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I'm trying to describe all the bifurcations in the two parameter Duffing oscillator:

$$\ddot{x} + ax + bx^3 = 0$$

In phase space with $y = \dot{x}$ I've found the origin to be a centre for $a>0$ and a saddle for $a<0$ and there are two other fixed points $x_1 = (\frac{-a}{b})^{1/2}$ and $x_2 = -(\frac{-a}{b})^{1/2}$ - these exist only when $a,b$ have opposite sign.

I can see that holding $b$ fixed and varying $a$ we have a sort of "pitchfork" bifurcation as the two saddles coalesce at the origin. However can we call this a pitchfork bifurcation given that the stability along the x axis is always marginal and there is no change of stability?

Secondly, I'm not sure if there is a bifurcation if $a$ and $b$ cross the origin together. How can a saddle turn into a centre?

Thank you for any help!

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1 Answer 1

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I am going to work out the analysis here in the answer, and see if I am able to make sense of your question in the process.

We are given the two parameter oscillator:

$$ \ddot{x} + ax + bx^3 = 0$$

We can rewrite this as follows:

$$ \begin{align} \dot{x} &= y \\ \dot{y} & = -ax - b{x^3} \end{align} $$

I might refer to $\mathbf{f}(x, y) = (\dot{x}, \dot{y})$ later.

Fixed points occur when nullclines intersect. That is:

$$ \begin{align} &\quad\; \dot{x} = 0 \wedge \dot{y} = 0 \\ &\equiv y = 0 \wedge 0 = -ax -bx^3 \\ &\equiv y = 0 \wedge 0 = x(-a - bx^2) \\ &\equiv y = 0 \wedge \left(x = 0 \vee x = \pm\sqrt{-a/b}\right) \\ &\equiv (y = 0 \wedge x = 0) \vee \left(y = 0 \wedge x = \sqrt{-a/b}\right) \vee \left(y = 0 \wedge x = -\sqrt{-a/b}\right) \end{align} $$

So, there are three fixed points. $\mathbf{p}_1^* = (0, 0)$ is independent of the parameters (i.e. exists for all values of the parameter). The other two ($\mathbf{p}_2^* = (\sqrt{-a/b}, 0)$ and $\mathbf{p}_3^* = (-\sqrt{-a/b}, 0)$) depend on the parameters. In particular, they only exist if $ab < 0$.

We can determine the topological type of hyperbolic fixed points using linear stability analysis. Consider the Jacobian of the problem at hand:

$$ \begin{align} \mathbb{J}(\mathbf{f}) = \left[\begin{array}{cc} \partial\dot{x}/\partial x & \partial\dot{x}/\partial y \\ \partial\dot{y}/\partial x & \partial\dot{y}/\partial y \end{array}\right] = \left[\begin{array}{cc} 0 & 1 \\ -3bx^2 -a & 0 \end{array}\right] \end{align} $$

Call the determinant of the Jacobian $\Delta$. Call the trace of the Jacobian $\tau$. We can use $\Delta$ and $\tau$ to figure out the topological type of hyperbolic fixed points.

Let us evalulate the Jacobian at the fixed points. At $\mathbf{p}_1^*$:

$$ \begin{align} \mathbb{J}(\mathbf{f})\mid_{(0, 0)} &= \left[\begin{array}{cc} 0 & 1 \\ -a & 0 \end{array}\right] \\ \Delta_1 &= a \\ \tau_1 &= 0 \end{align} $$

At $\mathbf{p}_2^*$:

$$ \begin{align} \mathbb{J}(\mathbf{f})\mid_{(\sqrt{-a/b}, 0)} &= \left[\begin{array}{cc} 0 & 1 \\ 2a & 0 \end{array}\right] \\ \Delta_2 &= -2a \\ \tau_2 &= 0 \end{align} $$

At $\mathbf{p}_3^*$:

$$ \begin{align} \mathbb{J}(\mathbf{f})\mid_{(-\sqrt{-a/b}, 0)} &= \left[\begin{array}{cc} 0 & 1 \\ 2a & 0 \end{array}\right] \\ \Delta_3 &= -2a \\ \tau_3 &= 0 \end{align} $$

Note that $\Delta_i$ only depend on $a$! When $a < 0$, then we a saddle one at $\mathbf{p}_1^*$ while $\mathbf{p}_2^*$, $\mathbf{p}_3^*$ are linear centres only -- that is, they are non-hyperbolic (the eigenvalues of the Jacobian are pure imaginary), and we cannot say much more about it without further analysis.

