Lyapunov coefficient for function f(x) = 1/(1+x)

In summary, the conversation discusses the use of function f(x) = 1/(1+x) to investigate chaos and the potential calculation of a Lyapunov coefficient. The function is found to converge to the Golden Ratio minus 1 when used as an iteration, making it difficult to calculate a meaningful Lyapunov Exponent.
  • #1
Sasho Andonov
21
0
May I use function f(x) = 1/(1+x) to investigate chaos?
I am trying to understand chaos using this function, but things are not going as I expected...
Could you please advice me how I can calculate Lyapunov coefficient for this function?
 
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  • #2
It isn't clear how you intend to use this function. Are you using it to iterate something? Are you using it as a mapping function somehow?

As an iteration ##x_{i+1} = \frac{1}{1+x_i}##, it converges to ##\frac{\sqrt{5}-1}{2}##, which is the Golden Ratio minus 1. The Golden Ratio crops up a lot in chaos theory. To calculate a meaningful Lyapunov Exponent, you would need some sort of chaotic time series. Because this function converges, the Lyapunov Exponent would always be negative.
 

1. What is a Lyapunov coefficient?

The Lyapunov coefficient is a mathematical concept used to analyze the stability of a dynamical system. It measures the rate of divergence or convergence of a system's trajectories.

2. How is the Lyapunov coefficient calculated?

The Lyapunov coefficient is calculated using the Lyapunov exponent, which is the limit of the logarithm of the ratio of the distances between nearby trajectories of the system. It can also be calculated using the derivative of a system's state variables.

3. What does a positive Lyapunov coefficient indicate?

A positive Lyapunov coefficient indicates that a system is unstable, as it shows that nearby trajectories are diverging from each other. This means that small changes in initial conditions can lead to significant differences in the system's behavior over time.

4. What does a negative Lyapunov coefficient indicate?

A negative Lyapunov coefficient indicates that a system is stable, as it shows that nearby trajectories are converging towards each other. This means that small changes in initial conditions will not have a significant impact on the system's behavior over time.

5. How is the Lyapunov coefficient used in chaos theory?

In chaos theory, the Lyapunov coefficient is used to classify systems as chaotic or non-chaotic. A positive Lyapunov coefficient is an indicator of chaos, while a negative coefficient indicates stability. It is also used to study the sensitivity of chaotic systems to initial conditions.

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