Bifurcation Diagrams: How to Plot Stability

[matpo39]

I have a qustion about bifucation diagrams in the book they use the DE: dy/dt= y*(a-y^2)
the critical points for, a<0: dne a=0:y=0 and a>0: y=0,+-sqrt(a)
then they ploted critical points as a function of a in the ay plane and the equation of the graph looks like a=y^2 and the graph looks like a normal x=y^2 graph except that everything below y=0 is dashed and everything above is solid line. Is this because above a=0 the DE is stable and below a=0 it is unstable? also if this is the case, would for other functions as well where you just plot the function of a and dash the parts of the graph that are unstable?

thanks

 

[member 553083]

Yes, this is correct. When plotting a bifurcation diagram, the dashed lines typically indicate unstable regions and the solid lines indicate stable regions. This is true for other functions as well, as long as the critical points can be determined from the function of a and the stability of each region can be determined.

 

[wais]

for your question. The bifurcation diagram is a useful tool for visualizing the stability of a dynamical system, like the one described in your question. The dashed and solid lines on the diagram represent the stability or instability of the critical points for different values of a.

In general, a bifurcation occurs when there is a qualitative change in the behavior of the system as a parameter, in this case a, is varied. In the case of a>0, the critical points are stable, meaning that the system will tend towards these points over time. However, for a<0, the critical point at y=0 becomes unstable, meaning that small perturbations will cause the system to diverge from this point. This is why the dashed line is used to represent this region on the bifurcation diagram.

To answer your question, yes, this concept can be applied to other functions as well. The general idea is to plot the critical points as a function of a and then use a dashed line to indicate the region where the system becomes unstable. This allows us to easily visualize the stability of the system for different values of the parameter.

I hope this helps to clarify the concept of bifurcation diagrams for you. If you have any further questions, please don't hesitate to ask.