Instabilities and local bifurcations. Elements of theory G erard - PowerPoint PPT Presentation

Instabilities and local bifurcations. Elements of theory G´ erard Iooss IUF, Universit´ e de Nice, Laboratoire J.A.Dieudonn´ e, Parc Valrose, F-06108 Nice Cedex02 M.Haragus, G.Iooss. Local bifurcations, center manifolds, and normal forms in infinite dimensional systems (329p.). Springer UTX, 2011 G. Iooss (IUF, Univ. Nice) water waves 1 / 34

Bifurcations in dimension 1 du dt = f ( u , µ ) , f (0 , 0) = 0 , ∂ f ∂ u (0 , 0) = 0 , µ parameter Saddle - node bifurcation Assume f is C k , k ≥ 2, in a neighborhood of (0 , 0), and ∂ 2 f ∂ f ∂µ (0 , 0) =: a � = 0 , ∂ u 2 (0 , 0) =: 2 b � = 0 . As ( u , µ ) → (0 , 0), f has the expansion f ( u , µ ) = a µ + bu 2 + o ( | µ | + u 2 ) u u u u 0 µ 0 µ 0 µ 0 µ a > 0 , b < 0 a > 0 , b > 0 a < 0 , b < 0 a < 0 , b > 0 G. Iooss (IUF, Univ. Nice) water waves 2 / 34

Dim 1 - Pitchfork bifurcation Assume f is C k , k ≥ 3, in a neighborhood of (0 , 0), and satisfies ∂µ∂ u (0 , 0) =: a � = 0 , ∂ 3 f ∂ 2 f f ( − u , µ ) = − f ( u , µ ) , ∂ u 3 (0 , 0) =: 6 b � = 0 . Hence, as ( u , µ ) → (0 , 0), f has the expansion f ( u , µ ) = a µ u + bu 3 + o [ | u | ( | µ | + u 2 )], u = 0 is an equilibrium for all µ . u u u u µ µ µ µ 0 0 0 0 a > 0 , b < 0 a > 0 , b > 0 a < 0 , b < 0 a < 0 , b > 0 G. Iooss (IUF, Univ. Nice) water waves 3 / 34

Dim 2 - Hopf bifurcation in R 2 du dt = F ( u , µ ) , F (0 , 0) = 0 , F is C k , k ≥ 3, in a neighborhood of (0 , 0). Define L := D u F (0 , 0). Assume L has a pair of complex conjugated purely imaginary eigenvalues ± i ω , ω > 0: L ζ = i ωζ, L ζ = − i ωζ. Normal form theorem (seen later): for any integer p ≤ k , and any µ sufficiently small, there exists a polynomial Φ µ of degree p in ( A , A ), with complex coefficients functions of µ , taking values in R 2 , such that Φ 0 (0 , 0) = 0 , ∂ A Φ 0 (0 , 0) = 0 , ∂ A Φ 0 (0 , 0) = 0 , u = A ζ + A ζ + Φ µ ( A , A ) , A ∈ C , transforms the system into the differential equation dA dt = i ω A + AQ ( | A | 2 , µ ) + o ( | A | p ) , Q polynomial in | A | 2 , Q (0 , 0) = 0 G. Iooss (IUF, Univ. Nice) water waves 4 / 34

Hopf bifurcation - continued dA dt = i ω A + A ( a µ + b | A | 2 ) + o ( | A | ( | µ | + | A | 2 )) , Assume a r � = 0 and b r � = 0. Truncated system: set A = re i φ , dr r ( a r µ + b r r 2 ) (pitchfork bifurcation for radial part) = dt d φ ω + a i µ + b i r 2 , (frequency of bifurcated periodic solution) = dt A A A 0 µ case a r > 0, b r < 0 G. Iooss (IUF, Univ. Nice) water waves 5 / 34

Hyperbolic situation in R n du dt = F ( u ) , F (0) = 0 , DF (0) = L M + + + C O M - - 0 σ− σ+ left: spectrum of L , center: linear situation, right: nonlinear situation G. Iooss (IUF, Univ. Nice) water waves 6 / 34

