric function.Since the leaf function and the trigonometric function are used in combination,a highly accurate solution of the Duffing equation can be easily obtained based on the data of leaf functions.In this study,seven types of the exact solutions are derived from leaf functions;the derivation of the seven exact solutions is detailed in the paper.Finally,waves obtained by the exact solutions are graphically visualized with the numerical results.

Damped and Divergence Exact Solutions for the Duffing Equation Using Leaf Functions and Hyperbolic Leaf Functions

According to the wave power rule,the second derivative of a functionχ(t)with respect to the variable t is equal to negative n times the functionχ(t)raised to the power of 2n?1.Solving the ordinary differential equations numerically results in waves appearing in the figures.The ordinary differential equation is very simple;however,waves,including the regular amplitude and period,are drawn in the figure.In this study,the function for obtaining the wave is called the leaf function.Based on the leaf function,the exact solutions for the undamped and unforced Duffing equations are presented.In the ordinary differential equation,in the positive region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes negative.Therefore,in the case that the curves vary with the time under the conditionχ(t)>0,the gradient dχ(t)/d constantly decreases as time increases.That is,the tangential vector on the curve of the graph(with the abscissa and the ordinate χ(t)changes from the upper right direction to the lower right direction as time increases.On the other hand,in the negative region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes positive.The gradient d χ(t)/d constantly increases as time decreases.That is,the tangent vector on the curve changes from the lower right direction to the upper right direction as time increases.Since the behavior occurring in the positive region of the variable χ(t)and the behavior occurring in the negative region of the variableχ(t)alternately occur in regular intervals,waves appear by these interactions.In this paper,I present seven types of damped and divergence exact solutions by combining trigonometric functions,hyperbolic functions,hyperbolic leaf functions,leaf functions,and exponential functions.In each type,I show the derivation method and numerical examples,as well as describe the features of the waveform.

A New Method for the Exact Solution of Duffing Equation

A lot of methods, such as Jacobian elliptic function analysis, are used to look for the explicit exact solution of Duffing differential equation. The key of the analysis is to construct quotient trigonometric function, and then nonlinear algebraic equation set theory and method are used for the solution of some kinds of nonlinear Duffing differential equation. In this paper, the exact solution of Duffing equation is obtained by using constant variation method, making use of the formula to solve cubic equations and general solution of the homogeneous equation of Duffing equation with appropriate Constant m and function f(t) .

Existence and Uniqueness of 2π-Periodic Solution About Duffing Equation and Its Numerical Method

A new provement of the existence and uniqueness about periodic boundary value Duffing equation is established by using global inverse function theorem. An algorithm for solving differential equation that has a large convergence domain is given. Finally, a numerical example is given.

A STUDY OF THE STATIC AND GLOBAL BIFURCATIONS FOR DUFFING EQUATION

In this paper, the static and global bifurcations of the forced Duffing equation have been studied by means of the averaged system. Bifurcation condition has been obtained in the whole parametric space. The change of the phase plane structure has been investigated.

A SIMPLIFIED PROOF TO A THEOREM BY DING TONG-REN ON PERIODIC SOLUTIONS OF DUFFING EQUATIONS

In 1982, DING Tong-ren gave a basic theorem about existence of periodic solutions of Duffing equations with double resonance. A simplified proof will be given by making use of the Leray-Schauder principle.

Finding Periodic Solutions of High Order Duffing Equations via Homotopy Method

This paper presents a detailed analysis of finding the periodic solutions for the high order Duffing equation x^（2n） ＋ g（x） = e（t） （n ≥ 1）. Firstly, we give a constructive proof for the existence of periodic solutions via the homotopy method. Then we establish an efficient and global convergence method to find periodic solutions numerically.

Harmonic Solutions of Duffing Equation with Singularity via Time Map

This paper is devoted to the study of second-order Duffing equation with singularity at the origin, where? tends to positive infinity as , and the primitive function as . By applying the phase-plane analysis methods and Poincaré-Bohl theorem, we obtain the existence of harmonic solutions of the given equation under a kind of nonresonance condition for the time map.

Bifurcations and Chaos in the Duffing Equation with One Degenerate Saddle Point and Single External Forcing

In this paper, we study the Duffing equation with one degenerate saddle point and one external forcing and obtain the criteria of chaos of Duffing equation under periodic perturbation through Melnikov method. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit the more new complex dynamical behaviors, including homoclinic bifurcation, bifurcation diagrams, maximum Lyapunov exponents diagrams, phase portraits and Poincaré maps.

Sinusoidal Time-Dependent Power Series Solution to Modified Duffing Equation

Duffing equation,, has a standard well-known ex-act solution [1]. Approximate solutions to this equation also are available [2]. Reference [3] [4] introduces a sinusoidal time-dependent Power Series solution. Applying this method successfully we investigate the approximate solution of the modified Duffing equations,, for n = 4 and 5. Symbolic manipulative utilities of a Computer Algebra System (CAS) specifically Mathematica [5] extensively is used investigating the results.

Stability Analysis of Periodic Solutions of Some Duffing’s Equations

In this paper, some stability results were reviewed. A suitable and complete Lyapunov function for the hard spring model was constructed using the Cartwright method. This approach was compared with the existing results which confirmed a superior global stability result. Our contribution relies on its application to high damping door constructions. (2010 Mathematics Subject Classification: 34B15, 34C15, 34C25, 34K13.)

A Study of Periodic Solution of a Duffing’s Equation Using Implicit Function Theorem

In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Un-der appropriate conditions around the origin, a unique periodic solution was obtained.

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.

A 1/3 PURE SUBHARMONIC SOLUTION AND TRANSIENT PROCESS FOR THE DUFFING'S EQUATION

The 1/3 subharmonic solution for the Duffing’s equation is investigated by using the methods of harmonic balance and numerical integration. The sensitivity of parameter variation for the transient process and the transient process for the perturbance initial conditions are studied. Over and above, the precision of numerical integration method is discussed and the numerical integration method is compared with the harmonic balance method. Finally, asymptotical stability of the pure subharmonic oscillations element is inspected.

Phase Portraits and Traveling Wave Solutions of a Fractional Generalized Reaction Duffing Equation

In this paper, we study the traveling wave solutions of the fractional generalized reaction Duffing equation, which contains several nonlinear conformable time fractional wave equations. By the dynamic system method, the phase portraits of the fractional generalized reaction Duffing equation are given, and all possible exact traveling wave solutions of the equation are obtained.

IMPERFECT BIFURCATION OF SYSTEMS WITH SLOWLY VARYING PARAMETERS AND APPLICATION TO DUFFING'S EQUATION

A new method was proposed for essentially studying the imperfect bifurcation problem of nonlinear systems with a slowly varying parameter. By establishing some theorems on the solution approximated by that of the linearized system, the delayed bifurcation transition and jump phenomena of the time_dependent equation were analyzed. V_function was used to predict the bifurcation transition value. Applying the new method to analyze the Duffing's equation, some new results about bifurcation as well as that about the sensitivity of the solutions with respect to initial values and parameters are obtained.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.