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

Transcription

1 Chapter 6 Center manifold reduction The previous chaper gave a rather detailed description of bifurcations of equilibria and fixed points in generic one-parameter families of ODEs and maps with minimal possible dimension of the state space. These results are also applicable to general n-dimensional systems because of the existence of a low-dimensional invariant manifold near the bifurcation, on which all interesting dynamics in the state space is concentrated. The present chapter is devoted to the constructive definition of this invariant Center Manifold and to a proof that reduction to it yields all the relevant information. 6.1 Center manifolds for maps Consider a C k -smooth map ( x y ) ( Bx + f(x, y) Cy + g(x, y) ), (6.1) where x R nc+nu, y R ns, and f(x, y) and g(x, y) have neither constant nor linear terms. Suppose that the (n c + n u ) (n c + n u ) matrix B has n c eigenvalues with λ = 1 and n u eigenvalues with λ > 1, while all n s eigenvalues of the n s n s matrix C satisfy λ < 1. Note, that the eigenvalues with λ = 1 are often called critical eigenvalues. Let n = n s + n c + n u. 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 R n. Then the map (6.1) has an invariant manifold W cu = {(x, h(x)) : x R nc+nu }, where h : R nc+nu R ns is a bounded and Lipschitz map satisfying h(0) = 0. If, in addition, f and g are C k -functions (k 1) for which all derivatives up to order k have small bounds and the k-th derivative has a small Lipschitz bound then h is a C k map satisfying h x (0) =

2 242 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, R ns and R nc+nu are such that α = C < 1, β = B 1 < 1 α. (6.2) Write (6.1) in the form u Au + R(u) = ( P(x, y) Q(x, y) ), (6.3) where u = ( x y ) R nc+nu R ns, A = ( B 0 0 C ) ( f(u), R(u) = g(u) ). On R n 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 R n. 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 M 0 of maps H : R nc+nu R ns, satisfying the following conditions: (i) H C(R nc+nu, R ns ) and H = sup H(x) 1; x R nc+nu (ii) H(x 1 ) H(x 2 ) x 1 x 2 for all x 1,2 R nc+nu ; (iii) H(0) = 0. The set M 0 is a complete metric space with respect to the distance ρ(h 1, H 2 ) = H 1 H 2. The invariance under the map (6.3)of the (Lipschitz) manifold for given H M 0 means that W = { (x, H(x)) : x R nc+nu} Q(ξ, H(ξ)) = H(P(ξ, H(ξ))) (6.6)

3 6.1. CENTER MANIFOLDS FOR MAPS 243 H T(H) H(x) [T(H)](x) H(ξ) 0 ξ 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 R nc+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 R nc+nu and each H M 0, there is a unique ξ = S(x, H), such that Indeed, from H M 0 and (6.4) we have and Theorem 3.20 applies, since From Theorem 3.20 we also obtain x = P(ξ, H(ξ)) = Bξ + f(ξ, H(ξ)). (6.7) f(ξ 1, H(ξ 1 )) f(ξ 2, H(ξ 2 )) L ξ 1 ξ 2, L < β 1 = B 1 1. S(x 1, H) S(x 2, H) 1 β 1 L x 1 x 2 (6.8) for all x 1, x 2 R nc+nu. Define now the Hadamard Graph Transform H T(H) by the formula (see Figure 6.1) or, equivalently, [T(H)](x) = Q(S(x, H), H(S(x, H))) [T(H)](x) = CH(S(x, H)) + g(s(x, H), H(S(x, H))). (6.9)

4 244 CHAPTER 6. CENTER MANIFOLD REDUCTION Clearly, the equation H = T(H) is equivalent to (6.6), since they are related by the transformation x = P(ξ, H(ξ)) with the inverse ξ = S(x, H). (Step 3) We prove now that T(M 0 ) M 0. (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)](x 1 ) [T(H)](x 2 ) α β 1 L x 1 x 2 + L max( S(x 1, H) S(x 2, H), Note that α + L β 1 L follows from (6.5). (iii) Finally, it is obvious that S(0, H) = 0 and hence H(S(x 1, H)) H(S(x 2, H)) ) α + L β 1 L x 1 x 2. [T(H)](0) = 0. (Step 4) Now we verify that T is a contraction. Indeed, for any two H 1,2 M 0 and any x R nc+nu, set Then ξ 1 = S(x, H 1 ), ξ 2 = S(x, H 2 ). ξ 1 ξ 2 = B 1 (f(ξ 1, H 1 (ξ 1 )) f(ξ 2, H 2 (ξ 2 ))) βl max( ξ 1 ξ 2, H 1 (ξ 1 ) H 1 (ξ 2 ) + H 1 (ξ 2 ) H 2 (ξ 2 ) ) ( (α + L) 1 + βl ) H 1 H 2. 1 βl Thus, With this estimate, (6.9) gives ξ 1 ξ 2 βl 1 βl H 1 H 2. T(H 1 ) T(H 2 ) (α + L)( ξ 1 ξ 2 + H 1 H 2 ) α + L 1 βl H 1 H 2. This implies that T is a contraction, since α + L 1 βl < 1

