How these two are related? What kind of bifurcation leads to chaos?
Let me quote Wikipedia here:
Chaos :
Robert L. Devaney says that, to classify a dynamical system as chaotic, it must have these properties:
Bifurcation:
A bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.
Hi again, Wajid!
Actually, the answer to this question can be found in the answers to the previous question. Namely, Filipe Martins mentioned Sharkovsky's theorem, which has a special case "Period Three Theorem" for one-dimensional systems.
In short, if the chain of the bifurcations eventually produces at least one period 3 cycle, the system immediately possesses regular cycles of every other length as well as completely chaotic cycles.
PS "topologically mixing" may be omitted sometimes. Consider the irrational rotations of the unit circle, for example. This system is ergodic, but not mixing.
Hi Wajid Ali !
The dynamic system and chaos can be related as follows: when a dynamical system, described by a set of parameterized differential equations, changes qualitatively, as a function of an external parameter, the nature of its long-time limiting behavior in terms of fixpoints or limit cycles, one speaks of a bifurcation.
On the types of bifurcations and chaos can be found in more detail in the book: Gros, Claudius. Complex and adaptive dynamical systems. Chapter 2
Hello W. Ali!
The appearance of chaos in a system is usually associated with systems whose mathematical model has a parameter that can vary. For a given range of values of this parameter, the system will exhibit a chaotic behavior. The point that marks this change is called a bifurcation point.
Hi Wajid Ali,
I recommend the book "Elements of Applied Bifurcation Theory" By Y. Kuznetsov.
The phase portrait of a dynamical system varies with the parameters. A bifurcation occurs when, as the parameter(s) pass through a critical value, a phase portrait that is topologically nonequivalent appears.
In this book you can also find the first bifurcations on a route to chaos.
Bifurcation as a Precursor to Chaos: Bifurcation refers to a qualitative change in the behavior of a dynamical system as a parameter is varied. It often involves the creation or destruction of periodic orbits (stable or unstable) in the system. Bifurcations can be thought of as points of transition in the system's behavior. Before a system exhibits chaotic behavior, it may go through a series of bifurcations.
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