Is this form of the Duffing equation correct?

Question

A complex dynamical system that describes a steel beam deflected toward two magnets is the Duffing equation: \begin{aligned} \frac{\text d x}{\text d t} &= y \\ \frac{\text d y}{\text d t} &= -y + rx - x^3 \end{aligned} where $r$ is a control parameter.

I'm looking through a problem sheet and the form for the Duffing equation appears to be wrong?

I'm trying to find stationary solutions for the system and find bifurcations and so on. The control parameter $r$ should be in front of the $y$ term according to the sources I have seen? With this form I can't get a solution that seems reasonable. Could anyone tell me if this is indeed a mistake?

Answer 1

Injecting the first equation in the second one, we get $$ \ddot{x} + \dot{x} - rx + x^3 = 0 \, , $$ which is the equation of a free (unforced) Duffing oscillator. Here, the control parameter $r$ is defined as the linear part of the stiffness.

The equilibrium points of the system satisfy $x \left(x^2 - r\right) = 0$, i.e. $x=0$ or $x = \pm\sqrt{r}$. The Jacobian matrix of the system is $$ J(x) = \left[ \begin{array}{cc} 0 & 1 \\ r - 3x^2 & -1 \end{array} \right] $$ Evaluated at the origin, we have the eigenvalues $$ \text{Sp}\, J(0) = \left\lbrace -\frac{\sqrt{1+4r} + 1}{2}, \frac{\sqrt{1+4r} - 1}{2} \right\rbrace . $$ If $r > 0$, one eigenvalue is strictly positive, and the origin is unstable. Otherwise, if $r<0$, the origin is asymptotically stable (both eigenvalues have negative real parts). Evaluated at the other equilibrium values, we have for $r>0$, $$ \text{Sp}\, J(\pm\sqrt{r}) = \left\lbrace -\frac{\sqrt{1-8r} + 1}{2}, \frac{\sqrt{1-8r} - 1}{2} \right\rbrace . $$ A similar stability analysis can be performed which shows that both equilibrium points are asymptotically stable. Here is a view of the bifurcation diagram:

Answer 2

The wikipedia version is $$\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t).$$ Obviously, for your task $γ=0$. Now set (assuming $δ>0$) $$ x(t)=c\,u(δt),~ \dot x =cδ\,\dot u(δt),~\ddot{x}(t)=cδ^2\,\ddot u(δt) $$ Inserting one gets $$ δ^2\,\ddot u(δt)+δ^2\,\dot u(δt)+α\,u(δt)+c^2β\,u(δt)^3=0 $$ so that setting $s=δt$, $c=δ/\sqrtβ$, $r=-α/δ^2$ one gets the normal form $$ \ddot u(s)+\dot u(s)-r\,u(s)+u(s)^3=0 $$ which is used in the question.

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One could also change the scaling in time and space to get the new parameters have $α=\pm 1$, $β=1$, then the remaining characteristic parameter is $δ$. Set $x(t)=c\,u(\sqrt{|α|}t)$, $s=\sqrt{|α|}t$ $$ |α|\,\ddot u(s)+\sqrt{|α|}δ\,\dot u(s)+α\,u(s)+c^2β\,u(s)^3=0 $$ now set $c=\sqrt{\frac{|α|}β}$, $\kappa=\fracδ{\sqrt{|α|}}$ to get the normal form $$ \ddot u(s)+κ\,\dot u(s)+\text{sign}(α)\,u(s)+u(s)^3=0 $$ which is what you were expecting from your other source.