5 6.1. CENTER MANIFOLDS FOR MAPS 245 This proves the existence of the unique global Lipschitz-continuous center-unstable manifold given by the graph of h, where h M 0 is the fixed point of T. (Step 5) To prove that h C k (R nc+nu, R ns ), introduce a set: M k = { H C k (R nc+nu, R ns ) : H(0) = 0, sup x R nc+nu H x (j)(x) 1, j = 0, 1,..., k, H x (j)(x) H x (j)(y) x y for all x, y R nc+nu, j = k }, which is a complete metric space with respect to the distance corresponding to the norm H k, = max H x (j), 0 j k for each k = 0, 1, 2,.... Here x (j) = x j 1 1 x j 2 2 x jn n, j = j 1 + j j n, so that H x (j) = j H x j 1 1 x j 2 2 x jn n We do not give the proof for general k, but indicate the main steps for the case k = 1. Let L 1 = Lip(R u ) be a Lipschitz bound for the derivative. In the following we will impose several smallness conditions on the constants K, L, L 1. First, note that the Lipschitz Inverse Function Theorem 3.20 can be extended to show C k -smoothness of the inverse function if the given function is of class C k. Therefore, the function ξ = S(x, H) defined by (6.7) is C k -smooth. Differentiating (6.7) and suppressing the arguments ξ and (ξ, H(ξ)) we find (cf. (6.8)) As before this yields. I n = (B + f x + f y H x )S x. (6.10) S x γ = Then differentiation of [T(H)](x) leads to Using (6.11) this implies the bound 1 β 1 L. (6.11) T(H) x = CH x S x + g x S x + g y H x S x. (6.12) T(H) x α + L β 1 L. (6.13) Next we prove a Lipschitz estimate for S x. Consider x 1, x 2 R nc+nu and write Sx j = S x (x j, H), fx j = f x (S(x j, h), H(S(x j, H))),... for short. From (6.10) and (6.7) we then obtain the estimate = S 1 x S 2 x = B 1 [ f 1 x(s 1 x S 2 x) + (f 1 x f 2 x)s 2 x + f 1 yh 1 x(s 1 x S 2 x) + f 1 y (H1 x H2 x )S2 x + (f1 y f2 y )H2 x f2 x] β [L + L 1 γ + L 1 + Lγ x 1 x 2 + L 1 Lγ].

6 246 CHAPTER 6. CENTER MANIFOLD REDUCTION Therefore, S x (x 1, H) S x (x 2, H) L S,x x 1 x 2, (6.14) where L S,x = Lγ(β 1 L L 1 γl 1 (1 + L)) 1. From this inequality and (6.12) we obtain a Lipschitz estimate for T(H) x T(H) 1 x T(H)2 x = CH1 x (S1 x S2 x ) + C(H1 x H2 x )S2 x + g 1 x(s 1 x S 2 x) + (g 1 x g 2 x)s 2 x + g 1 yh 1 x(s 1 x S 2 x) + g 1 y (H1 x H2 x )S2 x + (g1 y g2 y )H2 x S2 x (αl S,x + αγ + LL S,x + L 1 γ 2 + LL S,x + Lγ + L 1 γ 2 ) x 1 x 2. Since αγ < 1 this gives the Lipschitz constant 1 for L, L 1 sufficiently small. In order to prove contraction we consider ξ j = S(x, H j ), j = 1, 2 for two functions H 1,2 M 1. Let us write Sx 1 = S x (x, H 1 ), fx 1 = f x (S 1, H 1 (S 1 )),... and use fx f 1 x 2 L 1 (1+γ) H 1 H 2. Then (6.10) leads to an estimate of = Sx 1 S2 x as follows hence = B 1 (f 1 x (S1 x S2 x ) + (f1 x f2 x )S2 x + f1 y H 1,x(S 1 x S2 x ) + f 1 y(h 1,x H 2,x )S 2 x + (f 1 y f 2 y)h 2,x S 2 x β [L + L 1 (1 + Lγ) H 1 H 2 + L + Lγ H 1 H 2 + L 1 (1 + Lγ)γ H 1 H 2 ], S x (x, H 1 ) S x (x, H 2 ) L S,H H 1 H 2, (6.15) where L S,H = (L 1 (1 + Lγ)(1 + γ) + Lγ) (β 1 2L) 1. Finally, we use αγ < 1 and arrive at a contraction with respect to T(H 1 ) x T(H 2 ) x = CH 1,x (S 1 x S2 x ) + C(H 1,x H 2,x )S 2 x + g1 x (S1 x S2 x ) + (g 1 x g2 x )S2 x + g1 y H 1,x(S 1 x X2 x ) + g1 y (H 1,x H 2,x )S 2 x [αl S,H + αγ + 2LL S,H + 2L 1 (1 + Lγ)γ + Lγ] H 1 H 2 1,. (Step 6) To prove that h x (0) = 0, observe that from (6.6) now follows Since σ(b) σ(c) =, h x (0) = 0. Ch x (0) h x (0)B = 0. Similar to our approach for the Grobman-Hartman Theorem in Chapter 3.3 we now set up a local version of Theorem 6.1. Theorem 6.3 (Existence of a Local Center-Unstable Manifold) Assume that the functions f and g in (6.1) are of class C k+1 for some k 1 and satisfy f(0, 0) = 0, g(0, 0) = 0, f u (0, 0) = 0, g u (0, 0) = 0. Then there exists a C k map h : R nc+nu R ns and an ε > 0 such that W cu ε = {(x, h(x)) : x R nc+nu, x ε} is conditionally invariant for the map (6.1). Moreover, h(0) = 0 and h x (0) = 0.

