Lecture 3

Dynamical System Reduction:The Center Manifold

3.1 Center Manifold Theorem

We consider a system with isolated equalibrium at 0 in the form

_u = X(u)

= JX(0) +R(u), (3.1)

where R(u) = O(u2). We further assume the eigenvalues λ satisfy Reλ · 0. LetT be a non-singular linear transformation such that

T ¡1JX(0)T =

·Ac

As

¸,

where Ac and As are the blocks in the canonical form whose diagonals containthe eigenvalues with Reλ = 0 and Reλ < 0, respectively. Set mc = dimEc (Ec iscalled the center eigenspace) and ms = dimEs. We write

u = T£xy

¤, where x 2 Rmc , y 2 Rms ,

then the system (3.1) becomes

_x = Acx+ r1(x, y)

_y = Asy + r2(x, y), (3.2)

where ·r1(x, y)

r2(x, y)

¸= T¡1R(T

·x

y

¸).

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In this form Ec(0) = f(x, y) : y = 0g and Es(0) = f(x, y) : x = 0g. Note also

r1(0, 0) = 0, Jr1(0, 0) = 0

r2(0, 0) = 0, Jr2(0, 0) = 0.

Theorem 3.1 (Center Manifold Theorem) There exists a C1¡center man-ifold

W cloc(0) = f(x, y) : y = h(x), jxj < δ, h(0) = 0, Jh(0) = 0g

such that the dynamics of (3.2) restricted to the center manifold are given by

_x = Acx+ r1(x, h(x)).

There are a number of proofs of this result in the literature; some key ideasare as follows.

1. Find a change of variable ξ = y ¡ h(x) so that the system (3.2) becomes

_x = Acx+ f(x, ξ)_ξ = Asξ + ξg(x, ξ).

Since ξ = 0 is to be the invariant set, the dynamics on this set reduce to

_x = Acx + f (x, 0).

In order to …nd the change of variables we assume that all functions involvedin the original system are analytic and seek a power series representationfor h(x).

2. If we consider a surface y = h(x), then _y = Jh(x) _x and substituting from(3.2) we arrive at

Ash(x) + r2(x, h(x)) = Jh(x) [Acx + r1(x, h(x))] .

Expressed di¤erently we have

N(h) = Jh(x) [Acx + r1(x, h(x))]¡ Ash(x) ¡ r2(x, h(x)) = 0. (3.3)

This latter equation is quasi-linear and may be studied via the method ofcharacteristics.

Our approach to the center manifold is a blending of both ideas through thefollowing theorem due to Carr1 .

1J.Carr, Applications of Center Manifold Theory, Springer-Verlag (1981).

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Theorem 3.2 Let ψ : Rmc ! Rms be a C1 function with ψ(0) = 0 and Jψ(0) = 0.If N(ψ(x)) = O(jxjq) (jxj ! 0) for some q > 1, then

jh(x)¡ ψ(x)j = O(jxjq) (jxj ! 0).

Hence we can approximate the center manifold to any degree of approximationby solving the N-equation to the same degree of approximation. Power seriestechniques are particularly useful here.

Example 3.1 Consider the system of equations

_x = xy

_y = ¡y ¡ x2,

which is already in canonical form. The origin is the equilibrium point and

JX(0, 0) =

·0 00 ¡1

¸.

We look for y = h(x) = ax2 + bx3 + cx4 + dx5 + O(x6). Then

_y = h0(x) _x = xh0(x)h(x)

= 2a2x4 +5abx5 +O(x6)

and

_y = ¡h(x) ¡ x2

= ¡(a + 1)x2 ¡ bx3¡ cx4 ¡ dx5 +O(x6).

Comparing the two expressions we deduce that a = ¡1, b = 0, c = ¡2, d = 0,and our center manifold approximation is

y = h(x) = ¡x2 ¡ 2x4 + O(x6).

The dynamics are then governed by the equation

_x = ¡x3¡ 2x5 + O(x7).

For this last equation x = 0 is asymptotically stable. We conclude that (0, 0) isasymptotically stable for the original system.

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Example 3.2 Consider the system

_x = x2y ¡ x5

_y = ¡y+ x2.

Again (0, 0) is an equilibrium point and the system is in canonical form

JX(0, 0) =

·0 00 ¡1

¸.

Consider y = h(x) = ax2 + bx3 + O(x4). Then as in the previous example

_y = h0(x) _x

= 2a2x5 + [2a(b ¡ 1) + 3ab]x6 +O(x7)

and

_y = ¡h(x) + x2

= ¡(a ¡ 1)x2 ¡ bx3 +O(x4).

We deduce that a = 1 and b = 0. The center manifold is given by

y = h(x) = x2 +O(x4)

and the dynamics are governed by

_x = x4+ O(x5).

The later equation has x = 0 as unstable, hence, (0, 0) is unstable for the originalsystem.

Remark 3.1 One should note the following. If we approximate W cloc(0) in the

previous example by the tangent line Ec = f(x, y) : y = 0g, then one wouldclaim that the dynamics are governed by the equation _x = ¡x5. This would leadto the erroneous conclusion that the origin was stable. MORAL: tangent planeapproximation is not su¢cient.

Remark 3.2 Center manifolds are not unique. Consider the system

_x = x2

_y = ¡y.

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The obvious equilibrium is the origin and W s = fx = 0g, W c = fy = 0g. On theother hand, orbits of the system satisfy the ODE dy/dx = ¡y/x2 with generalsolution y(x) = ke1/x, where k is a constant. Thus the curves

y(x) =

½ke1/x if x < 0

0 if x ¸ 0,

form a one parameter family of center manifolds. Consequently, one can ask inusing power series expansions to approximate a center manifold, which centermanifold is being approximated? Theorem 3.2 indicates that the above exampleis generic, i.e., two center manifolds di¤er by order O(jxjq) for any q > 0. Thisalso says that the dynamics on two center manifolds will be the same.

