Duffing equation

The parameters in the above equation are: • \delta controls the amount of damping, • \alpha controls the linear stiffness, • \beta controls the amount of non-linearity in the restoring force; if \beta=0, the Duffing equation describes a damped and driven simple harmonic oscillator, • \gamma is the amplitude of the periodic driving force; if \gamma=0 the system is without a driving force, and • \omega is the angular frequency of the periodic driving force. The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. The restoring force provided by the nonlinear spring is then \alpha x + \beta x^3. When \alpha>0 and \beta>0 the spring is called a hardening spring. Conversely, for \beta it is a softening spring (still with \alpha>0). Consequently, the adjectives hardening and softening are used with respect to the Duffing equation in general, dependent on the values of \beta (and \alpha). The number of parameters in the Duffing equation can be reduced by two through scaling (in accord with the Buckingham π theorem), e.g. the excursion x and time t can be scaled as: \tau = t \sqrt{\alpha} and y = x \alpha/\gamma, assuming \alpha is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied). Then: \ddot{y} + 2 \eta\, \dot{y} + y + \varepsilon\, y^3 = \cos(\sigma\tau), where • \eta = \frac{\delta}{2\sqrt{\alpha}}, • \varepsilon = \frac{\beta\gamma^2}{\alpha^3}, and • \sigma = \frac{\omega}{\sqrt{\alpha}}. The dots denote differentiation of y(\tau) with respect to \tau. This shows that the solutions to the forced and damped Duffing equation can be described in terms of the three parameters (\varepsilon, \eta, and \sigma) and two initial conditions (i.e. for y(t_0) and \dot{y}(t_0)). ==Methods of solution==

In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well: • Expansion in a Fourier series may provide an equation of motion to arbitrary precision. • The x^3 term, also called the Duffing term, can be approximated as small and the system treated as a perturbed simple harmonic oscillator. • The Frobenius method yields a complex but workable solution. • Any of the various numeric methods such as Euler's method and Runge–Kutta methods can be used. • The homotopy analysis method (HAM) has also been reported for obtaining approximate solutions of the Duffing equation, also for strong nonlinearity. In the special case of the undamped (\delta = 0) and undriven (\gamma = 0) Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions. ==Boundedness of the solution for the unforced oscillator==

Undamped oscillator Multiplication of the undamped and unforced Duffing equation, \gamma = \delta = 0, with \dot{x} gives: \begin{align} & \dot{x} \left( \ddot{x} + \alpha x + \beta x^3 \right) = 0 \\[1ex] \Longrightarrow {} & \frac{\mathrm{d}}{\mathrm{d}t} \left[ \frac 1 2 \left( \dot{x} \right)^2 + \frac 1 2 \alpha x^2 + \frac 1 4 \beta x^4 \right] = 0 \\[1ex] \Longrightarrow {} & \frac 1 2 \left( \dot{x} \right)^2 + \frac 1 2 \alpha x^2 + \frac 1 4 \beta x^4 = H, \end{align} with a constant. The value of is determined by the initial conditions x(0) and \dot{x}(0). The substitution y=\dot{x} in H shows that the system is Hamiltonian: \begin{align} &\dot{x} = + \frac{\partial H}{\partial y}, \qquad \dot{y} = - \frac{\partial H}{\partial x} \\[1ex] \Longrightarrow {} & H = \tfrac 1 2 y^2 + \tfrac 1 2 \alpha x^2 + \tfrac 1 4 \beta x^4. \end{align} When both \alpha and \beta are positive, the solution is bounded: \begin{align} & \dot{x} \left( \ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 \right) = 0 \\[1ex] \Longrightarrow{}& \frac{\mathrm{d}}{\mathrm{d}t} \left[ \frac 1 2 \left( \dot{x} \right)^2 + \frac 1 2 \alpha x^2 + \frac 1 4 \beta x^4 \right] = -\delta\, \left(\dot{x}\right)^2 \\[1ex] \Longrightarrow{}& \frac{\mathrm{d}H}{\mathrm{d}t} = -\delta\, \left(\dot{x}\right)^2 \le 0, \end{align} since \delta \ge 0 for damping. Without forcing the damped Duffing oscillator will end up at (one of) its stable equilibrium point(s). The equilibrium points, stable and unstable, are at \alpha x + \beta x^3 = 0. If \alpha>0 the stable equilibrium is at x=0. If \alpha and \beta > 0 the stable equilibria are at x = +\sqrt{-\alpha/\beta} and x = -\sqrt{-\alpha/\beta}. ==Frequency response==

The forced Duffing oscillator with cubic nonlinearity is described by the following ordinary differential equation: \ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t). The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. For a linear oscillator with \beta=0, the frequency response is also linear. However, for a nonzero cubic coefficient \beta, the frequency response becomes nonlinear. Depending on the type of nonlinearity, the Duffing oscillator can show hardening, softening or mixed hardening–softening frequency response. Anyway, using the homotopy analysis method or harmonic balance, one can derive a frequency response equation in the following form: ==Examples==

Some typical examples of the time series and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are shown in the figures below. The forcing amplitude increases from \gamma = 0.20 to The other parameters have the values: \delta = 0.3 and The initial conditions are x(0) = 1 and \dot{x}(0) = 0. The red dots in the phase portraits are at times t which are an integer multiple of the period ==References==