7 6.1. CENTER MANIFOLDS FOR MAPS 247 Remark: Recall from Chapter 3 that conditional invariance means that any point (x, h(x)) in Wε cu with image (ξ, η) such that ξ ε satisfies (ξ, η) Wε cu, i.e. η = h(ξ). Definition 6.4 The set Wε cu fixed point (0, 0) of (6.1). is called a local center unstable manifold of the Proof of Theorem 6.3: Instead of (6.3), consider a map ( ) 1 u Au + χ ε u R(u), (6.16) where ε > 0 and χ C (R n, R) is a standard cut-off function with χ(u) = 1 for 0 u 1 and χ(u) = 0 for u 2. The map (6.16) coincides with (6.3) for u R n satisfying u ε. With the scaling u εu the mapping (6.16) transforms into u Au + R ε (u), (6.17) where u = ( x y ), R ε (u) = χ(u) 1 ε R(εu). We apply Theorem 6.1 to this map and verify (6.4) for constants K ε, L ε that can be made arbitrarily small. Note that l ε := sup{ R u (v) : v 2ε} 0 as ε 0 and, by the mean value theorem, R ε (u) χ(u) 1 0 R u (εtu)udt 2 χ l ε =: K ε, R ε,u (u) = χ u (u) 1 ε R(εu) + χ(u)r u(εu) ( χ u + χ )l ε =: L ε. Therefore, the conditions (6.5) are satisfied for ε sufficiently small. Finally, with h ε being the fixed point from Theorem 6.1 corresponding to R ε set h(x) = εh ε ( 1 ε x ). (6.18) Then global invariance of {(x, h ε (x)); x R nc+nu } yields conditional invariance of Wε cu. For the derivatives up to order k of R ε one obtains bounds that tend to 0 as ε 0. For example, R ε,uu (u) l ε ( χ uu + 2 χ u ) + ε χ sup{ R uu (v) : v 2ε}. Finally h x (0) = 0 directly follows from h ε,x (0) = 0. Theorem 6.5 (Existence of a Center Manifold) Under the assumptions of Theorem 6.3, the map (6.1) has a locally defined invariant manifold W c = {(ξ, h c (ξ)) : ξ R nc, ξ ǫ}, where ǫ > 0 is sufficiently small and h c : R nc R ns+nu is a C k -map satisfying h c (0) = 0, h c ξ (0) = 0.

8 248 CHAPTER 6. CENTER MANIFOLD REDUCTION Proof: Applying Theorem 6.3 to a map that is inverse to the restriction of the map (6.1) to its invariant center-unstable manifold, we get a manifold W c with all mentioned above properties. 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 no longer true for the local center-unstable manifold. In general Wε cu depends on ε and on the cut-off function. However, one can show that the derivatives of the possible functions h agree at the origin up to the given order of differentiability. The same remarks apply to the center manifold. (2) It can happen that ε 0, as k. Thus, there are C maps having no C center manifolds. However, for analytic mappings analytic center manifolds do exist. (3) It is possible to close the smoothness gap in Theorem 6.3 and prove that h is in fact a C k+1 map. However, this needs a much more elaborate argument. Theorem 6.7 (Reduction Principle) Consider a map ( ) ( ) ξ Bξ + f(ξ, η),, (6.19) η Cη + g(ξ, η) where ξ R nc, η R ns+nu, the n c n c matrix B has n c eigenvalues with λ = 1, while all eigenvalues of the (n s + n u ) (n s + n u ) matrix C satisfy λ 1, and the functions 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 ) ( Bξ + f(ξ, h c (ξ)), Cη where h c represents the center manifold W c given by Theorem 6.5. ), (6.20) The maps for ξ and η are decoupled in (6.20). Therefore, the map (6.20) is locally topologically conjugate (0, 0) to a map ( ) ( ) ξ b(ξ), η Cη where ξ b(ξ) is any map that is locally topologically conjugate near ξ = 0 to ξ Bξ+f(ξ, h c (ξ)). Indeed, the conjugating homeomorphism can be constructed as the direct product of a conjugating homeomorphism in the ξ-space and the identity map in the η-space. Proposition 6.8 If ξ = 0 is a stable fixed point of the restriction of (6.19) to its center manifold ξ Bξ + f(ξ, h c (ξ)), ξ R nc, and n u = 0 (i.e. all eigenvalues of C satisfy λ < 1), then (ξ, η) = (0, 0) is a stable fixed point of (6.19).

9 6.2. CENTER MANIFOLDS FOR ODES Center manifolds for ODEs Consider a C k -smooth system { ẋ = Bx + f(x, y), ẏ = Cy + g(x, y), (6.21) where x R nc+nu, y R ns, and f(x, y) and g(x, y) have neither constant nor linear terms. Suppose that the matrix B has n c critical eigenvalues (i.e. eigenvalues with Re λ = 0) and n u eigenvalues with Re λ > 0, while all n s eigenvalues of the matrix C 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 bounds and Lipschitz bounds on R n. Then there exists an invariant manifold W cu = {(x, h(x)) : x R nc+nu }, where h : R nc+nu R ns is bounded, globally Lipschitz and satisfies h(0) = 0. Moreover, h is in C k and satisfies h x (0) = 0 if f and g are C k functions for some k 1 with small derivatives up to order k and with a small Lipschitz bound for the k-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 ) R nc+nu R ns, A = ( B 0 0 C ) ( f(u), r(u) = g(u) ) and assume r κ, r(u) r(v) l u v (6.23) for all u, v R nu+nc. Since r has a global Lipschitz bound, the system generates a global solution flow Φ t (u). Then define R t (u) as the difference to the linearized flow ( ) e Φ t (u) = e ta + R t tb 0 (u) = 0 e tc + R t (u). (6.24) We show that Theorem 6.1 applies to this map Φ t (u) for all 0 < t 2. (Step 2) By Lemma?? there exist Lyapunov norms 1 on R ns and 2 on R nc+nu and numbers 0 < b < a such that α(t) = e tc 2 e at, β(t) = e tb 1 e bt, t 0. (6.25)

10 250 CHAPTER 6. CENTER MANIFOLD REDUCTION Clearly, α(t) < β(t) 1 for all t > 0. In what follows we use these adapted norms and their extension ( ) x u = max( x 1, y 2 ), u =. y (Step 3) From the variation of constants formula, t R t (u) = e (t s)a r(φ s (u))ds κ o t 0 e (t s) A ds = κ et A 1 A = K(t). Therefore, for κ sufficiently small α(t) + K(t) e at + κ et A 1 A < 1, 0 < t 2. Further note that the variation of constants formula implies Φ t (u) Φ t (v) e ta (u v) + e t A u v + l which by Gronwall s Lemma leads to t 0 t 0 e (t s)a (r(φ s (u)) r(φ s (v)))ds e (t s) A Φ s (u) Φ s (v) ds, Φ t (u) Φ t (v) e t( A +l) u v. (6.26) In this way we obtain the Lipschitz estimate for R t R t (u) R t (v) = t 0 t 0 e (t s)a (r(φ s (u)) r(φ s (v)))ds e A (t s) l Φ s (u) Φ s (v) ds e t A (e tl 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 + 2e t( A +b) (e tl 1) < 1 for all 0 < t 2 and for sufficiently small l. Finally, note that the third condition in (6.5) follows from the second since β(t) > 1. (Step 4) By Theorem 6.1 the Hadamard graph transform T t corresponding to Φ t has a unique fixed point H t in M 0 for 0 < t 2. Now we prove H t = H s for all 0 < t, s 1. From the flow property of Φ t one finds T t T s = T t+s = T s T t for 0 < t, s 1. Indeed T s (H s ) = H s and, therefore, T t (H s ) = T t (T s (H s )) = T s (T t (H s )).

