Dynamical Systems, Basin of Attraction Proof

In summary, the problem is asking to show that the basin of attraction of x1 under f is equal to f applied to the basin of attraction of x1. This can be proven by considering the definition of the basin of attraction and using a proof by contradiction method. The key is to understand that x1 is a periodic point with another k-period point x2, and f(x1) is equal to x2.
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mathgeekgirl
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1. Homework Statement

Suppose that f: I->I is a continuous and onto map on an interval I. Let x1 be an asymptotically stable periodic point of period k>=2. Show that Ws(f(x1))=f(Ws(x1))



2. Homework Equations
Ws(x), the basin of attraction of x1 is defined as {x: lim (n->infinity) f^n(x)=x1}. Note f^n is f(f(f(f(f(x), or f composed n times.




3. The Attempt at a Solution

I think that since x1 is a periodic point, so there is another k-period point I called x2 for which f(x1)=x2. So then the right side becomes the Basin of Attraction for x2. But I'm not sure how to describe the basin of attraction of x1 under f.
 
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  • #2
I tried to prove by contradiction and assumed that Ws(f(x1)) is not equal to f(Ws(x1)) but since I'm not sure how to define the basin of attraction of x1 under f, I'm not sure how to proceed. Any suggestions?
 

What is a dynamical system?

A dynamical system is a mathematical model used to describe the behavior of a system over time. It consists of a set of equations that govern the evolution of the system's state variables.

What is the basin of attraction in a dynamical system?

The basin of attraction is the set of initial conditions that will lead to a particular attractor in a dynamical system. It represents the range of possible states that the system can reach and stay in over time.

How is the basin of attraction proof used?

The basin of attraction proof is used to analyze the stability of a dynamical system and determine the range of initial conditions that will lead to a particular behavior. It helps to understand the long-term behavior of a system and make predictions about its future states.

What is the difference between a stable and unstable basin of attraction?

A stable basin of attraction indicates that the system will converge towards a specific attractor regardless of its initial conditions. An unstable basin of attraction means that the system may converge towards different attractors depending on its initial conditions.

What are the limitations of the basin of attraction proof?

The basin of attraction proof assumes that the system is linear and time-invariant, which may not always be the case in real-world systems. It also does not account for external factors or disturbances that may affect the behavior of the system.

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