Hyperbolic situation in R n continued u = X + Y , X = P + u ∈ E + , Y = P − u ∈ E − dX = L + X + P + R ( X + Y ) dt dY = L − Y + P − R ( X + Y ) dt Unstable manifold M + : solve in u ( t ) , t ≤ 0, with u ( t ) → 0 as t → −∞ � t � t u ( t ) = e L + t X + e L + ( t − s ) P + R ( u ( s )) ds + e L − ( t − s ) P − R ( u ( s )) ds 0 −∞ Then, by implicit function theorem, u ( t ) = Φ + ( X , t ), and u (0) = Φ + ( X , 0) = X + Ψ + ( X ), with Ψ + ( X ) ∈ E − G. Iooss (IUF, Univ. Nice) water waves 7 / 34

Center manifold in R n Pliss 1964, Kelley 1967, Lanford 1973, Henry 1981, Mielke 1988, Kirrmann 1991, Vanderbauwhede - Iooss 1992 du dt = Lu + R ( u , µ ) , ( u , µ ) ∈ R n × R m , R (0 , 0) = 0 , D u R (0 , 0) = 0 . spectrum of L = σ = σ − ∪ σ 0 Hypothesis: σ 0 = finite number of eigenvalues of finite mutiplicities sup λ ∈ σ − λ < − γ < 0 (gap assumption) R n = E 0 ⊕ E − , u = X + Y , X = P 0 u , Y = P − u 0 0 µ left: linear case for µ = 0, asymptotic solutions ∈ E 0 , right: non linear case G. Iooss (IUF, Univ. Nice) water waves 8 / 34

Center Manifolds- idea of proof Theorem: { u = u 0 + Ψ ( u 0 , µ ) , ( u 0 , µ ) ∈ E 0 × R m } M µ = C k ( O 0 , E − ) , O 0 neighb of 0 in E 0 × R m ∈ Ψ Ψ (0 , 0) = 0 , D u 0 Ψ (0 , 0) = 0 . M µ locally invariant and locally attracting . Idea of proof: Even though u ( t ) stays bounded for t ∈ R , the first term and the integral below with L 0 may grow polynomially in t as t → −∞ . � t � t u ( t ) = e L 0 t X + e L 0 ( t − s ) P 0 R ( u ( s )) ds + e L − ( t − s ) P − R ( u ( s )) ds . 0 −∞ Need of a (smooth) ”cut-off” function on E 0 , modifying and making the system linear for its part in E 0 , outside a ball of small radius. This allows to work in a space of functions growing at infinity. New complications due to the fact that we deal with such functions (which may grow at −∞ with a small exponential). G. Iooss (IUF, Univ. Nice) water waves 9 / 34

Center Manifolds in infinite dimensions du dt = Lu + R ( u , µ ) R (0 , 0) = 0 , D u R (0 , 0) = 0 L linear bounded Z → X , Z cont. embedded in X (both Hilbert spaces) R : ( Z × R m ) → X of class C k , k ≥ 2 in a neighborhood of 0 Hypothesis: (i) (gap assumption) spectrum σ of L = σ 0 ∪ σ − , For λ ∈ σ 0 , Re λ = 0, sup λ ∈ σ − Re λ < − γ < 0; (ii) σ 0 = finite number of eigenvalues of finite mutiplicities G. Iooss (IUF, Univ. Nice) water waves 10 / 34

Center Manifolds in infinite dimensions - continued Hypothesis on the linearized system || ( i ω I − L ) − 1 || L ( X ) ≤ C | ω | for ω ∈ R , | ω | large. Then the following properties (iii) and (iv) are satisfied. Define: E 0 = P 0 X = P 0 Z , Z h = P h Z , X = E 0 ⊕ X h , Z = E 0 ⊕ Z h , η ∈ [0 , γ ] (iii) du h dt = L h u h + f , f ∈ C 0 ( R , X ) , sup t ∈ R e η t || f ( t ) || X < ∞ , Then, there exists a unique u h = K h f , such that K h f ∈ C 0 ( R , Z ) , sup t ∈ R e η t || K h f ( t ) || Z < C ( η ) sup t ∈ R e η t || f ( t ) || X , C ( η ) continuous on [0 , γ ]. (iv) du h dt = L h u h , u | t =0 ∈ Z h . Then, there exists a unique u h ∈ C 0 ( R + , Z h ) , || u h || Z ≤ c η e − η t , t ≥ 0. G. Iooss (IUF, Univ. Nice) water waves 11 / 34

