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Chapter 6

Center manifold reduction

The previous chaper gave a rather detailed description of bifurcations of equilibriaand fixed points in generic one-parameter families of ODEs and maps with minimalpossible dimension of the state space. These results are also applicable to generaln-dimensional systems because of the existence of a low-dimensional invariant man-ifold near the bifurcation, on which all interesting dynamics in the state space isconcentrated.

The present chapter is devoted to the constructive definition of this invariantCenter Manifold and to a proof that reduction to it yields all the relevant informa-tion.

6.1 Center manifolds for maps

Consider a Ck-smooth map(

x

y

)

7→

(Bx + f(x, y)Cy + g(x, y)

)

, (6.1)

where x ∈ Rnc+nu , y ∈ Rns, and f(x, y) and g(x, y) have neither constant nor linearterms. Suppose that the (nc + nu) × (nc + nu) matrix B has nc eigenvalues with|λ| = 1 and nu eigenvalues with |λ| > 1, while all ns eigenvalues of the ns × ns

matrix C satisfy |λ| < 1. Note, that the eigenvalues with |λ| = 1 are often calledcritical eigenvalues. Let n = ns + nc + nu.

Theorem 6.1 (Existence of a Global Center-Unstable Manifold)Assume f(0, 0) = 0, g(0, 0) = 0, and that the functions f and g have sufficiently

small bounds and sufficiently small Lipschitz bounds Lip(f), Lip(g) on Rn. Then themap (6.1) has an invariant manifold

W cu = {(x, h(x)) : x ∈ Rnc+nu},

where h : Rnc+nu → Rns is a bounded and Lipschitz map satisfying h(0) = 0.If, in addition, f and g are Ck-functions (k ≥ 1) for which all derivatives up to

order k have small bounds and the k-th derivative has a small Lipschitz bound thenh is a Ck map satisfying hx(0) = 0.

241

242 CHAPTER 6. CENTER MANIFOLD REDUCTION

Definition 6.2 W cu is called a center–unstable manifold of the fixed point (0, 0)of (6.1).

Proof of Theorem 6.1:

(Step 1) We can assume that the norms on, respectively, Rns and Rnc+nu aresuch that

α = ‖C‖ < 1, β = ‖B−1‖ <1

α. (6.2)

Write (6.1) in the form

u 7→ Au + R(u) =

(P (x, y)Q(x, y)

)

, (6.3)

where

u =

(x

y

)

∈ Rnc+nu × R

ns, A =

(B 00 C

)

, R(u) =

(f(u)g(u)

)

.

On Rn we use the norm

‖u‖ = max{‖x‖, ‖y‖}.

By assumption we have constants K, L ≥ 0 such that

‖R(u)‖ ≤ K, ‖R(u) − R(v)‖ ≤ L‖u − v‖ (6.4)

for all u, v ∈ Rn. We impose the following smallness conditions

α + K < 1, β(α + 2L) < 1, α + (1 + β)L < 1. (6.5)

(Step 2) Introduce a set M0 of maps H : Rnc+nu → Rns , satisfying the followingconditions:

(i) H ∈ C(Rnc+nu , Rns) and

‖H‖∞ = supx∈Rnc+nu

‖H(x)‖ ≤ 1;

(ii) ‖H(x1) − H(x2)‖ ≤ ‖x1 − x2‖ for all x1,2 ∈ Rnc+nu ;

(iii) H(0) = 0.The set M0 is a complete metric space with respect to the distance ρ(H1, H2) =‖H1 − H2‖∞.

The invariance under the map (6.3)of the (Lipschitz) manifold

W ={(x, H(x)) : x ∈ R

nc+nu}

for given H ∈ M0 means that

Q(ξ, H(ξ)) = H(P (ξ, H(ξ))) (6.6)

6.1. CENTER MANIFOLDS FOR MAPS 243

H(x)

[T (H)](x)

H(ξ)

H

ξ

T (H)

0 x

Figure 6.1: Hadamard’s Graph Transform: x = P (ξ, H(ξ)) and [T (H)](x) =Q(ξ, H(ξ)).

should hold for all ξ ∈ Rnc+nu . Rewrite (6.6) as a fixed-point equation

T (H) = H,

where T is such that the graph of H is mapped to the graph of T (H) by the mapping(6.3), see Figure 6.1. Accordingly the graph of a fixed point of T is invariant under(6.3).

The formal definition of T proceeds as follows. For each x ∈ Rnc+nu and eachH ∈ M0, there is a unique ξ = S(x, H), such that

x = P (ξ, H(ξ)) = Bξ + f(ξ, H(ξ)). (6.7)

Indeed, from H ∈ M0 and (6.4) we have

‖f(ξ1, H(ξ1)) − f(ξ2, H(ξ2))‖ ≤ L‖ξ1 − ξ2‖,

and Theorem 3.20 applies, since

L < β−1 = ‖B−1‖−1.

From Theorem 3.20 we also obtain

‖S(x1, H) − S(x2, H)‖ ≤1

β−1 − L‖x1 − x2‖ (6.8)

for all x1, x2 ∈ Rnc+nu.Define now the Hadamard Graph Transform H 7→ T (H) by the formula

[T (H)](x) = Q(S(x, H), H(S(x, H)))

(see Figure 6.1) or, equivalently,

[T (H)](x) = CH(S(x, H)) + g(S(x, H), H(S(x, H))). (6.9)


Clearly, the equation H = T (H) is equivalent to (6.6), since they are related by thetransformation x = P (ξ, H(ξ)) with the inverse ξ = S(x, H).

(Step 3) We prove now that T (M0) ⊂ M0.(i) Using α + K < 1, we have

‖[T (H)](x)‖ ≤ ‖CH(S(x, H))‖+ ‖g(S(x, H), H(S(x, H)))‖

≤ α‖H‖∞ + K ≤ 1.

(ii) From (6.4) and (6.8) we obtain the Lipschitz estimate

‖[T (H)](x1) − [T (H)](x2)‖ ≤α

β−1 − L‖x1 − x2‖

+ L max(‖S(x1, H) − S(x2, H)‖,

‖H(S(x1, H)) − H(S(x2, H))‖)

≤α + L

β−1 − L‖x1 − x2‖.

Note that α + L ≤ β−1 − L follows from (6.5).(iii) Finally, it is obvious that S(0, H) = 0 and hence

[T (H)](0) = 0.

(Step 4) Now we verify that T is a contraction. Indeed, for any two H1,2 ∈ M0

and any x ∈ Rnc+nu, set

ξ1 = S(x, H1), ξ2 = S(x, H2).

Then

‖ξ1 − ξ2‖ = ‖B−1(f(ξ1, H1(ξ1)) − f(ξ2, H2(ξ2)))‖

≤ βL max(‖ξ1 − ξ2‖, ‖H1(ξ1) − H1(ξ2) + H1(ξ2) − H2(ξ2)‖)

≤ (α + L)

(

1 +βL

1 − βL

)

‖H1 − H2‖∞.

Thus,

‖ξ1 − ξ2‖ ≤βL

1 − βL‖H1 − H2‖∞.

With this estimate, (6.9) gives

‖T (H1) − T (H2)‖∞ ≤ (α + L)(‖ξ1 − ξ2‖ + ‖H1 − H2‖∞) ≤α + L

1 − βL‖H1 − H2‖∞.