11 6.2. CENTER MANIFOLDS FOR ODES 251 This means that T t (H s )) is a fixed point of T s so that by uniqueness T t (H s ) = H s. Then the uniqueness of the fixed point of T t gives H s = H t. Therefore, all the functions H t, 0 < t 1 coincide. This proves that the global center-unstable manifold W = W cu defined by h = H t is invariant under the flow Φ t, 0 < t 1. Since Φ n = } Φ 1 Φ 1 {{ Φ} 1, n times the graph of h is invariant under Φ n. For arbitrary t, we write Φ t = Φ [t] Φ t [t] and note that the graph of h is invariant under both Φ t [t] and Φ [t]. We conclude that we have invariance under Φ t without any restriction on t 0. (Step 5) As in Step 5 of the proof of Theorem 6.1 we indicate how to obtain estimates for the derivative of R t (u) with respect to u. The bound R t u(u) L(t) follows from the Lipschitz estimate of R t above. It remains to establish the Lipschitz bound for R t u. Differentiating Rt (u) gives R t u(u) = t 0 e (t s)a r u (Φ s (u))φ s u(u)ds. (6.27) Let κ 1 be a bound for r u and l 1 be a Lipschitz constant for r u. Similar to (6.26) one first establishes an estimate for Φ t u Then the representation (6.27) gives R t u (u) Rt u (v) t Φ t (u) Φ t (v) (l + l 1 )e t A etl 1 u v. l 0 ( e (t s) A l 1 e s( A +l) + κ 1 (l 1 + l)e s A esl 1 l ) ( e t A e tl 1 l 1 + l l 1 + κ 1 (e tl tl 1) l l 2 u v. This estimate shows that we can achieve a small Lipschitz constant. ) ds u v Theorem 6.10 (Existence of a local Center-Unstable Manifold) Suppose that f(0, 0) = 0, g(0, 0) = 0, f u (0, 0) = 0, g u (0, 0) = 0 and that f, g are of class C k+1 with k 1 Then the system (6.21) has a conditionally invariant local center-unstable manifold W cu ε = {(x, h(x)) : x R nc+nu, x ε}, where ε > 0 is sufficiently small and h is a C k function.

12 252 CHAPTER 6. CENTER MANIFOLD REDUCTION 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 system u = Au + r ε (u), where r ε (u) = χ(u) 1 ε r(εu). The proof of Theorem 6.3 shows that r ε, r ε,u have small global bounds and Lipschitz bounds as well. Again the global center-unstable manifold of the cut-off system leads to a conditionally invariant local center-unstable manifold of the original system (6.6). 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 locally defined invariant manifold W c = {(ξ, h c (ξ)) : ξ R nc, ξ ǫ}, where ǫ > 0 is sufficiently small and h c : R nc R ns+nu is a C k -map satisfying h c (0) = 0, h c ξ(0) = 0. 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 R nc, η R ns+nu, the n c n c matrix B has n c critical eigenvalues with Re λ = 0, while all eigenvalues of the (n s +n u ) (n s +n u ) matrix C satisfy Re λ 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, h c (ξ)), (6.29) η = Cη. The systems for ξ and η are decoupled in (6.29). Therefore, the system (6.29) is locally topologically conjugate (0, 0) to a system { ξ = b(ξ), η = Cη.

13 6.3. CRITICAL NORMAL FORMS ON CENTER MANIFOLDS 253 where ξ = b(ξ) is any system that is locally topologically conjugate near ξ = 0 to ξ = Bξ + f(ξ, h c (ξ)). Moreover, the second equation in the last system can be substituted by the standard saddle, i.e. the linear system { ηs = η s, η s R ns, (6.30) η u = +η u, η u R nu. 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 its center manifold ξ = Bξ + f(ξ, h c (ξ)), ξ R nc, and all eigenvalues of C satisfy Re λ < 0, then (ξ, η) = (0, 0) is a stable equilibrium of (6.21). 6.3 Critical normal forms on center manifolds We now address the problem of computing in practice the coefficients of normal forms of restrictions of multidimensional ODEs and maps to the corresponding critical center manifolds at codim 1 bifurcations of equilibria and fixed points. The resulting formulas are simple and allow one to perform all computations in the original coordinates, without any preliminary transformation. Consider a smooth system of ODEs u = Au + F(u), u R n, (6.31) where the matrix A has n c eigenvalues with zero real parts, and write the Taylor expansion for F at u 0 = 0 as F(u) = 1 2 B(u, u) C(u, u, u) + O( u 4 ), where B : R n R n R n and C : R n R n R n R n are multilinear functions with the components: B i (p, q) = n j,k=1 2 F i (0) u j u k p j q k, C i (p, q, r) = n j,k,l=1 3 F i (0) u j u k u l p j q k r l, 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 C n : the range R(L) = {v C n : v = Lu for some u C n } and the null-space N(L) = {w C n : Lw = 0}.