Exercise 3.1 Study the dynamics near the origin via the center manifold for eachof the following.(a) _x = ¡x + y2, _y = ¡ sin x.(b) _x = 1/2x + y + x2y, _y = x+ 2y+ y2.(c) _x = ¡x ¡ y + z2, _y = 2x+ y ¡ z2, _z = x+ 2y ¡ z.

3.2 Parameter Dependant Systems

The above realm of thought also applies in situations where the system is para-meter dependant. We assume the form

_x = Acx + r1(x, y, ε)

_y = Asy + r2(x, y, ε), (3.4)

where ε 2 Rk (control parameters) and

r1(0, 0, 0) = 0, Jr1(0, 0, 0) = 0

r2(0, 0, 0) = 0, Jr2(0, 0, 0) = 0.

Note, the Jacobians here include derivatives in the ε¡variables. The matrices Ac

and As are as in the previous section and do not depend on the parameters ε.In order to apply the Center Manifold theorem, we enlarge the system to

include di¤erential equations for the parameters.

_x = Acx + r1(x, y, ε)

_ε = 0

_y = Asy + r2(x, y, ε). (3.5)

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Applying the theorem we obtain a local center manifold of form

W cloc(0, 0) = f(x, ε, y) : y = h(x, ε), jxj < δ, jεj < δ 0

h(0, 0) = 0, Jh(0, 0) = 0g

such that the dynamics of the above system and those of (3.4) reduce to

_x = Acx + r1(x, h(x, ε), ε)

_ε = 0.

In this case, the computation leading to the quasi-linear partial di¤erential equa-tion involves

_y = Jh,x(x, ε) _x + Jh,ε(x, ε)_ε.

Hence, this equation takes the same form as before:

N(h(x, ε)) = Dh,x(x, ε) [Acx+ r1(x, h(x, ε), ε)]¡ Ash(x, ε)¡ r2(x, h(x, ε), ε) = 0.

It is important to realize here that the analog of Theorem 3.2 is available here;indeed, takes the same form. Therefore, the power series techniques we used inprevious examples …nd a natural home in this parameter dependant case.

Example 3.3 (Lorenz) Consider the Lorenz equations written in the form

_x = σ(y ¡ x)

_y = ηx + x ¡ y ¡ xz

_z = ¡βz + xy.

Here η = ρ¡1, where ρ is the usual parameter in the Lorenz system. We consider σand β as …xed and are interested in the dynamics near η = 0. Simple computationgives the equilibria:

(¹x, ¹y, ¹z) =

½(0, 0, 0) if η 2 R(§p

βη,§pβη, η) if η > 0

.

We now perform the linearization of the system in the (x, y, z) variables, treatingη as an auxiliary variable, i.e., when η = 0,

JX(0, 0, 0) =

24

¡σ σ 01 ¡1 00 0 ¡β

35 ,

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whose eigenvalues are λ = 0, ¡(σ +1), ¡β, with corresponding eigenvectors24110

35 ,

24

σ¡10

35 ,

24001

35 .

We now put the system in canonical form using the matices

T =

241 σ 01 ¡1 00 0 1

35 , T ¡1 =

1

1 + σ

241 σ 01 ¡1 00 0 1 + σ

35

24

xyz

35 = T

24

uvw

35 =

24

u + σvu ¡ v

w

35 .

We have,24_u_v_w

35 =

240 0 00 ¡(1 + σ) 00 0 ¡β

35

24

uvw

35+ T¡1

24

0(η ¡ w)(u+ σv)(u+ σv)(u ¡ v)

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and carrying through the calculations, (3.5) in this case becomes:

_u = 0u+σ

1 + σ(η ¡ w)(u+ σv)

_η = 0·_v_w

¸=

· ¡(1 + σ) 00 ¡β

¸ ·vw

¸+

1

1 + σ

· ¡(η ¡ w)(u+ σv)(1 + σ)(u+ σv)(u ¡ v)

¸.

The center manifold will take the form

W cloc = f(u, v, w, η) : v = h1(u, η), w = h2(u, η),

hi(0, 0) = 0, Jhi(0, 0) = 0g.

The ideas of the two previous examples lead to:

∂h1∂u

σ

1 + σ(η ¡ h2) (u + σh1) = ¡(1 + σ)h1 +

(η ¡ h2) (u+ σh1)

1 + σ∂h2∂u

σ

1 + σ(η ¡ h2) (u + σh1) = ¡βh2 ¡ (u+ σh1)(u ¡ h1).

Substituting the power series form

h1(u, η) = a1u2 + a2uη+ a3η

2 +H.O.T

h2(u, η) = b1u2 + b2uη + b3η

2 +H.O.T

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and comparing the coe¢cients of u2 and uη gives

u2 :

½a1(1 + σ) = 0βb1 ¡ 1 = 0 , uη :

½a2(1 + σ) + 1

1+σ = 0βb2 = 0

.

Up to higher order terms the center manifolds are given by

v = ¡ 1

(1 + σ)2uη and w =

1

βu2,

and the dynamics near the origin are governed by the equations

_u =σ

1 + σ

µηu ¡ u3

β

¶

_η = 0

The equilibrium u = 0 is stable for η · 0 and unstable for η > 0. Hence there isan exchange of stability bifurcation at η = 0. The u equation exhibits a pitchforkbifurcation at η = 0 having two other stable equilibria u = §p

βη for η > 0.We say that the equilibria are captured on the center manifold w = u2/2. It isnatural to wonder if the higher order terms e¤ect this bifurcation. We will see inthe forthcoming discussion of normal forms that they do not.

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