This implies that T is a contraction, since

α + L

1 − βL< 1


This proves the existence of the unique global Lipschitz-continuous center-unstablemanifold given by the graph of h, where h ∈ M0 is the fixed point of T .

(Step 5) To prove that h ∈ Ck(Rnc+nu , Rns), introduce a set:

Mk = { H ∈ Ck(Rnc+nu , Rns) : H(0) = 0,supx∈Rnc+nu ‖Hx(j)(x)‖ ≤ 1, |j| = 0, 1, . . . , k,

‖Hx(j)(x) − Hx(j)(y)‖ ≤ ‖x − y‖ for all x, y ∈ Rnc+nu,

|j| = k },

which is a complete metric space with respect to the distance corresponding to thenorm

‖H‖k,∞ = max0≤|j|≤k

‖Hx(j)‖∞,

for each k = 0, 1, 2, . . .. Here x(j) = xj11 x

j22 · · ·xjn

n , |j| = j1 + j2 + · · ·+ jn, so that

Hx(j) =∂|j|H

∂xj11 ∂x

j22 · · ·∂x

jnn

.

We do not give the proof for general k, but indicate the main steps for the case k = 1.Let L1 = Lip(Ru) be a Lipschitz bound for the derivative. In the following we willimpose several smallness conditions on the constants K, L, L1 . First, note that theLipschitz Inverse Function Theorem 3.20 can be extended to show Ck-smoothnessof the inverse function if the given function is of class Ck.

Therefore, the function ξ = S(x, H) defined by (6.7) is Ck-smooth. Differentiat-ing (6.7) and suppressing the arguments ξ and (ξ, H(ξ)) we find (cf. (6.8))

In = (B + fx + fyHx)Sx. (6.10)

As before this yields

‖Sx‖ ≤ γ =1

β−1 − L. (6.11)

Then differentiation of [T (H)](x) leads to

T (H)x = CHxSx + gxSx + gyHxSx. (6.12)

Using (6.11) this implies the bound

‖T (H)x‖ ≤α + L

β−1 − L. (6.13)

Next we prove a Lipschitz estimate for Sx. Consider x1, x2 ∈ Rnc+nu and writeSj

x = Sx(xj , H), f jx = fx(S(xj, h), H(S(xj, H))), . . . for short. From (6.10) and (6.7)

we then obtain the estimate

∆ = ‖S1x − S2

x‖

= ‖B−1[f 1

x(S1x − S2

x) + (f 1x − f 2

x)S2x + f 1

y H1x(S

1x − S2

x)

+ f 1y (H1

x − H2x)S2

x + (f 1y − f 2

y )H2xf

2x

]‖

≤ β [L∆ + L1γ∆ + L1∆ + Lγ‖x1 − x2‖ + L1Lγ] .


Therefore,|Sx(x1, H) − Sx(x2, H)‖ ≤ LS,x‖x1 − x2‖, (6.14)

where LS,x = Lγ(β−1 −L−L1 − γL1(1 + L))−1. From this inequality and (6.12) weobtain a Lipschitz estimate for T (H)x

‖T (H)1x − T (H)2

x‖ = ‖CH1x(S1

x − S2x) + C(H1

x − H2x)S2

x

+ g1x(S

1x − S2

x) + (g1x − g2

x)S2x + g1

yH1x(S1

x − S2x)

+ g1y(H

1x − H2

x)S2x + (g1

y − g2y)H

2xS2

x‖

≤ (αLS,x + αγ + LLS,x + L1γ2 + LLS,x

+ Lγ + L1γ2)‖x1 − x2‖.

Since αγ < 1 this gives the Lipschitz constant 1 for L, L1 sufficiently small.In order to prove contraction we consider ξj = S(x, Hj), j = 1, 2 for two functions

H1,2 ∈ M1. Let us write S1x = Sx(x, H1), f

1x = fx(S

1, H1(S1)), . . . and use ‖f 1

x−f 2x‖ ≤

L1(1+γ)‖H1−H2‖∞. Then (6.10) leads to an estimate of ∆ = ‖S1x −S2

x‖ as follows

∆ = ‖B−1(f 1x(S1

x − S2x) + (f 1

x − f 2x)S2

x + f 1y H1,x(S

1x − S2

x)

+ f 1y (H1,x − H2,x)S

2x + (f 1

y − f 2y )H2,xS

2x‖

≤ β [L∆ + L1(1 + Lγ)‖H1 − H2‖∞ + L∆

+ Lγ‖H1 − H2‖∞ + L1(1 + Lγ)γ‖H1 − H2‖∞] ,

hence‖Sx(x, H1) − Sx(x, H2)‖ ≤ LS,H‖H1 − H2‖∞, (6.15)

where LS,H = (L1(1 + Lγ)(1 + γ) + Lγ) (β−1 − 2L)−1. Finally, we use αγ < 1 andarrive at a contraction with respect to ‖ · ‖∞

‖T (H1)x − T (H2)x‖ = ‖CH1,x(S1x − S2

x) + C(H1,x − H2,x)S2x + g1

x(S1x − S2

x)

+ (g1x − g2

x)S2x + g1

yH1,x(S1x − X2

x) + g1y(H1,x − H2,x)S

2x‖

≤ [αLS,H + αγ + 2LLS,H + 2L1(1 + Lγ)γ + Lγ] ‖H1 − H2‖1,∞.

(Step 6) To prove that hx(0) = 0, observe that from (6.6) now follows

Chx(0) − hx(0)B = 0.

Since σ(B) ∩ σ(C) = ∅, hx(0) = 0.2

Similar to our approach for the Grobman-Hartman Theorem in Chapter 3.3 wenow set up a local version of Theorem 6.1.

Theorem 6.3 (Existence of a Local Center-Unstable Manifold) Assume thatthe functions f and g in (6.1) are of class Ck+1 for some k ≥ 1 and satisfy

f(0, 0) = 0, g(0, 0) = 0, fu(0, 0) = 0, gu(0, 0) = 0.

Then there exists a Ck map h : Rnc+nu → Rns and an ε > 0 such that

W cuε = {(x, h(x)) : x ∈ R

nc+nu , ‖x‖ ≤ ε}

is conditionally invariant for the map (6.1). Moreover, h(0) = 0 and hx(0) = 0.


Remark: Recall from Chapter 3 that conditional invariance means that any point(x, h(x)) in W cu

ε with image (ξ, η) such that ‖ξ‖ ≤ ε satisfies (ξ, η) ∈ W cuε , i.e.

η = h(ξ).

Definition 6.4 The set W cuε is called a local center–unstable manifold of the

fixed point (0, 0) of (6.1).

Proof of Theorem 6.3: Instead of (6.3), consider a map

u 7→ Au + χ

(1

εu

)

R(u), (6.16)

where ε > 0 and χ ∈ C∞(Rn, R) is a standard “cut-off” function with χ(u) = 1 for0 ≤ ‖u‖ ≤ 1 and χ(u) = 0 for ‖u‖ ≥ 2. The map (6.16) coincides with (6.3) foru ∈ R

n satisfying ‖u‖ ≤ ε. With the scaling u 7→ εu the mapping (6.16) transformsinto

u 7→ Au + Rε(u), (6.17)

where

u =

(x

y

)

, Rε(u) = χ(u)1

εR(εu).

