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1.4 Physical Quantities

2.1.1 Bifurcations

Meiss [1] offers a definition of a bifurcation as,

Bifurcation: a qualitative change in dynamics occurring upon a small change in a parameter.

Many different types of changes in dynamics can occur in (2.1) as a consequence of varying system parameters, see [1, Chapter 8] for a treatment of some of these. Some bifurcations occur as a single parameter is varied while others occur only as a consequence of varying multiple parameters at once. The minimum number of parameters which must be varied for a given bifurcation to occur is denoted theCodimension of the bifurcation.

For the systems investigated in the present work the codimension 1 saddle-center bifurcation was the only bifurcation observed away from domain boundaries in all but a single case8. In the single exceptional case a co-dimension 2 bifurcation was identified. Due to its frequent occurrence the saddle-center bifurcation is covered in detail here, while the co-dimension 2 bifurcation is treated specifically in chapter 8 when it is encountered.

Saddle-Center Bifurcation: This bifurcation corresponds to the creation or annihilation of two critical points as a single parameter is varied. One of the critical points is a saddle point and the other is a center.

In order to illustrate such a bifurcation consider a Taylor expansion of the vorticity given by,

In [2] Brøns shows that if one applies transformations to the dynamical system in con-sideration such that the degenerate critical point undergoing a bifurcation is centred at the origin and the coordinate axis are twisted and stretched appropriately the following theorem holds9,

Theorem 1: Let a10, a01, a11, a20 of (2.7)and ˜an0 for n < N−1 be small parameters, and assume that non-degeneracy conditions a02 6= 0 and ˜aN0 6= 0. Then there is a coordinate transformation that brings ω into the normal form,

ω= σ

8In [2, part III] Brøns shows that for a flow away from a boundary this is the only possible codimension 1 bifurcation.

9This isTheorem 5in [2] formulated forωinstead of Ψ.

2.1 Dynamical Systems Theory 2 THEORY

a˜N0 =aN0+ a non-linear combination of anm of lower order then N. (2.11) Using this transformed expression for ω it is easy to see that all critical points near the degenerate point for the system (2.5) are situated along the transformed x-axis and that there are at mostN −1 of them,

x˙ = ∂ω

∂y =σy= 0⇔y= 0, y˙=−∂ω

∂x =−fx(x) = 0. (2.12) The Jacobian of (2.5) at a critical point, determines its type and is given by,

J( ˙x,y)˙ =

"

0 σ

−fxx(x) 0

#

. (2.13)

From (2.13) one can determine the type of a critical point by considering the eigenvalues given by,λp−σfxx(x). This leaves three options. Either the critical point is degenerate allowing for a bifurcation to happen. The second option is −σfxx(x) >0 which given two real eigenvalues of opposite sign, corresponding to a saddle as the critical point has one attracting eigendirection and one repelling eigendirection. The last option is−σfxx(x)<0 which gives two purely imaginary eigenvalues of opposite sign which corresponds to a center, i.e. either a maximum or minimum in the vorticity.

For a co-dimension 1 bifurcation one has N = 3, i.e. the terms of order lower then four determine the behaviour of the system. From Theorem 1 this gives σ = 1 and reduces the expression for ω to,

ω= 1

2y2+f(x) +O(4), (2.14)

f(x) =c1x+ 1 3x3.

Using (2.12) and (2.13) it can be seen that the occurrence of a bifurcation as c1 is varied may be illustrated in one dimension aroundx= 0 by considering the derivative off(x),

fx(x, c1) =c1+x2, (2.15)

From (2.15) it is seen that the origin is a degenerate critical point for the system (2.5) when c1 = 0. It is easy to see that in general critical points exist for (2.5) at,

x=±√

−c1fx(x, c1) = 0. (2.16)

For c1 = 0 we see that fx(0,0) = 0 and ∂x fx(0,0) = fxx(0,0) = 0 while ∂x22fx(0,0) = fxxx(0,0) = 2 and ∂c

1fx(0,0) = 1. The first two conditions are known as singularity

conditions, the next as a non-degeneracy condition and the last as a transversality condition.

If these conditions are fulfilled then the bifurcation which occurs at x= 0 when c1 is varied is a saddle-center bifurcation [1, Corollary 8.4]. For the 1D case the singularity conditions simply state that at x = 0, fx(x,0) is zero valued, has a horizontal tangent and the non-degeneracy condition that fx(x,0) has non-zero curvature. The transversality condition states that increasing/decreasingc1 around zero increases/decreases the value offxatx= 0 which is necessary for a change in system behaviour to occur.

From (2.16) it may then be seen that forc1 <0 two critical points exist whereas forc1 = 0 only a single (degenerate) critical point exists and forc1>0 no critical points exist. I.e. the saddle-center bifurcation occurs just asc1 passes zero.

The change in the number of critical points asc1 is varied is illustrated on figure 2.1.

−1 −0.5 0.5 1 1.5 2

−1 1 2 3

fx(x, c1= 0) fx(x, c1>0)

fx(x, c1<0)

x fx

Figure 2.1: Illustration of the creation of two critical points asc1 is varied across zero. The figure can also be used to illustrate how the search for extrema and saddles in vorticity can be performed using the nullclines. Consider the x-axis as an ωx nullcline and searching along this to identify ωy = 0 corresponding to fx = 0.

A bifurcation diagram for the situation showing the annihilation of a center (vortex in the flow) and a saddle asc1 is varied across zero is presented on figure 2.2.

Figure 2.2: Illustration of the creation/annihilation of a vortex (center) and saddle point as the parameterc1 is varied. The figure is borrowed from [2, Figure 3.2] courtesy of Brøns.

The first illustration in figure 2.2 corresponds to the situation offx(x, c1<0) shown in figure 2.1. Here a saddle and a center exist. The second illustration in figure 2.2 corresponds to

2.1 Dynamical Systems Theory 2 THEORY

the situation of fx(x, c1 = 0) shown in figure 2.1, which corresponds to the bifurcation point where the critical point becomes degenerate. Lastly, the third illustration in figure 2.2 corresponds to the situation offx(x, c1 >0) shown in figure 2.1 where no critical points exists.

Note: A final and important remark on the parameterc1 is that it is a mathematical para-meter which depends on one or more of the physical parapara-meters of the system. Thus for our model problem of the cylinder near the moving wall we have that c1 may depend on time, Reynolds number and D/G-ratio. This means that c1(Re, D/G, t0) may change as either of the three parameters are varied causing the co-dimension 1 bifurcation.