Critical value of homoclinic bifurcation

In summary, to find the value of parameter A at which the homoclinic bifurcation occurs, you can use numerical methods such as Newton's method or the bisection method.
  • #1
LorenaGr
1
0
Hello,
I hope to find some help.
I have equation:
upload_2017-11-13_15-17-24.png
and I need to find the value of parameter A at which the homoclinic bifurcation occurs.
I know, that in this case A=-0.34, but I solution, how to find it.
Can you help me, what numerical method I should use to find A?

Thank you very much.
 

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  • #2
One possible approach would be to use numerical methods such as Newton's method or the bisection method to find the root of the equation, where the root is the value of A at which the homoclinic bifurcation occurs. These methods involve iteratively approximating the root of the equation given an initial guess. You can read more about the different numerical methods here: https://en.wikipedia.org/wiki/Root-finding_algorithm.
 

1. What is a critical value of homoclinic bifurcation?

A critical value of homoclinic bifurcation is a value at which a homoclinic orbit (a trajectory that approaches and then returns to the same point in phase space) changes from being stable to unstable, or vice versa. It is an important concept in the study of dynamical systems and chaos theory.

2. How is the critical value of homoclinic bifurcation calculated?

The critical value of homoclinic bifurcation can be calculated using mathematical techniques such as stability analysis, where the stability of the homoclinic orbit is determined by the eigenvalues of the system's Jacobian matrix. Alternatively, it can also be estimated through numerical simulations of the system.

3. What are the implications of a critical value of homoclinic bifurcation?

A critical value of homoclinic bifurcation can signal a transition in the behavior of a dynamical system. It can lead to the creation of a chaotic attractor, where the system exhibits unpredictable and sensitive behavior, or the destruction of a stable orbit, leading to instability and potentially causing the system to deviate from its intended trajectory.

4. How is the critical value of homoclinic bifurcation related to chaos theory?

The critical value of homoclinic bifurcation is an important concept in chaos theory, as it marks a threshold at which a system can exhibit chaotic behavior. In particular, it is closely related to the concept of a strange attractor, which is a fractal structure that characterizes chaotic systems.

5. Can the critical value of homoclinic bifurcation be controlled or manipulated?

In some cases, it may be possible to control or manipulate the critical value of homoclinic bifurcation through external parameters or feedback mechanisms. However, this is highly dependent on the specific system being studied and is an ongoing area of research in the field of dynamical systems and chaos theory.

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