We apply Theorem 6.1 to this map and verify (6.4) for constants Kε, Lε that canbe made arbitrarily small. Note that ℓε := sup{‖Ru(v)‖ : ‖v‖ ≤ 2ε} → 0 as ε → 0and, by the mean value theorem,

‖Rε(u)‖ ≤ |χ(u)|‖

∫ 1

0

Ru(εtu)udt‖ ≤ 2‖χ‖∞ℓε =: Kε,

‖Rε,u(u)‖ = ‖χu(u)1

εR(εu) + χ(u)Ru(εu)‖ ≤ (‖χu‖∞ + ‖χ‖∞)ℓε =: Lε.

Therefore, the conditions (6.5) are satisfied for ε sufficiently small. Finally, with hε

being the fixed point from Theorem 6.1 corresponding to Rε set

h(x) = εhε

(1

εx

)

. (6.18)

Then global invariance of {(x, hε(x)); x ∈ Rnc+nu} yields conditional invariance ofW cu

ε . For the derivatives up to order k of Rε one obtains bounds that tend to 0 asε → 0. For example,

‖Rε,uu(u)‖ ≤ ℓε(‖χuu‖∞ + 2‖χu‖∞) + ε‖χ‖∞ sup{‖Ruu(v)‖ : ‖v‖ ≤ 2ε}.

Finally hx(0) = 0 directly follows from hε,x(0) = 0.2

Theorem 6.5 (Existence of a Center Manifold)Under the assumptions of Theorem 6.3, the map (6.1) has a locally defined invariantmanifold

W c = {(ξ, hc(ξ)) : ξ ∈ Rnc , ‖ξ‖ ≤ ǫ},

where ǫ > 0 is sufficiently small and hc : Rnc → Rns+nu is a Ck-map satisfying

hc(0) = 0, hcξ(0) = 0.


Proof: Applying Theorem 6.3 to a map that is inverse to the restriction of themap (6.1) to its invariant center-unstable manifold, we get a manifold W c with allmentioned above properties. 2

Definition 6.6 W c is called a center manifold of the fixed point (0, 0) of (6.1).

Remarks:(1) While the global center-unstable manifold in Theorem 6.1 is unique, this is nolonger true for the local center-unstable manifold. In general W cu

ε depends on ε andon the cut-off function. However, one can show that the derivatives of the possiblefunctions h agree at the origin up to the given order of differentiability. The sameremarks apply to the center manifold.(2) It can happen that ε → 0, as k → ∞. Thus, there are C∞ maps having noC∞ center manifolds. However, for analytic mappings analytic center manifolds doexist.(3) It is possible to close the smoothness gap in Theorem 6.3 and prove that h is infact a Ck+1 map. However, this needs a much more elaborate argument. 3

Theorem 6.7 (Reduction Principle) Consider a map(

ξ

η

)

7→

(Bξ + f(ξ, η),Cη + g(ξ, η)

)

, (6.19)

where ξ ∈ Rnc, η ∈ Rns+nu, the nc × nc matrix B has nc eigenvalues with |λ| = 1,while all eigenvalues of the (ns + nu) × (ns + nu) matrix C satisfy |λ| 6= 1, and thefunctions f and g have neither constant nor linear terms.

The map (6.19) is locally topologically conjugate near (0, 0) to the map(

ξ

η

)

7→

(Bξ + f(ξ, hc(ξ)),

Cη

)

, (6.20)

where hc represents the center manifold W c given by Theorem 6.5. 2

The maps for ξ and η are decoupled in (6.20). Therefore, the map (6.20) is locallytopologically conjugate (0, 0) to a map

(ξ

η

)

7→

(b(ξ)Cη

)

,

where ξ 7→ b(ξ) is any map that is locally topologically conjugate near ξ = 0 toξ 7→ Bξ+f(ξ, hc(ξ)). Indeed, the conjugating homeomorphism can be constructed asthe direct product of a conjugating homeomorphism in the ξ-space and the identitymap in the η-space.

Proposition 6.8 If ξ = 0 is a stable fixed point of the restriction of (6.19) to itscenter manifold

ξ 7→ Bξ + f(ξ, hc(ξ)), ξ ∈ Rnc ,

and nu = 0 (i.e. all eigenvalues of C satisfy |λ| < 1), then (ξ, η) = (0, 0) is a stablefixed point of (6.19).

6.2. CENTER MANIFOLDS FOR ODES 249

6.2 Center manifolds for ODEs

Consider a Ck-smooth system

{x = Bx + f(x, y),y = Cy + g(x, y),

(6.21)

where x ∈ Rnc+nu , y ∈ Rns, and f(x, y) and g(x, y) have neither constant nor linearterms. Suppose that the matrix B has nc critical eigenvalues (i.e. eigenvalues withRe λ = 0) and nu eigenvalues with Re λ > 0, while all ns eigenvalues of the matrixC satisfy Re λ < 0.

Theorem 6.9 (Existence of a global Center-Unstable Manifold)Assume that f(0, 0) = 0, g(0, 0) = 0 and that f and g have sufficiently small boundsand Lipschitz bounds on Rn. Then there exists an invariant manifold

W cu = {(x, h(x)) : x ∈ Rnc+nu},

where h : Rnc+nu 7→ R

ns is bounded, globally Lipschitz and satisfies h(0) = 0.Moreover, h is in Ck and satisfies hx(0) = 0 if f and g are Ck functions for somek ≥ 1 with small derivatives up to order k and with a small Lipschitz bound for thek-th derivatives.

Proof of Theorem 6.9:

(Step 1) Write (6.21) in the form

u = Au + r(u), (6.22)

where

u =

(x

y

)

∈ Rnc+nu × R

ns, A =

(B 00 C

)

, r(u) =

(f(u)g(u)

)

and assume‖r‖∞ ≤ κ, ‖r(u) − r(v)‖ ≤ ℓ‖u − v‖ (6.23)

for all u, v ∈ Rnu+nc . Since r has a global Lipschitz bound, the system generates aglobal solution flow Φt(u). Then define Rt(u) as the difference to the linearized flow

Φt(u) = etA + Rt(u) =

(etB 00 etC

)

+ Rt(u). (6.24)

We show that Theorem 6.1 applies to this map Φt(u) for all 0 < t ≤ 2.

(Step 2) By Lemma ?? there exist Lyapunov norms ‖ ·‖1 on Rns and ‖ ·‖2 on Rnc+nu

and numbers 0 < b < a such that

α(t) = ‖etC‖2 ≤ e−at, β(t) = ‖e−tB‖1 ≤ ebt, t ≥ 0. (6.25)


Clearly, α(t) < β(t)−1 for all t > 0. In what follows we use these adapted normsand their extension

‖u‖ = max(‖x‖1, ‖y‖2), u =

(x

y

)

.

(Step 3) From the variation of constants formula,

‖Rt(u)‖ = ‖

∫ t

o

e(t−s)Ar(Φs(u))ds‖ ≤ κ

∫ t

0

e(t−s)‖A‖ds = κet‖A‖ − 1

‖A‖= K(t).

Therefore, for κ sufficiently small

α(t) + K(t) ≤ e−at + κet‖A‖ − 1

‖A‖< 1, 0 < t ≤ 2.

