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I am doing a presentation on bifurcations and would like physical examples to go along with each type of bifurcation but I am unable to find or think of any good example of a simple Saddle Node Bifurcation.

The most basic saddle node-bifurcation can be described as:

Any type of system that has no equilibrium or steady state solutions initially but bifurcates as a parameter is varied into having 2 solutions, 1 stable, another unstable.

or vice versa, any system that has 2 solutions that vanish as a parameter is varied.

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2 Answers 2

Suppose you have a small ball in a periodic potential. The period of the potential is much larger than the ball size. (You can see it as a bead in a wash-board). This system is in a water (or any viscous liquid). Then minima (maxima) correspond to nodes (saddles).If you tilt the wash-board, then at angles larger than some threshold, there are no minima and maxima anymore. The ball falls down along the board, because no equilibrium points exist.

In fact, any system, that is equivalent to damped motion of a particle in a potential of the specific type, will have saddle-node bifurcation. The corresponding equation is $$ {d^2 x \over dt^2} + \gamma {d x \over dt} + {d U(x, C) \over dx} = 0, \quad (1) $$ where $\gamma >0$ is the dissipation parameter, $C$ is parameter of the potential. The potential $U(x, C)$ should satisfy two conditions. (i) It should have at least 2 extrema (maximum and minimum) for some range of $C$. (ii) At some value of $C$, the positions of maximum and minimum coinside, and disappear.

Dissipation "transforms" a center point to a node (for large dissipation) via a focus (for small dissipation). In the example, water plays a role of dissipation. It is clear that saddle-node bifurcation is related closely to saddle-center and sadlle-focus bifurcations.

Another (more physical) example is a Josephson (superconductor-insulator-superconductor) junction under constant current $I$. The corresponding equation (in dimensionless form) is $$ {d^2 \phi \over dt^2} + \gamma {d \phi \over dt} + \sin(\phi) = I, \quad (2) $$ where $\phi$ is the difference of quantum phases of the two SCs. The corresponding potential is $U(x,I)= 1 - \cos(\phi) -I \phi$. For some value of I, saddle-node bifurcation occurs in Eq.(2).

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I strongly suggest Steven H. Strogatz' book Nonlinear Dynamics and Chaos, which has several simple examples of all basic bifurcations in a few different fields. An interesting one in biology is that of fireflies synchronizing their light-flashing.

As a 1D physical example (for which the term saddle-node bifurcation seems a bit odd because saddles and nodes are really higher-dimensional fixed points but the mechanism is exactly the same in 1D), Strogatz turns to the problem of an overdamped pendulum driven by a constant torque $\Gamma$. (so a pendulum immersed in some viscous fluid like oil or honey and connected to some motor applying a constant torque to it) If $L$ is the length of the pendulum, $m$ its mass and $\theta$ the angle between the pendulum and the vertical direction, then Newton's law yields

$$mL^2\ddot{\theta} + b\dot{\theta} + mgL\sin{\theta} = \Gamma$$

where $b$ is a viscous damping factor. Now, in the overdamped limit of large $b$, the first term (the inertia term) can be neglected in comparison to the others and we get the equation

$$b\dot{\theta} + mgL\sin{\theta} = \Gamma.$$

We can simplify the analysis by transforming the problem into dimensionless variables. We can do this by dividing by a torque. A good choice here is to divide by $mgL$, yielding the following differential equation:

$$\frac{b}{mgL}\dot{\theta} = \frac{\Gamma}{mgL} - \sin{\theta}.$$

Subsequently substituting $\tau = \frac{mgL}{b}$ and $\gamma = \frac{\Gamma}{mgL}$ yields the dimensionless expression:

$$\theta' = \gamma - \sin{\theta}$$

where $\theta' = \frac{d\theta}{d\tau}$. Now it's easy to see that this system undergoes a saddle-node bifurcation as $\gamma$ varies.

  • For $\gamma > 1$, $\theta'$ is never zero, meaning the pendulum keeps overturning continuously, with no fixed points around. Physically, since $\gamma$ is the ratio of the constant applied torque to the magnitude of the gravitational torque, this means gravity is never able to cancel out the applied torque completely. So we should expect no fixed points.
  • For $\gamma = 1$, $\theta'$ is identically zero for $\theta = \pi/2$, meaning there is a fixed point for the pendulum hanging horizontally.
  • For $\gamma < 1$, $\theta'$ has two zero's symmetrically located around $\theta = \pi/2$, meaning there are two fixed points now, one stable and one unstable. To find out which one is stable, you can consider the sign of $\theta'$ in either one, but on physical grounds it is already clear that the lower one (below $\pi/2$ so below the horizontal) is the stable one. Especially if we consider what happens if $\gamma$ decreases even further toward $0$.
  • For $\gamma = 0$, $\theta'$ is just a sinusoidal function, so it has a zero at $\theta = 0$ and one at $\theta = \pi$ (inverted pendulum). Obviously, the inverted pendulum is unstable, so our conclusion for the stability of the fixed points was correct.

From the above analysis, it is clear that the saddle-node bifurcation occurs at $\gamma = 1$, where two fixed points are born (or, equivalently if we approach $\gamma \rightarrow 1^-$, where a stable and an unstable fixed point collide and mutually annihilate).

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