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Boundedness of solutions for semilinear duffing equations
Applied Mathematics and Computation, 2002The author studies the boundedness of all solutions of the equation \(x''+n^{2}x+f(x) = p(t)\). The main result is the following theorem: Suppose \(f(x) \in C^{\infty } ({\mathbb R}), p(t)\in C^{6} ({\mathbb R}/2 \pi {\mathbb Z})\). If \(\int^{2\pi }_{0} p(t)e^{-int}dt = 0\) and if \(f(x)\) satisfies the following conditions: the limits \(\lim \limits_{
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Duffing's equation and nonlinear resonance
International Journal of Mathematical Education in Science and Technology, 2003The phenomenon of nonlinear resonance (sometimes called the ‘jump phenomenon’) is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto an ...
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Solution of the Duffing Equation
1994Consider the Duffing equation with variable excitation and constant coefficients α, β, γ $$\begin{gathered} {\text{u''}} + \alpha u' + \beta u + \gamma {u^3} = \delta (t) \hfill \\ u(0) = {c_0}{\text{ u'(0) = }}{{\text{c}}_1} \hfill \\ \end{gathered} % MathType!End!2!1! $$ δ(t) will be written as a series δ(t) = Σ n=0 ∞ δntn. Let L = d2/dt2. Then
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A General Solution of the Duffing Equation
Nonlinear Dynamics, 2006zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Almost periodic solutions for nonlinear duffing equations
Acta Mathematica Sinica, 1997Consider the Duffing differential equation \[ d^2x/dt^2- x+ x^3= f(t),\tag{\(*\)} \] where \(f\) is almost periodic. By using the theory of exponential dichotomy, the author first proves that \((*)\) has a unique bounded solution provided \(|f|\leq 8/27\).
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