# Relationship between Turing bifurcation, saddle-node bifurcation, and Hopf bifurcation?

Quoting from

http://jxshix.people.wm.edu/2009-harbin-course/mississippi-bifurcation-2.pdf

a Turing bifurcation occurs when for an ODE and related PDE

$u' = f(u,v), v' = g(u,v)$

$u_t = d_1 \nabla u + f(u,v), v_t = d_2 \nabla v + g(u,v)$

a constant solution

$u(x,t) = u_0, v(x,t) = v_0$

remains a stable steady state for the ODE and becomes an unstable steady state for the PDE.

Stable steady states are generally destabilized either by one real eigenvalue (of the associated Jacobian matrix evaluated in the steady state) crossing the imaginary axis or by a pair of two complex eigenvalues crossing the imaginary axis (i.e. the real part changes from negative to positive values).

Generally, one real eigenvalue crossing the imaginary axis is called saddle-node bifurcation. A pair of complex eigenvalues crossing the imaginary axis is referred to as Hopf bifuraction.

Are these two types of bifurcations exactly the ones that occur for the PDE (while the steady state remains stable for the ODE) in what is generally referred to as Turing bifurcation / Turing instability?

If so, what are the qualitative differences, if any, between the patterns formed past the Turing bifurcation?

Generally, one real eigenvalue crossing the imaginary axis is called saddle-node bifurcation. A pair of complex eigenvalues crossing the imaginary axis is referred to as Hopf bifuraction.

This is true, however, some additional conditions have to be met to identify the bifurcation as saddle-node or Hopf. This is less important with Hopf bifurcation since the appearance of two purely imaginary eigenvalues in vast majority cases implies appearance (or disappearance) of a unique limit cycle, but in the case of simple real eigenvalue crossing the imaginary axis there are at least two more cases which are often met in applications: transcritical bifurcation and pitchfork bifurcation. At a qualitative level:

• saddle-node bifurcation corresponds to the appearance "out of nowhere" two steady states, one is stable, another is unstable
• transcritical bifurcation corresponds to the case when two steady states exchange, upon meeting each other, their stabilities
• pitchfork bifurcation corresponds when from one, say stable, equilibrium, two more stable equilibria appear and the original one becomes unstable.

The exact conditions which type you encounter depend on the type of the normal form of the equation you consider. See, e.g, this book.

Are these two types of bifurcations exactly the ones that occur for the PDE (while the steady state remains stable for the ODE) in what is generally referred to as Turing bifurcation / Turing instability?

Yes, in the spatially explicit system all these four types can appear in a similar way how they appear in local systems described by ODEs.

If so, what are the qualitative differences, if any, between the patterns formed past the Turing bifurcation?

In the case of simple zero eigenvalue what is usually expected is an appearance of a spatially non-homogeneous steady state (or more than one). This steady state can be stable generating spatially non-homogeneous patterns, which are used in various problems of pattern formation and morphogenesis. A lot of details are given, e.g., in James Murray's book Mathematical Biology, together with some technical details. Hopf bifurcation yields not only spatially non-homogeneous solutions, but also temporally periodic oscillations.

P.S. And final remark. The name "Hopf bifurcation" is quite unfortunate. The fact that periodic solutions appear under parameter change when two eigenvalues cross imaginary axis
was known to the father of qualitative analysis of dynamical systems -- Poincaré. The exact statements of the theorem in the planar case, together with proofs, were given by Andronov. The main contribution of Hopf is generalization of this situation to $n$-dimensional case. So the correct name would be Poincaré-Andronov-Hopf bifurcation, or, in my opinion, a much better option is the bifurcation of the birth of limit cycle :)

Generally, one real eigenvalue crossing the imaginary axis is called saddle-node bifurcation. A pair of complex eigenvalues crossing the imaginary axis is referred to as Hopf bifuraction.

Your said is true but not very preciesly, because the PDE is a infinite dynamical system, that is to say, PDE system has infinite pair of eigenvalues corresponding to the serious of basis, while ODE system only has a pair of eigenvalues.

Are these two types of bifurcations exactly the ones that occur for the PDE (while the steady state remains stable for the ODE) in what is generally referred to as Turing bifurcation / Turing instability?

No, only Turing bifurcation occur for PDE instable ODE stable. Hopf bifurcation is appearing with PDE and ODE simultaneously unstable, that is to say, the diffusion term is no contribution to the Hopf bifurcation. In fact, Turing bifurcation is not the really saddle-node point, just like the saddle-node point having a real eigenvalue 0. Turing bifurcation onset generates the fixed spatial wavenumber $k$, while the Hopf bifurcation generates the time periodic limit circle not the spatially wavelength. Hopf bifurcation in PDE generates the travelling wave solution in the neighborhood of Hopf critical point, like the classical Landau equation, i.e. the wave shape maintain on change but the wave front will move forward with time. Turing bifurcation will converge to a spatially periodic stable solution after a long time, although it has a transient time oscillation.

p.s. The Hopf bifurcation's time periodic limit cycle has been proved by the Hopf bifurcation theorey, as Artem's said. But the Turing bifurcation's spatially periodic stable solution has not been proved by the partial differential equations route.