Bifurcations and bifurcation diagrams
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- Опубліковано 14 жов 2024
- (Lecture 3.4) A bifurcation diagram tells us how the qualitative behavior of solutions to a different equation can change as a parameter changes. In this lecture we look at proto-typical examples of saddle-node, transcritical, pitchfork, and fold bifurcations. (Remark: from Example 3 onward, my stylus was giving me trouble, sorry about the hand-writing!)
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In detail, we explore bifurcations in differential equations, particularly in first order autonomous systems. We focus on how changing a parameter in a model can lead to significant qualitative changes in the system's behavior.
A bifurcation occurs when a small change in a parameter causes a qualitative change in the system. In our differential equation 𝑑𝑦/𝑑𝑡=𝜇−𝑦2, we observe how varying the parameter 𝜇 leads to different behaviors. For example, no real equilibrium solutions exist when 𝜇 is negative, but as 𝜇 increases, we see the emergence of equilibrium solutions, indicating a bifurcation.
We construct phase lines for different values of 𝜇. When 𝜇=−4 and 𝜇=−2, there are no equilibrium solutions, and the system's behavior is uniform. However, at 𝜇=0, we observe an equilibrium at zero, indicating a node. Increasing 𝜇 further, to 𝜇=1 and 𝜇=4, we find two equilibrium solutions, forming a source and a sink.
The bifurcation diagram visualizes how the system evolves as we vary 𝜇. We plot the equilibrium solutions vertically against 𝜇 horizontally, revealing a bifurcation structure. For instance, at 𝜇=1, the node splits into a source and a sink, which is a significant qualitative change.
We analyze different values of 𝜇 and their effects on the system. As 𝜇 changes, we observe the emergence, splitting, and disappearance of equilibrium solutions. These changes are neatly captured in the bifurcation diagram, demonstrating the concept of bifurcation vividly.
1. Saddle-Node Bifurcation: This occurs when a parameter change causes the emergence or disappearance of two equilibrium solutions, which are a source and a sink. It is characterized by a parabolic shape in the bifurcation diagram. This type of bifurcation was observed in the example with 𝑑𝑦/𝑑𝑡=𝜇−𝑦2 when we transitioned from 𝜇 negative (no equilibrium solutions) to 𝜇=0 and 𝜇 positive (one or two equilibrium solutions, respectively).
2. Transcritical Bifurcation: In this bifurcation, two equilibrium solutions exchange their stability as the parameter crosses a critical value. It was demonstrated in the example with 𝑑𝑦/𝑑𝑡=𝜇𝑦−𝑦^2, where the equilibrium solutions 𝑦=0 and 𝑦=𝜇 exchange stabilities as 𝜇 changes sign.
3. Pitchfork Bifurcation: This type of bifurcation is illustrated in the example 𝑑𝑦/𝑑𝑡=𝜇𝑦−𝑦^3. It features a single equilibrium solution that splits into three as the parameter 𝜇 passes through a critical value. Specifically, when 𝜇 is positive, there are three equilibrium solutions (one source and two sinks), and when 𝜇 is negative, there is only one (a sink).
4. Fold Bifurcation: Also known as a cusp or blue sky bifurcation, this is exemplified in the final example. It is characterized by the appearance and disappearance of equilibrium solutions as the parameter changes, resembling a fold in the diagram.
These bifurcations illustrate how small changes in a system's parameters can lead to significant changes in its qualitative behavior, particularly in the number and stability of equilibrium solutions.
#mathematics #bifurcation #differentialequation #differentialequations #ordinarydifferentialequations #parameters #bifurcationdiagram #dynamicalsystems
#MathModeling #ParameterVariation #mathlectures
Thank you for going over multiple different equations instead of just doing the easiest ones
Of course! I am glad it was helpful.
Dear Bevin, I extend my sincere gratitude for your invaluable video; it proved to be exceptionally helpful and effective. Please accept my warmest regards. Thank you
I'm so pleased it was helpful, and I appreciate the kind words! Best wishes.
The clearest lecture video out there!
Wow, thanks!
it's pity there are no new videos. You explain so well. Please record videos on qualitative theory of differential equations
There will be in about a month, I’m traveling this semester (so, on hiatus).
@@bevinmaultsby Thank you ! Have a good day !
Dear professor, thanks for the detailed explanation you gave. I found it too helpful.
You’re welcome! Have a great day.
You have no I idea how this helped with my assignment thank you so much😂
Happy to help!
literally helped me a day before my exam, grazie
Prego! Good luck on your exam!
@@bevinmaultsby thank you!!
Fantastic explanation, thank you
You're very welcome!
Hi, Dr thanks for these worthy lectures. Where can i find these pdf lectures, please.
thank you so muchh for the amazing explanation!!
You're very welcome!
thanks for so nicely explaining.....
You're very welcome!
100000000₂th like.
I wish! Thank you for watching :)
Really nice video...my assignment grade thanks you.🎉
Most welcome 😊 !