Further note that the variation of constants formula implies

‖Φt(u) − Φt(v)‖ ≤ ‖etA(u − v)‖ + ‖

∫ t

0

e(t−s)A(r(Φs(u)) − r(Φs(v)))ds‖

≤ et‖A‖‖u − v‖ + ℓ

∫ t

0

e(t−s)‖A‖‖Φs(u) − Φs(v)‖ds,

which by Gronwall’s Lemma leads to

‖Φt(u) − Φt(v)‖ ≤ et(‖A‖+ℓ)‖u − v‖. (6.26)

In this way we obtain the Lipschitz estimate for Rt

‖Rt(u) − Rt(v)‖ = ‖

∫ t

0

e(t−s)A(r(Φs(u)) − r(Φs(v)))ds‖

≤

∫ t

0

e‖A‖(t−s)ℓ‖Φs(u) − Φs(v)‖ds

≤ et‖A‖(etℓ − 1)‖u − v‖ = L(t)‖u − v‖.

Then we can satisfy the second condition in (6.5)

β(t)(α(t) + 2L(t)) = e(b−a)t + 2et(‖A‖+b)(etℓ − 1) < 1

for all 0 < t ≤ 2 and for sufficiently small ℓ. Finally, note that the third conditionin (6.5) follows from the second since β(t) > 1.

(Step 4) By Theorem 6.1 the Hadamard graph transform Tt corresponding to Φt

has a unique fixed point Ht in M0 for 0 < t ≤ 2. Now we prove Ht = Hs for all0 < t, s ≤ 1. From the flow property of Φt one finds Tt ◦ Ts = Tt+s = Ts ◦ Tt for0 < t, s ≤ 1. Indeed Ts(Hs) = Hs and, therefore,

Tt(Hs) = Tt(Ts(Hs)) = Ts(Tt(Hs)).

6.2. CENTER MANIFOLDS FOR ODES 251

This means that Tt(Hs)) is a fixed point of Ts so that by uniqueness Tt(Hs) =Hs. Then the uniqueness of the fixed point of Tt gives Hs = Ht. Therefore, allthe functions Ht, 0 < t ≤ 1 coincide. This proves that the global center-unstablemanifold W = W cu defined by h = Ht is invariant under the flow Φt, 0 < t ≤ 1.Since

Φn = Φ1 ◦ Φ1 ◦ · · · ◦ Φ1︸︷︷︸

n times

,

the graph of h is invariant under Φn. For arbitrary t, we write Φt = Φ[t] ◦Φt−[t] andnote that the graph of h is invariant under both Φt−[t] and Φ[t]. We conclude thatwe have invariance under Φt without any restriction on t ≥ 0.

(Step 5) As in Step 5 of the proof of Theorem 6.1 we indicate how to obtain estimatesfor the derivative of Rt(u) with respect to u. The bound

‖Rtu(u)‖ ≤ L(t)

follows from the Lipschitz estimate of Rt above. It remains to establish the Lipschitzbound for Rt

u. Differentiating Rt(u) gives

Rtu(u) =

∫ t

0

e(t−s)Aru(Φs(u))Φs

u(u)ds. (6.27)

Let κ1 be a bound for ru and ℓ1 be a Lipschitz constant for ru. Similar to (6.26)one first establishes an estimate for Φt

u

‖Φt(u) − Φt(v)‖ ≤ (ℓ + ℓ1)et‖A‖ etℓ − 1

ℓ‖u − v‖.

Then the representation (6.27) gives

‖Rtu(u) − Rt

u(v)‖ ≤

∫ t

0

e(t−s)‖A‖

(

ℓ1es(‖A‖+ℓ) + κ1(ℓ1 + ℓ)es‖A‖ esℓ − 1

ℓ

)

ds‖u − v‖

≤ et‖A‖

(

ℓ1etℓ − 1

ℓ+ κ1

ℓ1 + ℓ

ℓ2(etℓ − tℓ − 1)

)

‖u − v‖.

This estimate shows that we can achieve a small Lipschitz constant. 2

Theorem 6.10 (Existence of a local Center-Unstable Manifold) Supposethat

f(0, 0) = 0, g(0, 0) = 0, fu(0, 0) = 0, gu(0, 0) = 0

and that f, g are of class Ck+1 with k ≥ 1 Then the system (6.21) has a conditionallyinvariant local center-unstable manifold

W cuε = {(x, h(x)) : x ∈ R

nc+nu , ‖x‖ ≤ ε},

where ε > 0 is sufficiently small and h is a Ck function.


Proof of Theorem 6.10:Take the cut-off function χ from the proof of Theorem 6.3 and replace (6.22) by

the scaled systemu = Au + rε(u),

where

rε(u) = χ(u)1

εr(εu).

The proof of Theorem 6.3 shows that rε, rε,u have small global bounds and Lipschitzbounds as well. Again the global center-unstable manifold of the cut-off system leadsto a conditionally invariant local center-unstable manifold of the original system(6.6). 2

As in the discrete-time case, we can prove the following result.

Theorem 6.11 (Existence of a Center Manifold) The system (6.21) has a lo-cally defined invariant manifold

W c = {(ξ, hc(ξ)) : ξ ∈ Rnc , ‖ξ‖ ≤ ǫ},

where ǫ > 0 is sufficiently small and hc : Rnc → R

ns+nu is a Ck-map satisfying

hc(0) = 0, hcξ(0) = 0.

2

Finally, we formulate without proof the Reduction Principle for ODEs.

Theorem 6.12 (Reduction Principle) Consider a system

{ξ = Bξ + f(ξ, η),η = Cη + g(ξ, η),

(6.28)

where ξ ∈ Rnc , η ∈ Rns+nu, the nc × nc matrix B has nc critical eigenvalues withRe λ = 0, while all eigenvalues of the (ns+nu)×(ns+nu) matrix C satisfy Re λ 6= 0,and the functions f and g are smooth and have neither constant nor linear terms.

The system (6.28) is locally topologically conjugate near (0, 0) to the system

{

ξ = Bξ + f(ξ, hc(ξ)),η = Cη.

(6.29)

2

The systems for ξ and η are decoupled in (6.29). Therefore, the system (6.29) islocally topologically conjugate (0, 0) to a system

{

ξ = b(ξ),η = Cη.

6.3. CRITICAL NORMAL FORMS ON CENTER MANIFOLDS 253

where ξ = b(ξ) is any system that is locally topologically conjugate near ξ = 0to ξ = Bξ + f(ξ, hc(ξ)). Moreover, the second equation in the last system can besubstituted by the standard saddle, i.e. the linear system

{ηs = −ηs, ηs ∈ Rns,

ηu = +ηu, ηu ∈ Rnu.

(6.30)

As for the discrete-time case, we have the following result.

Proposition 6.13 If ξ = 0 is a stable equilibrium of the restriction of (6.28) to itscenter manifold

ξ = Bξ + f(ξ, hc(ξ)), ξ ∈ Rnc ,

and all eigenvalues of C satisfy Re λ < 0, then (ξ, η) = (0, 0) is a stable equilibriumof (6.21).

6.3 Critical normal forms on center manifolds

We now address the problem of computing in practice the coefficients of normalforms of restrictions of multidimensional ODEs and maps to the corresponding crit-ical center manifolds at codim 1 bifurcations of equilibria and fixed points. Theresulting formulas are simple and allow one to perform all computations in theoriginal coordinates, without any preliminary transformation.