14 254 CHAPTER 6. CENTER MANIFOLD REDUCTION We call two vectors u, v C n orthogonal and write u v, if their scalar product vanishes: u, v ū T v = 0. Denote by L the transposed matrix to the complexconjugate of L, i.e. L = L T. If L is real, L = L T. The matrix L is called the adjoint matrix for L. We have for all u, v C n. u, Lv = L u, v Lemma 6.14 (Fredholm s Decomposition) C n = R(L) N(L ) with R(L) N(L ), i.e., any vector x C n can be uniquely decomposed as x = v + w with v R(L), w N(L ), and w, v = 0. Proof: Consider an orthogonal complement W of R(L) in C n, so that C n = R(L) W and R(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 C n. Take u = L w. Then we have Thus, L w = 0 and w N(L ). 0 = w, v = w, Lu = w, LL w = L w, L w = L w 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 solvability condition. If L is nonsingular, then L is also nonsingular, so that N(L ) = 0 and Lu = v has a unique solution for any v R n, u = L 1 v. If L is singular, implying that both N(L) and N(L ) are nontrivial, but v satisfies the Fredholm solvability condition, then a solution u to Lu = v exists but is not unique. Indeed, u + ξ is another solution for any ξ N(L). Theorem 6.15 (Critical fold coefficient) Suppose λ 1 = 0 is a simple eigenvalue of A and assume that it has no other critical eigenvalues. Introduce vectors q, p R n, such that Aq = 0, A T p = 0, p, q = 1. Then the restriction of (6.31) to a one-dimensional center manifold W c (0) can be written in the form ξ = bξ 2 + O( ξ 3 ), ξ R, (6.32) where b = 1 p, B(q, q). (6.33) 2

15 6.3. CRITICAL NORMAL FORMS ON CENTER MANIFOLDS 255 Proof: Write (6.31) as u = Au B(u, u) + O( u 3 ), u R n. (6.34) The center manifold is one dimensional and can be represented as u = ξq ξ2 h 2 + O(ξ 3 ), ξ R, (6.35) for some h 2 R n. Using (6.34),(6.35), and (6.32), we get u = ξq + ξ ξh = bξ 2 q + ξ 3 h and u = Au B(u, u) +... = 1 2 ξ2 Ah ξ2 B(q, q) +... Comparing the ξ 2 -terms we find bq = 1 2 Ah B(q, q) 2 or Ah 2 = 2bq B(q, q). This linear system for h 2 is obviously singular but has a solution. The Fredholm solvability condition implies then that from which we obtain (6.33).. p, 2bq B(q, q) = 0, Theorem 6.16 (Critical Andronov-Hopf coefficient) Suppose λ 1,2 = ±iω 0, ω 0 > 0, is a simple pair of purely imaginary eigenvalues of A and assume that it has no other critical eigenvalues. Introduce vectors q, p C n, such that Aq = iω 0 q, A T p = iω 0 p, p, q = 1. Then the restriction of (6.31) to a two-dimensional center manifold W c (0) can be written in the form where η = iω 0 η + c 1 η η 2 + O( η 4 ), η C, (6.36) c 1 = 1 2 p, C(q, q, q) 2B(q, A 1 B(q, q)) + B( q, (2iω 0 I A) 1 B(q, q)). (6.37)

16 256 CHAPTER 6. CENTER MANIFOLD REDUCTION Proof: Write (6.31) as u = Au B(u, u) C(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 + η q η2 h 20 + η ηh η2 h η2 ηh , (6.39) where the dots denote all inessential terms. Here h ij C n. Using (6.38) and (6.36), we get by collecting the η 2 -terms in (6.38): (2iω 0 I A)h 20 = B(q, q). The matrix of this system is nonsingular, since 2iω 0 is not an eigenvalue of A. Thus, h 20 = (2iω 0 I A) 1 B(q, q). Collecting the η η-terms gives another nonsingular system or Ah 11 = B(q, q) h 11 = A 1 B(q, q). The η 2 -terms lead to h 02 = h 20, while collecting the coefficients in front of the η 2 η-term yields the linear system: (iω 0 I A)h 21 = C(q, q, q) + B( q, h 20 ) + 2B(q, h 11 ) 2c 1 q. This system is singular but has a solution. Thus, the Fredholm solvability condition must be satisfied: p, C(q, q, q) + B( q, h 20 ) + 2B(q, h 11 ) 2c 1 = 0, which gives (6.37). The first Lyapunov coefficient is, therefore, l 1 = 1 ω 0 Re c 1. Consider now a smooth map u Au + F(u), u R n, (6.40) where the matrix A has n c critical eigenvalues satisfying λ = 1, and F(u) = 1 2 B(u, u) C(u, u, u) + O( u 4 ), is as in (6.31).

17 6.3. CRITICAL NORMAL FORMS ON CENTER MANIFOLDS 257 Theorem 6.17 (Critical fold coefficient for maps) Suppose λ 1 = 1 is a simple eigenvalue of A and assume that it has no other critical eigenvalues. Introduce vectors q, p R n, such that Aq = q, A T p = p, p, q = 1. Then the restriction of (6.40) to a one-dimensional center manifold W c (0) can be written in the form ξ ξ + bξ 2 + O( ξ 3 ), ξ R, (6.41) where Proof: Write b = 1 p, B(q, q). (6.42) 2 f(h) = AH B(H, H) + O( H 3 ), and locally represent the center manifold W c as the graph of a function u = H(ξ), H : R R n, where H(ξ) = ξq h 2ξ 2 + O(ξ 3 ), ξ R, h 2 R n. The restriction of (6.40) to W c (0) is ξ G(ξ), where G(ξ) = ξ + bξ 2 + O(ξ 3 ). The invariance equation for the center manifold reads as f(h(ξ)) = H(G(ξ)) A(ξq+ 1 2 h 2ξ 2 + )+ 1 2 B(ξq+, ξq+ )+ = (ξ+bξ2 + )q+ 1 2 h 2(ξ+ ) 2 + The ξ 2 -terms give the equation for h 2 : (A I)h 2 = B(q, q) + 2bq. It is singular and its solvability implies (6.42). Theorem 6.18 (Critical flip coefficient) Suppose λ 1 = 1 is a simple eigenvalue of A and assume that it has no other critical eigenvalues. Introduce vectors q, p R n, such that Aq = q, A T p = p, p, q = 1. Then the restriction of (6.40) to a one-dimensional center manifold W c (0) can be written in the normal form ξ ξ + cξ 3 + O( ξ 4 ), ξ R, (6.43) where c = 1 6 p, C(q, q, q) 1 2 p, B(q, (A I) 1 B(q, q)). (6.44)

