We plot Bifurcation Diagram for different mappings. It after 8 or 16 period shows Chaotic behavior. But what does the shape of Bifurcation Diagram represents? For different mappings, we are having different shapes. But what does these shapes signifies?
Here period doubling means route to chaos. If you want to show chaotic behaviour, then either find period 3 cycle that represents chaos or determine Lyapunov exponents which should be positive. There are also some more quantities can show about existence of chaotic behaviour.
See any book on chaos and fractals.
I know it sir
But the important thing is that what does shape signify.
Henon map has different shape. Logistic has different.
In a dynamic system, a bifurcation occurs when a small smooth change made to the parameter values of the system, causes a sudden qualitative or topological change in its behavior. The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. or from 3 to 6 to 12 etc., or from 5 to 10 to 20 etc., and so on. Each of the bifurcation points is a period-doubling bifurcation.
That is true and I know it. But my question is not regarding the points, it is about the shape of diagram.
The shape of a bifurcation diagram suggests the qualitative behavior of a dynamic system as its parameters are varied. The diagram plots the steady-state solutions of the system as a function of the parameters.
The shape of the bifurcation diagram can indicate the presence of different types of bifurcations, such as saddle-node bifurcations, transcritical bifurcations, pitchfork bifurcations, and period-doubling bifurcations, among others. These bifurcations correspond to qualitative changes in the behavior of the system, such as the appearance or disappearance of fixed points, limit cycles, or chaos.
For example, a pitchfork bifurcation can indicate a change from a single stable fixed point to two stable fixed points and one unstable fixed point, while a period-doubling bifurcation can indicate a transition to chaos. Understanding the shape of the bifurcation diagram can therefore provide insight into the stability, behavior, and response of the dynamic system to changes in its parameters.
Great!
can you please share some basic material on this topic with me. My email id is:
The following video explains the bifurcation diagram. https://www.youtube.com/watch?v=JgLyk304MYk
different shapes give different information about how we transition in and out of chaos as one parameter increases. There is also a matlab code in the description.
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