Consider a smooth system of ODEs

u = Au + F (u), u ∈ Rn, (6.31)

where the matrix A has nc eigenvalues with zero real parts, and write the Taylorexpansion for F at u0 = 0 as

F (u) =1

2B(u, u) +

1

6C(u, u, u) + O(‖u‖4),

where B : Rn × Rn → Rn and C : Rn × Rn × Rn → Rn are multilinear functionswith the components:

Bi(p, q) =

n∑

j,k=1

∂2Fi(0)

∂uj∂uk

pjqk, Ci(p, q, r) =

n∑

j,k,l=1

∂3Fi(0)

∂uj∂uk∂ul

pjqkrl,

for i = 1, 2, . . . , n.In what follows, we use some results from Linear Algebra. Given an n × n

complex matrix L, introduce two linear subspaces of Cn: the range

R(L) = {v ∈ Cn : v = Lu for some u ∈ C

n}

and the null-spaceN(L) = {w ∈ C

n : Lw = 0}.


We call two vectors u, v ∈ Cn orthogonal and write u ⊥ v, if their scalar productvanishes: 〈u, v〉 ≡ uT v = 0. Denote by L∗ the transposed matrix to the complex-

conjugate of L, i.e. L∗ = LT. If L is real, L∗ = LT . The matrix L∗ is called the

adjoint matrix for L. We have

〈u, Lv〉 = 〈L∗u, v〉

for all u, v ∈ Cn.

Lemma 6.14 (Fredholm’s Decomposition) Cn = R(L) ⊕ N(L∗) with R(L) ⊥

N(L∗), i.e., any vector x ∈ Cn can be uniquely decomposed as x = v + w withv ∈ R(L), w ∈ N(L∗), and 〈w, v〉 = 0.

Proof: Consider an orthogonal complement W of R(L) in Cn, so that Cn = R(L) ⊕ W andR(L) ⊥ W . We are going to prove that W = N(L∗).

(1) Suppose that w ∈ N(L∗) meaning L∗w = 0. For any v = Lu ∈ R(L), holds

〈w, v〉 = 〈w, Lu〉 = 〈L∗w, u〉 = 〈0, u〉 = 0.

Hence, w ∈ W .

(2) Suppose now that w ∈ W or 〈w, v〉 = 0 for v = Lu ∈ R(L) with any u ∈ Cn. Takeu = L∗w. Then we have

0 = 〈w, v〉 = 〈w, Lu〉 = 〈w, LL∗w〉 = 〈L∗w, L∗w〉 = ‖L∗w‖2.

Thus, L∗w = 0 and w ∈ N(L∗). 2

Lemma 6.14 implies that a linear system Lu = v has a solution if and only if〈w, v〉 = 0 for all w satisfying L∗w = 0. This is known as the Fredholm solvabilitycondition. If L is nonsingular, then L∗ is also nonsingular, so that N(L∗) = 0 andLu = v has a unique solution for any v ∈ R

n, u = L−1v. If L is singular, implyingthat both N(L) and N(L∗) are nontrivial, but v satisfies the Fredholm solvabilitycondition, then a solution u to Lu = v exists but is not unique. Indeed, u + ξ isanother solution for any ξ ∈ N(L).

Theorem 6.15 (Critical fold coefficient) Suppose λ1 = 0 is a simple eigenvalueof A and assume that it has no other critical eigenvalues. Introduce vectors q, p ∈ Rn,such that

Aq = 0, AT p = 0, 〈p, q〉 = 1.

Then the restriction of (6.31) to a one-dimensional center manifold W c(0) can bewritten in the form

ξ = bξ2 + O(|ξ|3), ξ ∈ R, (6.32)

where

b =1

2〈p, B(q, q)〉. (6.33)


Proof: Write (6.31) as

u = Au +1

2B(u, u) + O(‖u‖3), u ∈ R

n. (6.34)

The center manifold is one dimensional and can be represented as

u = ξq +1

2ξ2h2 + O(ξ3), ξ ∈ R, (6.35)

for some h2 ∈ Rn. Using (6.34),(6.35), and (6.32), we get

u = ξq + ξξh2 + . . . = bξ2q + ξ3h2 + . . .

and

u = Au +1

2B(u, u) + . . . =

1

2ξ2Ah2 +

1

2ξ2B(q, q) + . . .

Comparing the ξ2-terms we find

bq =1

2Ah2 +

1

2B(q, q)

or

Ah2 = 2bq − B(q, q).

This linear system for h2 is obviously singular but has a solution. The Fredholmsolvability condition implies then that

〈p, 2bq − B(q, q)〉 = 0,

from which we obtain (6.33). 2.

Theorem 6.16 (Critical Andronov-Hopf coefficient) Suppose λ1,2 = ±iω0, ω0 >

0, is a simple pair of purely imaginary eigenvalues of A and assume that it has noother critical eigenvalues. Introduce vectors q, p ∈ Cn, such that

Aq = iω0q, AT p = −iω0p, 〈p, q〉 = 1.

Then the restriction of (6.31) to a two-dimensional center manifold W c(0) can bewritten in the form

η = iω0η + c1η|η|2 + O(|η|4), η ∈ C, (6.36)

where

c1 =1

2〈p, C(q, q, q) − 2B(q, A−1B(q, q)) + B(q, (2iω0I − A)−1B(q, q))〉. (6.37)


Proof: Write (6.31) as

u = Au +1

2B(u, u) +

1

6C(u, u, u) + O(‖u‖4), u ∈ R

n. (6.38)

There is a two-dimensional center manifold that we can parametrize with η ∈ C:

u = ηq + ηq +1

2η2h20 + ηηh11 +

1

2η2h02 +

1

2η2ηh21 + . . . , (6.39)

where the dots denote all inessential terms. Here hij ∈ Cn. Using (6.38) and (6.36),we get by collecting the η2-terms in (6.38):

(2iω0I − A)h20 = B(q, q).

The matrix of this system is nonsingular, since 2iω0 is not an eigenvalue of A. Thus,

h20 = (2iω0I − A)−1B(q, q).

Collecting the ηη-terms gives another nonsingular system

Ah11 = −B(q, q)

orh11 = −A−1B(q, q).

The η2-terms lead to h02 = h20, while collecting the coefficients in front of theη2η-term yields the linear system:

(iω0I − A)h21 = C(q, q, q) + B(q, h20) + 2B(q, h11) − 2c1q.

This system is singular but has a solution. Thus, the Fredholm solvability conditionmust be satisfied:

〈p, C(q, q, q) + B(q, h20) + 2B(q, h11) − 2c1〉 = 0,

which gives (6.37). 2

The first Lyapunov coefficient is, therefore,

l1 =1

ω0Re c1.

Consider now a smooth map

u 7→ Au + F (u), u ∈ Rn, (6.40)

where the matrix A has nc critical eigenvalues satisfying |λ| = 1, and

F (u) =1

2B(u, u) +

1

6C(u, u, u) + O(‖u‖4),

is as in (6.31).


Theorem 6.17 (Critical fold coefficient for maps) Suppose λ1 = 1 is a simpleeigenvalue of A and assume that it has no other critical eigenvalues. Introducevectors q, p ∈ Rn, such that

Aq = q, AT p = p, 〈p, q〉 = 1.

Then the restriction of (6.40) to a one-dimensional center manifold W c(0) can bewritten in the form

ξ 7→ ξ + bξ2 + O(|ξ|3), ξ ∈ R, (6.41)

where

b =1

2〈p, B(q, q)〉. (6.42)

Proof: Write

f(H) = AH +1

2B(H, H) + O(‖H‖3),

and locally represent the center manifold W c as the graph of a function u =H(ξ), H : R → Rn, where

H(ξ) = ξq +1

2h2ξ

2 + O(ξ3), ξ ∈ R, h2 ∈ Rn.

