Note on Bifurcation Normal Form

Basic Notions

Def Topological Equivalence: 2 dynamic systems are topological equivalence when there is a homeomorphism between their solutions.

Def Conjugate: Two maps are connected by

$$g=h^{-1}\circ f \circ h$$

.

• In some fields like group theory this is also called equivariance w.r.t. $h$
• For Discrete dynamic systems, map $f,g$ are topological equivalence when they are conjugate.
  • $$g=h^{-1}\circ f \circ h$$.
• For continuous dynamic systems, when the systems $\dot x=f(x)$, $\dot y=g(y)$ are topological equivalence, their flows are conjugate.
  • $\psi^t=h^{-1}\circ \phi^t \circ h$
  • $h(\psi^t(x))=\phi^t(h(x))$

Comments:

• For smooth coordinate change $y=h(x)$ in continuous time system $\dot y=g(y)$ and $\dot x=f(x)$. Then their vector fields suffice $f(x)=(\frac{dh}{dx})^{-1} g(h(x))$.
  • If this latter property is true over the space, the two systems are conjugates.

Locally topological equivalent:

Saddle-Node bifurcation

$$ \dot x=\alpha-x^2 $$

Pitchfork bifurcation

$$ \dot x = x(\alpha-x^2) $$

Flip Bifurcation

$$ \dot x=x(x-\alpha) $$

Hopf Bifurcation

The super critical Hopf Bifurcation

$$ \dot z=(\alpha+i)z-z\|z\|^2 $$

The sub critical Hopf Bifurcation

$$ \dot z=(\alpha+i)z+z\|z\|^2 $$