18 258 CHAPTER 6. CENTER MANIFOLD REDUCTION Proof: Expand f(h) = AH B(H, H) C(H, H, H) + O( H 4 ), and parametrize the center manifold W c (0) with u = H(ξ), where H(ξ) = ξq h 2ξ h 3ξ 3 + O(ξ 4 ), and ξ R, h 2,3 R n. The critical normal form is ξ = G(ξ) = ξ + cξ 3 + O(ξ 4 ). The ξ 2 -terms in the invariance equation f(h(ξ)) = H(G(ξ)) give for h 2 : (A I)h 2 = B(q, q). Since λ = 1 is not an eigenvalue of A, the matrix (A I) is nonsingular. Thus, h 2 = (A I) 1 B(q, q). The ξ 3 -terms in the invariancy equation give the linear system for h 3 : (A + I)h 3 = 6cq C(q, q, q) 3B(q, h 2 ). This system is singular, since (A + I)q = 0, so it has a solution only if p, 6cq C(q, q, q) 3B(q, h 2 ) = 0, which implies c = 1 6 p, C(q, q, q) p, B(q, h 2). Taking into account h 2 = (A I) 1 B(q, q), we obtain (6.44). Theorem 6.19 (Critical Neimark-Sacker coefficient) Suppose that λ 1,2 = e ±iθ 0, where e ikθ0 1, k = 1, 2, 3, 4, is a simple pair of purely imaginary eigenvalues of A and that A has no other critical eigenvalues. Introduce vectors q, p C n, such that Aq = e iθ 0 q, A T p = e iθ 0 p, p, q = 1. Then the restriction of (6.40) to a two-dimensional center manifold W c (0) can be written in the form where η e iθ 0 η + c 1 η η 2 + O( η 4 ), η C, (6.45) c 1 = 1 2 p, C(q, q, q) + B( q, (e2iθ 0 I A) 1 B(q, q)) + 2B(q, (I A) 1 B(q, q)). (6.46)

19 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 where and H(η, η) = ηq + η q + f(h(η, η)) = H(G(η, η)), (6.47) 1 j+k 3 1 j!k! h jkη j η k + O( η 4 ), f(h) = AH B(H, H) C(H, H, H) + O( H 4 ), Quadratic terms in (6.47) give G(η, η) = e iθ 0 η + c 1 η η 2 + O( η 4 ). h 20 h 11 = (e 2iθ 0 I A) 1 B(q, q), = (I A) 1 B(q, q). While the η 2 w-terms lead to the singular system (e iθ 0 I A)h 21 = C(q, q, q) + B( q, h 20 ) + 2B(q, h 11 ) 2c 1 q. The solvability of this system is equivalent to p, C(q, q, q) + B( q, h 20 ) + 2B(q, h 11 ) 2c 1 q = 0, so the cubic normal form coefficient can indeed be expressed by (6.46). Recall that the direction of the bifurcation of the closed invariant curve is determined by the sign of a = Re(e iθ 0 c 1 ). 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 R nc, η R ns+nu, α R m, 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 representable for small α as W c α = {(ξ, w(ξ, α) : ξ R nc, x ε}, where ε > 0 is sufficiently small and the map w : R nc R m R ns+nu is smooth. Moreover, w(ξ, 0) = h c (ξ), i.e. W0 c c coincides with a center manifold W from Theorem 6.11.

20 260 CHAPTER 6. CENTER MANIFOLD REDUCTION Proof: Consider the following extended system: ξ = P(ξ, η, α), η = Q(ξ, η, α), α = 0, (6.49) where ξ R nc, η R ns+nu, and α R m. The equilibrium (ξ, η, α) = (0, 0, 0) of (6.49) is nonhyperbolic and has n c + m eigenvalues with Re λ = 0 (m of them are equal to zero). Theorem 6.11 guarantees local existence of a (n c + m)-dimensional invariant center manifold in (6.49). This manifold is the union of n c -dimensional manifolds Wα c located in the invariant linear subspeces α = const of (6.49). Theorem 6.21 (Shoshitaishvilly, 1975) The system (6.48) is locally topologically equivalent near (ξ, η, α) = (0, 0, 0) to the system { ξ = P(ξ, w(ξ, α), α), (6.50) η = Cη. This theorem means that all essential events near the bifurcation parameter value occur on the invariant manifold W c α and are captured by the n c-dimensional restricted system: ξ = P(ξ, w(ξ, α), α), ξ R nc, α R m. (6.51) Obviously, this system can be substiututed in Theorem 6.21 by any smooth system ξ = b(ξ, α), ξ R nc, 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 dynamical systems 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 multidimensional systems Generic fold bifurcation in planar systems Consider a smooth planar system ẋ = f(x, α), x R 2, α R. (6.52) Assume that at α = 0 it has the equilibrium x 0 = 0 with one eigenvalue λ 1 = 0 and one eigenvalue λ 2 < 0. Theorem 6.20 gives the existence of a smooth, locally

21 6.5. BIFURCATIONS OF EQUILIBRIA AND CYCLES IN N-DIMENSIONAL ODES.261 defined, one-dimensional attracting invariant manifold Wα c At α = 0, equation (6.51) has the form for (6.52) for small α. ξ = bξ 2 + O(ξ 3 ). If b 0 and equation (6.51) depends generically on the parameter, then it is locally topologically equivalent to the normal form u = β + σu 2, where σ = sign b = ±1. Under these genericity conditions, Theorem 6.21 implies that (6.52) is locally topologically equivalent to the system { u = β + σu 2, (6.53) v = v. Equations (6.53) are decoupled. The resulting phase portraits are presented in Figure 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 nonhyperbolic saddle-node point, and disappear. There are no equilibria for β > 0. The same events happen in (6.52) on some one-dimensional, parameter-dependent, invariant manifold, that is locally attracting (see Figure 6.3). All the equilibria belong to this manifold. Figures 6.2 and 6.3 explain why the fold bifurcation is often called the saddle-node bifurcation. It should be clear how to generalize these considerations to cover the case λ 2 > 0, as well as the n-dimensional case.