The situation is reversed when $a < 0$, with $\mathbf{p}_1^*$ being a non-hyperbolic linear centre, and $\mathbf{p}_2^*, \mathbf{p}_3^*$ being saddles. The fact that things switch from being marginally stable to being linear centres should give us a clue that we might want to test for the existence of Hopf Bifurcations.

Let's see if there are any conserved quantities in the system that we can make use of in determining the actual stability of the linear centres. A quantity $E(x, y)$ is conserved if $\dot{E} = 0 = \nabla E \cdot \mathbf{f}$:

$$ \begin{align} &\quad\; \frac{\partial E}{\partial x}y + \frac{\partial E}{\partial y}(-ax - bx^3) = 0 \\ &\Rightarrow \frac{\partial E}{\partial x} = (ax + bx^3) \wedge \frac{\partial E}{\partial y} = y \end{align} $$

So, you can do partial integration and stuff (I am getting lazy, and will update my post later properly) and we should find that: $$E(x, y) = \frac{ax^2}{2} + \frac{bx^4}{4} + \frac{y^2}{2},$$

is a conserved quantity. This implies that trajectories of the phase point always lie on level sets of $E$. Using Wolfram Alpha with $a = -1$ and $b = 1$, we can see that there are clearly closed orbits around the minima of the conserved quantity (which correspond to the fixed points of $\mathbf{f}$):

enter image description here enter image description here

Now when $a = 1$ and $b = -1$ we see:

enter image description here enter image description here

So the origin flipped from being a saddle point, to a centre with orbits around it (as you can see from the contour map!).

When $a = -1 < 0$ and $b = -1 < 0$ we see:

enter image description here enter image description here

When $a = 1 > 0$ and $b = 1 > 0$ we see:

enter image description here enter image description here

So, we go from having marginal stability to oscillations.

The comments bear witness to our confusion regarding whether these are simply degenerate Hopf bifurcations, or whether the pitchfork-like nature of the bifurcations has some signficance worth a name. Ultimately, I had to ask my instructor since I was unable to find a good answer on my own. I quote him here:

The equation is conservative, or Hamiltonian. There is a theory of bifurcations for Hamiltonian systems, different from the "generic" bifurcation theory because of the additional structure. What the Duffing oscillator is experiencing is usually called a "Hamiltonian pitchfork" bifurcation (fix $b$, vary $a$ through $0$). See p.399 of: http://people.mbi.ohio-state.edu/golubitsky.4/reprintweb-0.5/output/papers/hamiltonian_87.pdf for example.

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  • $\begingroup$ Thank you for this very detailed answer! So are these Hopf bifurcations occurring as we vary $a$ and $b$ together? It seems something special happens if we vary $a$ and $b$ whilst keeping them the same sign $\endgroup$
    – Wooster
    Commented Apr 18, 2015 at 21:10
  • $\begingroup$ @Wooster That's right. I added a picture for that case since I missed it the first time around. We go from having a saddle point at the origin, to having a centre at the origin. $\endgroup$
    – bzm3r
    Commented Apr 18, 2015 at 21:16
  • $\begingroup$ Also for a Hopf bifurcation do we not need to see stable spirals turn into unstable spirals surrounded by limit cycles? $\endgroup$
    – Wooster
    Commented Apr 18, 2015 at 21:16
  • $\begingroup$ @Wooster Yeah, I was a little unsure about that myself, but seeing the definition here: en.wikipedia.org/wiki/…, well, that definitely matches our case. We see closed orbits appear (although not limit cycles), even though its not a limit cycle around a stable/unstable spiral. $\endgroup$
    – bzm3r
    Commented Apr 18, 2015 at 21:18
  • $\begingroup$ @Wooster You might also be seeing a type of "global bifurcation" maybe: en.wikipedia.org/wiki/Bifurcation_theory#Local_bifurcations? $\endgroup$
    – bzm3r
    Commented Apr 18, 2015 at 21:20

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