(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.
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23 сен 2024
