Transcritical bifurcation
In mathematics , a transcritical bifurcation is a local or global bifurcation of an ordinary differential equation . This type of branching only occurs when the system has a point that exists for all parameter values that can never be destroyed. When this point collides with another also equal, the two points exchange their stabilities, and continue to exist after the fork.
A transcritical bifurcation is a type of bifurcation that can be local, which means that it is characterized by an equilibrium that has its own value (or eigenvalue), the real part of which passes through zero.
The transcritical bifurcation arises in systems where there is a certain “trivial” base branch solution, which corresponds to x = 0, and which exists for all values of parameter C.
(This differs from the case of a chair-node branch, where solution branches exist locally only on one side of the branch point.).
There is a second solution branch x = C that traverses the first one at the branch point (x, C) = (0, 0).
When branches are crossed, one solution goes from stable to unstable, while the other goes from stable to unstable.
This phenomenon is known as a “stability exchange”. [one]
Summary
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- 1 Shape
- 2 Example 1
- 3 See also
- 4 References
Shape
The normal way is:
dx / dt = Cx ∓ x 2
The normal form of a fork is a simple dynamic system that is equivalent to all systems showing this fork. [2]
Example 1
The example of a transcritical bifurcation of the differential equation:
dx / dt = Cx – x 2
it has 2 equilibria: x = 0, x = C
with f (x, C) = Cx – x 2
df / dx (x, C) = C – 2x
df / dx (0, C) = C
df / dx (C, C) = -C
equilibrium x = 0 is stable , and not stable for C> 0, while equilibrium x = C is not stable for C <0 and stable for C> 0.
Note that while x = C is stable for C <0, it is not global stable, only local. (See the effect of negative disturbances of sizes larger than C).
