📂Dynamics

Heteroclinic Bifurcation

Definition

Heteroclinic Bifurcation is a bifurcation in which a heteroclinic orbit appears or disappears as the parameters of a dynamical system change.

Explanation

The heteroclinic bifurcation, as its name suggests, involves heteroclinic orbits. It is helpful to imagine it as a scenario where the manifold connecting two fixed points joins or breaks as parameters change. Since the indication of being part of a heteroclinic orbit cannot be observed by examining the neighborhood of the two fixed points alone, it is considered a global bifurcation.

Example ¹

$$\begin{align*} \dot{x}_{1} =& 1 - x_{1}^{2} - \alpha x_{1} x_{2} \\ \dot{x}_{2} =& x_{1} x_{2} + \alpha \left( 1 - x_{1}^{2} \right) \end{align*}$$ Let’s consider a system given as above. This system has two fixed points $\mathbf{x}_{1} = (-1,0)$ and $\mathbf{x}_{2} = (1, 0)$.

In this system, when $\alpha \ne 0$, there is no manifold connecting $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$, but when $\alpha = 0$, there exists a heteroclinic orbit exactly at $x_{2} = 0$ and $x_{1} \in [-1, 1]$.

See Also

• Homoclinic Orbit and Heteroclinic Orbit
• Homoclinic Bifurcation