The restriction of (6.40) to W c(0) is ξ 7→ G(ξ), where

G(ξ) = ξ + bξ2 + O(ξ3).

The invariance equation for the center manifold

f(H(ξ)) = H(G(ξ))

reads as

A(ξq+1

2h2ξ

2+· · ·)+1

2B(ξq+· · · , ξq+· · ·)+· · · = (ξ+bξ2+· · ·)q+

1

2h2(ξ+· · ·)2+· · ·

The ξ2-terms give the equation for h2:

(A − I)h2 = −B(q, q) + 2bq.

It is singular and its solvability implies (6.42). 2

Theorem 6.18 (Critical flip coefficient) Suppose λ1 = −1 is a simple eigen-value of A and assume that it has no other critical eigenvalues. Introduce vectorsq, p ∈ Rn, such that

Aq = −q, AT p = −p, 〈p, q〉 = 1.

Then the restriction of (6.40) to a one-dimensional center manifold W c(0) can bewritten in the normal form

ξ 7→ −ξ + cξ3 + O(|ξ|4), ξ ∈ R, (6.43)

where

c =1

6〈p, C(q, q, q)〉 −

1

2〈p, B(q, (A − I)−1B(q, q))〉. (6.44)


Proof: Expand

f(H) = AH +1

2B(H, H) +

1

6C(H, H, H) + O(‖H‖4),

and parametrize the center manifold W c(0) with u = H(ξ), where

H(ξ) = ξq +1

2h2ξ

2 +1

6h3ξ

3 + O(ξ4),

and ξ ∈ R, h2,3 ∈ Rn. The critical normal form is

ξ = G(ξ) = −ξ + cξ3 + O(ξ4).

The ξ2-terms in the invariance equation

f(H(ξ)) = H(G(ξ))

give for h2:(A − I)h2 = −B(q, q).

Since λ = 1 is not an eigenvalue of A, the matrix (A − I) is nonsingular. Thus,

h2 = −(A − I)−1B(q, q).

The ξ3-terms in the invariancy equation give the linear system for h3:

(A + I)h3 = 6cq − C(q, q, q) − 3B(q, h2).

This system is singular, since (A + I)q = 0, so it has a solution only if

〈p, 6cq − C(q, q, q) − 3B(q, h2)〉 = 0,

which implies

c =1

6〈p, C(q, q, q)〉+

1

2〈p, B(q, h2)〉.

Taking into account h2 = −(A − I)−1B(q, q), we obtain (6.44).

Theorem 6.19 (Critical Neimark-Sacker coefficient) Suppose that λ1,2 = e±iθ0,where

eikθ0 6= 1, k = 1, 2, 3, 4,

is a simple pair of purely imaginary eigenvalues of A and that A has no other criticaleigenvalues. Introduce vectors q, p ∈ Cn, such that

Aq = eiθ0q, AT p = e−iθ0p, 〈p, q〉 = 1.

Then the restriction of (6.40) to a two-dimensional center manifold W c(0) can bewritten in the form

η 7→ eiθ0η + c1η|η|2 + O(|η|4), η ∈ C, (6.45)

where

c1 =1

2〈p, C(q, q, q) + B(q, (e2iθ0I −A)−1B(q, q)) + 2B(q, (I −A)−1B(q, q))〉. (6.46)

6.4. FAMILIES OF CENTER MANIFOLDS 259

Proof: The invariancy of W c(0) represented as the graph of u = H(η, η) with η ∈ C

can be written in the form

f(H(η, η)) = H(G(η, η)), (6.47)

where

H(η, η) = ηq + η q +∑

1≤j+k≤3

1

j!k!hjkη

j ηk + O(|η|4),

f(H) = AH +1

2B(H, H) +

1

6C(H, H, H) + O(‖H‖4),

andG(η, η) = eiθ0η + c1η|η|

2 + O(|η|4).

Quadratic terms in (6.47) give

h20 = (e2iθ0I − A)−1B(q, q),

h11 = (I − A)−1B(q, q).

While the η2w-terms lead to the singular system

(eiθ0I − A)h21 = C(q, q, q) + B(q, h20) + 2B(q, h11) − 2c1q.

The solvability of this system is equivalent to

〈p, C(q, q, q) + B(q, h20) + 2B(q, h11) − 2c1q〉 = 0,

so the cubic normal form coefficient can indeed be expressed by (6.46). 2

Recall that the direction of the bifurcation of the closed invariant curve is deter-mined by the sign of

a = Re(e−iθ0c1).

6.4 Families of center manifolds

Consider a smooth parameter-dependent system of ODEs{

ξ = P (ξ, η, α),η = Q(ξ, η, α),

(6.48)

where ξ ∈ Rnc , η ∈ Rns+nu , α ∈ Rm, and suppose that (6.48) coincides with (6.28)at α = 0:

P (ξ, η, 0) = Bξ + f(ξ, η), Q(ξ, η, 0) = Cη + g(ξ, η).

Theorem 6.20 The system (6.48) has a family of invariant manifolds, locally rep-resentable for small ‖α‖ as

W cα = {(ξ, w(ξ, α) : ξ ∈ R

nc , ‖x‖ ≤ ε},

where ε > 0 is sufficiently small and the map

w : Rnc × R

m → Rns+nu

is smooth. Moreover, w(ξ, 0) = hc(ξ), i.e. W c0 coincides with a center manifold W c

from Theorem 6.11.


Proof: Consider the following extended system:

ξ = P (ξ, η, α),η = Q(ξ, η, α),α = 0,

(6.49)

where ξ ∈ Rnc , η ∈ Rns+nu, and α ∈ Rm. The equilibrium (ξ, η, α) = (0, 0, 0) of(6.49) is nonhyperbolic and has nc + m eigenvalues with Re λ = 0 (m of them areequal to zero). Theorem 6.11 guarantees local existence of a (nc + m)-dimensionalinvariant center manifold in (6.49). This manifold is the union of nc-dimensionalmanifolds W c

α located in the invariant linear subspeces α = const of (6.49). 2

Theorem 6.21 (Shoshitaishvilly, 1975) The system (6.48) is locally topologi-cally equivalent near (ξ, η, α) = (0, 0, 0) to the system

{ξ = P (ξ, w(ξ, α), α),η = Cη.

(6.50)

2

This theorem means that all “essential” events near the bifurcation parameter valueoccur on the invariant manifold W c

α and are captured by the nc-dimensional restrictedsystem:

ξ = P (ξ, w(ξ, α), α), ξ ∈ Rnc , α ∈ R

m. (6.51)

Obviously, this system can be substiututed in Theorem 6.21 by any smooth system

ξ = b(ξ, α), ξ ∈ Rnc , α ∈ R

m,

that is locally topologically equivalent to (6.51), while the second equation in (6.50)can be replaced by the standard saddle (6.30).

A theorem similar to Theorem 6.21 can be formulated for discrete-time dynamicalsystems generated by smooth maps.

6.5 Bifurcations of equilibria and cycles in n-dimensional

ODEs.

Let us apply Theorem 6.21 to the fold and Hopf bifurcations of equilibria in multi-dimensional systems.

