Lyapunov exponent -- Numerical calculations

In summary, the conversation discusses the calculation of the largest Lyapunov exponent in computational physics and its relationship to chaos in a system. It is noted that a positive LE indicates chaos, while a negative or zero LE implies no chaos. The difference between specific values of LE is also explored, with the conclusion that a more negative LE results in a faster return to the unperturbed path in phase space. The conversation also raises questions about the relationship between LE and motion, with the answer being that a negative LE does not always imply a periodic orbit.
  • #1
LagrangeEuler
717
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In computational physics is very often to calculate largest Lyapunov exponent. If largest Lyapunov exponent ##LE## is positive there is chaos in the system, if it is negative or zero there is no chaos in the system. But what can we say about some certain value of ##LE##. For example ##LE_1=-0.2## and ##LE_2=-0.4##. What is the difference between those particular values? Could we say something in small scales? Thanks for the answer.
 
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  • #2
The more negative the LE is, the more quickly the trajectory returns to its unperturbed path in that dimension of phase space.
 
  • #3
Thank you for the answer. Is it necessary that if LE is negative motion is periodic? And when LE is zero motion is quasiperiodic?
 
  • #4
Too many variables in play for a simple answer. Conservative or not? Winding numbers? Dimensions?

A negative LE does not imply a periodic orbit.
 
  • #5
Ok thanks. But periodic orbits imply that LE is negative. Right?
 
  • #6
LagrangeEuler said:
Ok thanks. But periodic orbits imply that LE is negative. Right?

Not always. There are cases of unstable periodic orbits where the LE is positive.

Try a google search for unstable periodic orbits. There are a lot of examples.
 

FAQ: Lyapunov exponent -- Numerical calculations

1. What is a Lyapunov exponent?

A Lyapunov exponent is a mathematical concept used to measure the rate of exponential divergence or convergence of nearby trajectories in a dynamical system. It is a measure of the system's sensitivity to initial conditions.

2. Why is it important to calculate Lyapunov exponents numerically?

Calculating Lyapunov exponents numerically allows for the analysis of complex systems that cannot be solved analytically. It also provides a more accurate estimation of the exponents, as they can be sensitive to small changes in initial conditions.

3. What are some common methods for numerically calculating Lyapunov exponents?

Some common methods for numerically calculating Lyapunov exponents include the Gram-Schmidt algorithm, the Wolf algorithm, and the Eckmann-Ruelle algorithm. Each of these methods has its own advantages and limitations, and the choice of method depends on the specific system being studied.

4. Can Lyapunov exponents be negative?

Yes, Lyapunov exponents can be negative. Negative exponents indicate that the nearby trajectories in the system are converging, rather than diverging. This can be seen in stable systems, where small changes in initial conditions do not significantly affect the overall behavior of the system.

5. How are Lyapunov exponents used in real-world applications?

Lyapunov exponents have applications in various fields, such as chaos theory, weather forecasting, and biological systems. They can help predict the behavior of chaotic systems, improve the accuracy of weather predictions, and understand the dynamics of biological systems, such as the human heart.

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