22 262 CHAPTER 6. CENTER MANIFOLD REDUCTION Generic Andronov-Hopf bifurcation in three-dimensional systems Consider a smooth system ẋ = f(x, α), x R 3, α R. (6.54) Assume that at α = 0 it has the equilibrium x 0 = 0 with eigenvalues λ 1,2 = ±iω 0, ω 0 > 0 and one negative eigenvalue λ 3 < 0. Theorem 6.20 gives the existence of a parameter-dependent, smooth, local two-dimensional attracting invariant manifold Wα c of (6.54) for small α. At α = 0 the restricted equation (6.51) can be written in complex form as ż = iω 0 z + g(z, z), z C, where g = O( z 2 ). If the Lyapunov coefficient l 1 (0) of this equation is nonzero and (6.51) depends generically on the parameter, then it is locally topologically equivalent to the normal form ż = (β + i)z + σz 2 z, where σ = sign l 1 (0) = ±1. Under these genericity conditions, Theorem 6.21 β < 0 β = 0 β > 0 Figure 6.4: Hopf bifurcation in the standard system (6.55) for σ = 1. β(α) < 0 β(α) = 0 β(α) > 0 Figure 6.5: Supercritical Hopf bifurcation in a generic three-dimensional system.

23 6.5. BIFURCATIONS OF EQUILIBRIA AND CYCLES IN N-DIMENSIONAL ODES.263 implies that (6.54) is locally topologically equivalent to the system { ż = (β + i)z + σz2 z, v = v. (6.55) The phase portrait of (6.55) is shown in Figure 6.2 for σ = 1. The supercritical Hopf bifurcation takes place in the invariant plane v = 0, which is attracting. The same events happen for (6.54) on some two-dimensional attracting manifold (see Figure 6.5). The construction can be generalized to arbitrary dimension n 3. A combination of the Poincaré map and the center manifold reduction allows us to describe bifurcations of limit cycles in generic n-dimensional ODEs depending on one parameter. Let L 0 be a limit cycle of a smooth system ẋ = f(x, α), x R n, α R, (6.56) at α = 0. Let P (α) denote the associated Poincaré map for nearby α; P (α) : Σ Σ, where Σ is a local cross-section to L 0. If some coordinates ξ = (ξ 1, ξ 2,...,ξ n 1 ) are introduced on Σ, then ξ = P (α) (ξ) can be defined to be the point of the next intersection with Σ of the orbit of (6.56) having initial point with coordinates ξ on Σ. The intersection of Σ and L 0 gives a fixed point ξ 0 for P (0) : P (0) (ξ 0 ) = ξ 0. As we know, map P (α) is smooth and locally invertible. Suppose that the cycle L 0 is nonhyperbolic, having n 0 multipliers on the unit circle. The center manifold theorems then give a parameter-dependent invariant manifold W c α Σ of P (α) on which the essential events take place. The Poincaré map P (α) is locally topologically equivalent to the suspension of its restriction to this manifold by the standard saddle map. Fix n = 3, for simplicity, and consider the implications of this result for the limit cycle bifurcations in R Fold bifurcation of limit cycles in R 3 Assume that at α = 0 the cycle has a simple multiplier µ 1 = 1 and its other multiplier satisfies 0 < µ 2 < 1. The restriction of P (α) to the invariant manifold W c α is a one-dimensional map, having a fixed point with µ 1 = 1 at α = 0. As has been shown, this generically implies the collision and disappearance of two fixed points of P (α) as α passes through zero. Under our assumption on µ 2, this happens on a one-dimensional attracting invariant manifold of P (α) ; thus, a stable and a saddle fixed point are involved in the bifurcation (see Figure 6.6 for an illustration). Each fixed point of the Poincaré map corresponds to a limit cycle of the continuous-time system. Therefore, two limit cycles (stable and saddle) collide and disappear in system (6.56) at this fold bifurcation of cycles Flip (period-doubling) bifurcation of limit cycles in R 3 Suppose that at α = 0 the cycle has a simple multiplier µ 1 = 1, while 1 < µ 2 < 0. Then, the restriction of P (α) to the invariant manifold will demonstrate generically

24 264 CHAPTER 6. CENTER MANIFOLD REDUCTION L 1 L 2 L0 α < 0 α = 0 α > 0 Figure 6.6: Fold bifurcation of limit cycles. L 0 L0 L 0 L 1 α < 0 α = 0 α > 0 Figure 6.7: Flip bifurcation of limit cycles. L 0 L 0 L 0 T 2 α > 0 α = 0 α > 0 Figure 6.8: Neimark-Sacker bifurcation of a limit cycle.