6.5.1 Generic fold bifurcation in planar systems

Consider a smooth planar system

x = f(x, α), x ∈ R2, α ∈ R. (6.52)

Assume that at α = 0 it has the equilibrium x0 = 0 with one eigenvalue λ1 = 0and one eigenvalue λ2 < 0. Theorem 6.20 gives the existence of a smooth, locally

6.5. BIFURCATIONS OF EQUILIBRIA AND CYCLES IN N -DIMENSIONAL ODES.261

defined, one-dimensional attracting invariant manifold W cα for (6.52) for small |α|.

At α = 0, equation (6.51) has the form

ξ = bξ2 + O(ξ3).

If b 6= 0 and equation (6.51) depends generically on the parameter, then it is locallytopologically equivalent to the normal form

u = β + σu2,

where σ = sign b = ±1. Under these genericity conditions, Theorem 6.21 impliesthat (6.52) is locally topologically equivalent to the system

{u = β + σu2,

v = −v.(6.53)

Equations (6.53) are decoupled. The resulting phase portraits are presented inFigure 6.2 for the case σ > 0. For β < 0, there are two hyperbolic equilibria in the

β < 0 β = 0 β > 0

Figure 6.2: Fold bifurcation in the standard system (6.53) for σ = 1.

β(α) < 0 β(α) = 0 β(α) > 0

Figure 6.3: Fold bifurcation in a generic planar system.

u-axis: a stable node and a saddle. They collide at β = 0, forming a nonhyperbolicsaddle-node point, and disappear. There are no equilibria for β > 0. The sameevents happen in (6.52) on some one-dimensional, parameter-dependent, invariantmanifold, that is locally attracting (see Figure 6.3). All the equilibria belong to thismanifold. Figures 6.2 and 6.3 explain why the fold bifurcation is often called thesaddle-node bifurcation. It should be clear how to generalize these considerations tocover the case λ2 > 0, as well as the n-dimensional case.


6.5.2 Generic Andronov-Hopf bifurcation in three-dimensional

systems

Consider a smooth system

x = f(x, α), x ∈ R3, α ∈ R. (6.54)

Assume that at α = 0 it has the equilibrium x0 = 0 with eigenvalues λ1,2 =±iω0, ω0 > 0 and one negative eigenvalue λ3 < 0. Theorem 6.20 gives the exis-tence of a parameter-dependent, smooth, local two-dimensional attracting invariantmanifold W c

α of (6.54) for small |α|. At α = 0 the restricted equation (6.51) can bewritten in complex form as

z = iω0z + g(z, z), z ∈ C,

where g = O(|z|2). If the Lyapunov coefficient l1(0) of this equation is nonzeroand (6.51) depends generically on the parameter, then it is locally topologicallyequivalent to the normal form

z = (β + i)z + σz2z,

where σ = sign l1(0) = ±1. Under these genericity conditions, Theorem 6.21

β > 0β = 0β < 0

Figure 6.4: Hopf bifurcation in the standard system (6.55) for σ = −1.

β(α) > 0β(α) = 0β(α) < 0

Figure 6.5: Supercritical Hopf bifurcation in a generic three-dimensional system.

6.5. BIFURCATIONS OF EQUILIBRIA AND CYCLES IN N -DIMENSIONAL ODES.263

implies that (6.54) is locally topologically equivalent to the system{

z = (β + i)z + σz2z,

v = −v.(6.55)

The phase portrait of (6.55) is shown in Figure 6.2 for σ = −1. The supercriticalHopf bifurcation takes place in the invariant plane v = 0, which is attracting. Thesame events happen for (6.54) on some two-dimensional attracting manifold (seeFigure 6.5). The construction can be generalized to arbitrary dimension n ≥ 3.

A combination of the Poincare map and the center manifold reduction allows usto describe bifurcations of limit cycles in generic n-dimensional ODEs depending onone parameter.

Let L0 be a limit cycle of a smooth system

x = f(x, α), x ∈ Rn, α ∈ R, (6.56)

at α = 0. Let P(α) denote the associated Poincare map for nearby α; P(α) : Σ → Σ,where Σ is a local cross-section to L0. If some coordinates ξ = (ξ1, ξ2, . . . , ξn−1)are introduced on Σ, then ξ = P(α)(ξ) can be defined to be the point of the nextintersection with Σ of the orbit of (6.56) having initial point with coordinates ξ onΣ. The intersection of Σ and L0 gives a fixed point ξ0 for P(0): P(0)(ξ0) = ξ0. As weknow, map P(α) is smooth and locally invertible.

Suppose that the cycle L0 is nonhyperbolic, having n0 multipliers on the unitcircle. The center manifold theorems then give a parameter-dependent invariantmanifold W c

α ⊂ Σ of P(α) on which the “essential” events take place. The Poincaremap P(α) is locally topologically equivalent to the suspension of its restriction tothis manifold by the standard saddle map.

Fix n = 3, for simplicity, and consider the implications of this result for the limitcycle bifurcations in R3.

6.5.3 Fold bifurcation of limit cycles in R3

Assume that at α = 0 the cycle has a simple multiplier µ1 = 1 and its othermultiplier satisfies 0 < µ2 < 1. The restriction of P(α) to the invariant manifold W c

α

is a one-dimensional map, having a fixed point with µ1 = 1 at α = 0. As has beenshown, this generically implies the collision and disappearance of two fixed pointsof P(α) as α passes through zero. Under our assumption on µ2, this happens on aone-dimensional attracting invariant manifold of P(α); thus, a stable and a saddlefixed point are involved in the bifurcation (see Figure 6.6 for an illustration). Eachfixed point of the Poincare map corresponds to a limit cycle of the continuous-timesystem. Therefore, two limit cycles (stable and saddle) collide and disappear insystem (6.56) at this fold bifurcation of cycles.

6.5.4 Flip (period-doubling) bifurcation of limit cycles in R3

Suppose that at α = 0 the cycle has a simple multiplier µ1 = −1, while −1 < µ2 < 0.Then, the restriction of P(α) to the invariant manifold will demonstrate generically


α < 0

L1 L2 L0

α > 0α = 0

Figure 6.6: Fold bifurcation of limit cycles.

α > 0α = 0α < 0

L1

L0L0L0

Figure 6.7: Flip bifurcation of limit cycles.

L0L0 L0

α > 0α = 0α > 0

T2

Figure 6.8: Neimark-Sacker bifurcation of a limit cycle.

6.6. REFERENCES 265

the period-doubling (flip) bifurcation: A cycle of period-2 appears or disappearsfor the map, while the fixed point changes its stability (see Figure 6.7, where thesupercritical case is illustrated). Since the manifold is attracting, the stable fixedpoint, for example, loses stability and becomes a saddle point, while a stable cycleof period-2 appears. The fixed points correspond to limit cycles of the relevantstability. The cycle of period-two points for the map corresponds to a unique stablelimit cycle in (6.56) with approximately twice the period of the “basic” cycle L0.The double-period cycle makes two big “excursions” near L0 before the closure.The exact bifurcation scenario is determined by the normal form coefficient of therestricted Poincare map evaluated at α = 0.

6.5.5 Neimark-Sacker (torus) bifurcation of limit cycles inR3

The last codim 1 bifurcation corresponds to the case when the multipliers are com-plex and simple and lie on the unit circle: µ1,2 = e±iθ0 . The Poincare map P(α)

then has a parameter-dependent, two-dimensional, invariant manifold on which aclosed invariant curve generically bifurcates from the fixed point (see Figure 6.8,where the supercritical bifurcation is shown). This closed curve corresponds to atwo-dimensional invariant torus T2 in (6.56). The bifurcation is determined bythe normal form coefficient of the restricted Poincare map at the critical parametervalue. The orbit structure on the torus T2 depends on the restriction of the Poincaremap to this closed invariant curve. Thus, generically, there are long-period cycles ofdifferent stability types located on the torus, which appear and disappear pair-wisevia fold bifurcations.

6.6 References

Bifurcations of stationary points and periodic orbits in one- and two-parameter fam-ilies of multidimensional ODEs and maps are treated in many textbooks, including[Arnol’d 1983, Guckenheimer & Holmes 1983, Arrowsmith & Place 1990, Shilnikovet al. 2001, Wiggins 2003, Kuznetsov 2004]. A useful summary is given in [Arnol’det al. 1994], while many technical issues are clarified in [Iooss 1979, Vanderbauwhede1989, Iooss & Adelmeyer 1992]. For an alternative approach to the bifurcation the-ory based on the Lyapunov-Schmidt reduction, see [Chow & Hale 1982, Iooss &Joseph 1990, Kielhofer 2004]

A direct proof of the existence of a local center manifold near a nonhyperbolicequilibrium in ODEs, that does not depend on the corresponding result for maps, isgiven in [Carr 1981]; a proof of Theorem 6.12 (Reduction Principle for ODEs) canbe found in [Kirchgraber & Palmer 1990].

Numerical methods for bifurcations of stationary points and periodic orbits inmultidimensional ODEs and maps are summarized in [Beyn, Champneys, Doedel,Govaerts, Kuznetsov & Sandstede 2002].


6.7 Exercises

E 6.7.1 (Andronov-Hopf bifurcation in 3D systems)

Check that each of the following feedback control systems1 has an equilibrium that exhibits anAndronov-Hopf bifurcation at µ = 0, and compute the first Lyapunov coefficient of the restrictedsystem on the center manifold:

(a)

x = µx − y,

y = µy + x + xz,

z = −z + x2.

(b)

x = µx − y − xz,

y = µy + x,

z = −z + y2 + x2z.

E 6.7.2 (Pitchfork bifurcation in Lorenz system)

Compute the second-order approximation to the family of one-dimensional center manifolds of theLorenz system2

x = −σx + σy,

y = −xz + rx − y,

z = xy − bz,

(6.57)

near the origin (x, y, z) = (0, 0, 0) for fixed (σ, b) and r close to r0 = 1. Then, calculate therestricted system up to third-order terms in ξ and analyse its bifurcation.

E 6.7.3 (Andronov-Hopf bifurcation in Lorenz system)

(a) Show that for fixed b > 0, σ > b + 1, and

r1 =σ(σ + b + 3)

σ − b − 1, (6.58)

a nontrivial equilibrium of (6.57) exhibits an Andronov-Hopf bifurcation.(b) Prove that this bifurcation is subcritical and, therefore, gives rise to a unique saddle limit

cycle for r < r1

(Hints: [Shilnikov et al. 2001, pp. 877–880](i) Write (6.57) as a single third-order equation

···

x +(σ + b + 1)x + b(1 + σ)x + bσ(1 − r)x =(1 + σ)x2

x+

xx

x− x2x − σx3.

(ii) Translate the origin to the equilibrium by introducing the new coordinate ξ = x − x0,where x0 =

√

b(r − 1), thus obtaining the equation

···

ξ +(σ + b + 1)ξ + [b(1 + σ) + x2

0]ξ + [bσ(1 − r) + 3σx2

0]ξ = f(ξ, ξ, ξ), (6.59)

where

f(ξ, ξ, ξ) = −3σx0ξ2 − 2x0ξξ +

1 + σ

x0

ξ2 +1

x0

ξξ − σξ3 − ξ2ξ −1 + σ

x2

0

ξξ2 −1

x2

0

ξξξ + · · ·

1Moon, F.C. and Rand, R.H. ‘Parametric stiffness control of flexible structures’, In: Proceedings

of the Workshop on Identification and Control of Flexible Space Structures, Vol. II, Jet PropulsionLaboratory Publication 85-29, Pasadena, CA, 1985, pp. 329-342.

2Lorenz, E. ‘Deterministic non-periodic flow’, J. Atmos. Sci. 20 (1963), 130-141.

6.7. EXERCISES 267

and the dots stand for all higher-order terms in (ξ, ξ, ξ).(iii) Rewrite (6.59) as a system

U = AU + F (U), U = (ξ, ξ, ξ)T ∈ R3. (6.60)

Find the eigenvector and the adjoint eigenvector of A corresponding to its purely imaginary eigen-values (when (6.58) is satisfied).

(iv) Compute the first Lyapunov coefficient l1 for (6.60) using (6.37). Substitute σ = σ∗+b+1and show that l1 is positive for all positive σ∗ and b.)

E 6.7.4 (No Neimark-Sacker bifurcation bifurcation in Lorenz system)

Prove that the Neimark-Sacker bifurcation of a limit cycle never occurs in (6.57), provided that(σ, r, b) are all positive.(Hint: Use the formula for the multiplier product and the fact that div f = −(σ + b + 1) < 0,where f is the vector field given by the right-hand side of (6.57).)

E 6.7.5 (Fold and flip bifurcations in Henon map)

Consider the Henon map3: (x

y

)

7→

(y

α − βx − y2

)

. (6.61)

(a) Find equations for the fold and flip bifurcation curves of fixed points in (6.61).(b) Prove that found fold and flip bifurcations, occuring in (6.61) under variation of parameter

α, are nondegenerate for fixed β 6= ±1.

E 6.7.6 (Duopoly model of Kopel)

Consider the following Kopel map from mathematical economics4:(

x

y

)

7→

((1 − ρ)x + ρµy(1 − y)(1 − ρ)y + ρµx(1 − x)

)

, (6.62)

where (µ, ρ) are positive parameters.(a) Find equations for period-doubling and Neimark-Sacker bifurcations of nonnegative fixed

points in (6.62).(b) Study the nondegeneracy of these bifurcations by computing the corresponding normal

forms.(c) Compute numerically bifurcation curves of fixed points, 2- and 4-cycles in the parameter

domain2.9 ≤ µ ≤ 3.8, 0.75 ≤ ρ ≤ 1.4 .

E 6.7.7 (Flip and Neimark-Sacker bifurcations in an adaptive control map)

(a) Demonstrate that the fixed point (x0, y0, z0) = (1, 1, 1−b−k) of the discrete-time dynamicalsystem5

x

y

z

7→

y

bx + k + yz

z −ky

c + y2(bx + k + zy − 1)

3Henon, M. ‘A two-dimensional mapping with a strange attractor’, J Comm. Math. Phys. 50(1976), 69-77.

4Kopel, M. ‘Simple and complex adjustment dynamics in Cournot duopoly models’, Chaos,

Solitons & Fractals, 12 (1996), 2031-2048.5Golden, M.P. and Ydstie, B.E. ‘Bifurcation in model reference adaptive control systems’,

Systems Control Lett. 11 (1988), 413-430.


exhibits a flip bifurcation at

bF = 1 −

[1

2+

1

4(c + 1)

]

k,

and a Neimark-Sacker bifurcation at

bNS = −c + 1

c + 2.

(b) Determine the direction of the period-doubling bifurcation that occurs as b increases andpasses through bF .

(c) Show that the Neimark-Sacker bifurcation in the system under variation of the parameterb can be either sub- or supercritical depending on the values of the parameter (c, k).

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