25 6.6. REFERENCES 265 the period-doubling (flip) bifurcation: A cycle of period-2 appears or disappears for the map, while the fixed point changes its stability (see Figure 6.7, where the supercritical case is illustrated). Since the manifold is attracting, the stable fixed point, for example, loses stability and becomes a saddle point, while a stable cycle of period-2 appears. The fixed points correspond to limit cycles of the relevant stability. The cycle of period-two points for the map corresponds to a unique stable limit cycle in (6.56) with approximately twice the period of the basic cycle L 0. The double-period cycle makes two big excursions near L 0 before the closure. The exact bifurcation scenario is determined by the normal form coefficient of the restricted Poincaré map evaluated at α = Neimark-Sacker (torus) bifurcation of limit cycles in R 3 The last codim 1 bifurcation corresponds to the case when the multipliers are complex and simple and lie on the unit circle: µ 1,2 = e ±iθ 0. The Poincaré map P (α) then has a parameter-dependent, two-dimensional, invariant manifold on which a closed invariant curve generically bifurcates from the fixed point (see Figure 6.8, where the supercritical bifurcation is shown). This closed curve corresponds to a two-dimensional invariant torus T 2 in (6.56). The bifurcation is determined by the normal form coefficient of the restricted Poincaré map at the critical parameter value. The orbit structure on the torus T 2 depends on the restriction of the Poincaré map to this closed invariant curve. Thus, generically, there are long-period cycles of different stability types located on the torus, which appear and disappear pair-wise via fold bifurcations. 6.6 References Bifurcations of stationary points and periodic orbits in one- and two-parameter families of multidimensional ODEs and maps are treated in many textbooks, including [Arnol d 1983, Guckenheimer & Holmes 1983, Arrowsmith & Place 1990, Shilnikov et al. 2001, Wiggins 2003, Kuznetsov 2004]. A useful summary is given in [Arnol d et al. 1994], while many technical issues are clarified in [Iooss 1979, Vanderbauwhede 1989, Iooss & Adelmeyer 1992]. For an alternative approach to the bifurcation theory based on the Lyapunov-Schmidt reduction, see [Chow & Hale 1982, Iooss & Joseph 1990, Kielhöfer 2004] A direct proof of the existence of a local center manifold near a nonhyperbolic equilibrium in ODEs, that does not depend on the corresponding result for maps, is given in [Carr 1981]; a proof of Theorem 6.12 (Reduction Principle for ODEs) can be found in [Kirchgraber & Palmer 1990]. Numerical methods for bifurcations of stationary points and periodic orbits in multidimensional ODEs and maps are summarized in [Beyn, Champneys, Doedel, Govaerts, Kuznetsov & Sandstede 2002].

26 266 CHAPTER 6. CENTER MANIFOLD REDUCTION 6.7 Exercises E (Andronov-Hopf bifurcation in 3D systems) Check that each of the following feedback control systems 1 has an equilibrium that exhibits an Andronov-Hopf bifurcation at µ = 0, and compute the first Lyapunov coefficient of the restricted system on the center manifold: (a) ẋ = µx y, ẏ = µy + x + xz, ż = z + x 2. (b) ẋ = µx y xz, ẏ = µy + x, ż = z + y 2 + x 2 z. E (Pitchfork bifurcation in Lorenz system) Compute the second-order approximation to the family of one-dimensional center manifolds of the Lorenz system 2 ẋ = σx + σy, ẏ = xz + rx y, (6.57) ż = xy bz, near the origin (x, y, z) = (0, 0, 0) for fixed (σ, b) and r close to r 0 = 1. Then, calculate the restricted system up to third-order terms in ξ and analyse its bifurcation. E (Andronov-Hopf bifurcation in Lorenz system) (a) Show that for fixed b > 0, σ > b + 1, and r 1 = 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 < r 1 (Hints: [Shilnikov et al. 2001, pp ] (i) Write (6.57) as a single third-order equation (1 + σ)ẋ2 x +(σ + b + 1)ẍ + b(1 + σ)ẋ + bσ(1 r)x = + ẋẍ x x x2 ẋ σx 3. (ii) Translate the origin to the equilibrium by introducing the new coordinate ξ = x x 0, where x 0 = b(r 1), thus obtaining the equation where ξ +(σ + b + 1) ξ + [b(1 + σ) + x 2 0 ] ξ + [bσ(1 r) + 3σx 2 0 ]ξ = f(ξ, ξ, ξ), (6.59) f(ξ, ξ, ξ) = 3σx 0 ξ 2 2x 0 ξ ξ σ x 0 ξ2 + 1 ξ ξ σξ 3 ξ σ ξ x 0 x 2 ξ ξ x 2 ξ ξ ξ Moon, 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 Propulsion Laboratory Publication 85-29, Pasadena, CA, 1985, pp Lorenz, E. Deterministic non-periodic flow, J. Atmos. Sci. 20 (1963),

27 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 R 3. (6.60) Find the eigenvector and the adjoint eigenvector of A corresponding to its purely imaginary eigenvalues (when (6.58) is satisfied). (iv) Compute the first Lyapunov coefficient l 1 for (6.60) using (6.37). Substitute σ = σ +b+1 and show that l 1 is positive for all positive σ and b.) E (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 (Fold and flip bifurcations in Hénon map) Consider the Hénon map 3 : ( x y ) ( ) y α βx y 2. (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 ±1. E (Duopoly model of Kopel) Consider the following Kopel map from mathematical economics 4 : ( ) ( ) x (1 ρ)x + ρµy(1 y), (6.62) y (1 ρ)y + ρµx(1 x) 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 domain 2.9 µ 3.8, 0.75 ρ 1.4. E (Flip and Neimark-Sacker bifurcations in an adaptive control map) (a) Demonstrate that the fixed point (x 0, y 0, z 0 ) = (1, 1, 1 b k) of the discrete-time dynamical system 5 x y z z y bx + k + yz ky (bx + k + zy 1) c + y2 3 Hénon, M. A two-dimensional mapping with a strange attractor, J Comm. Math. Phys. 50 (1976), Kopel, M. Simple and complex adjustment dynamics in Cournot duopoly models, Chaos, Solitons & Fractals, 12 (1996), Golden, M.P. and Ydstie, B.E. Bifurcation in model reference adaptive control systems, Systems Control Lett. 11 (1988),

28 268 CHAPTER 6. CENTER MANIFOLD REDUCTION exhibits a flip bifurcation at and a Neimark-Sacker bifurcation at [ ] 1 b F = k, 4(c + 1) b NS = c + 1 c + 2. (b) Determine the direction of the period-doubling bifurcation that occurs as b increases and passes through b F. (c) Show that the Neimark-Sacker bifurcation in the system under variation of the parameter b can be either sub- or supercritical depending on the values of the parameter (c, k).