Reduced system for asymptotic dynamics and Symmetries du 0 = L 0 u 0 + P 0 R ( u 0 + Ψ ( u 0 , µ ) , µ ) := f ( u 0 , µ ) dt f (0 , 0) = 0 , D u 0 f (0 , 0) = L 0 , spectrum of L 0 : σ 0 Frequent case: 0 is a solution of the system for any µ R (0 , µ ) = 0, hence Ψ (0 , µ ) = 0 , f (0 , µ ) = 0 and the linear operator A µ := D u 0 f (0 , µ ) has the eigenvalues close to the imaginary axis of the linearized operator L µ := L + D u R (0 , µ ) G. Iooss (IUF, Univ. Nice) water waves 12 / 34

Reduced system for asymptotic dynamics and Symmetries du 0 = L 0 u 0 + P 0 R ( u 0 + Ψ ( u 0 , µ ) , µ ) := f ( u 0 , µ ) dt f (0 , 0) = 0 , D u 0 f (0 , 0) = L 0 , spectrum of L 0 : σ 0 Frequent case: 0 is a solution of the system for any µ R (0 , µ ) = 0, hence Ψ (0 , µ ) = 0 , f (0 , µ ) = 0 and the linear operator A µ := D u 0 f (0 , µ ) has the eigenvalues close to the imaginary axis of the linearized operator L µ := L + D u R (0 , µ ) Presence of symmetry TLu = LTu , TR ( u , µ ) = R ( Tu , µ ) T | E 0 := T 0 is an isometry Then T Ψ ( u 0 , µ ) = Ψ ( T 0 u 0 , µ ) , for u 0 ∈ E 0 T 0 f ( u 0 , µ ) = f ( T 0 u 0 , µ ) . G. Iooss (IUF, Univ. Nice) water waves 12 / 34

Computation of center manifold and reduced system NB. We compute Taylor expansions, in powers of ( u 0 , µ ) ∈ E 0 × R m D u 0 Ψ ( u 0 , µ ) du 0 dt = du h dt replace du 0 by L 0 u 0 + P 0 R ( u 0 + Ψ ( u 0 , µ ) , µ ), dt and replace du h by L h Ψ ( u 0 , µ ) + P h R ( u 0 + Ψ ( u 0 , µ ) , µ ) dt and identify powers of ( u 0 , µ ). G. Iooss (IUF, Univ. Nice) water waves 13 / 34

Computation of center manifold and reduced system NB. We compute Taylor expansions, in powers of ( u 0 , µ ) ∈ E 0 × R m D u 0 Ψ ( u 0 , µ ) du 0 dt = du h dt replace du 0 by L 0 u 0 + P 0 R ( u 0 + Ψ ( u 0 , µ ) , µ ), dt and replace du h by L h Ψ ( u 0 , µ ) + P h R ( u 0 + Ψ ( u 0 , µ ) , µ ) dt and identify powers of ( u 0 , µ ). Example: quadratic order in u 0 : f ( u 0 , µ ) = L 0 u 0 + P 0 R 2 , 0 ( u 0 ) + P 0 R 0 , 1 ( µ ) + h . o . t . , h.o.t. depends on Ψ D u 0 Ψ 2 , 0 ( u 0 ) L 0 u 0 − L h Ψ 2 , 0 ( u 0 ) = P h R 2 , 0 ( u 0 ) leads to � ∞ e L h t P h R 2 , 0 ( e − L 0 t u 0 ) dt . Ψ 2 , 0 ( u 0 ) = 0 This may become tedious, and may lead to a complicate vector field in E 0 , in case of dimension > 1, specially if orders > 2 are required. Our purpose now is to simplify the reduced system, in using Symmetries and Normal form theory. G. Iooss (IUF, Univ. Nice) water waves 13 / 34

Normal forms Poincar´ e, Birkhoff, Arnold, Belitskii, Elphick et al... p ≥ 2 , ∃ polynomial Φ µ : E 0 → E 0 , of degree p and a neighborhood O 0 of 0 in E 0 × R m , such that the local change of variable in E 0 u 0 = v 0 + Φ µ ( v 0 ) transforms the reduced system into a new system where N µ is a polynomial of degree p such that dv 0 dt = L 0 v 0 + N µ ( v 0 ) + ρ ( v 0 , µ ) , N 0 (0) = 0 , D v 0 N 0 (0) = 0 e L ∗ 0 t N µ ( v 0 ) N µ ( e L ∗ 0 t v 0 ) , ∀ ( t , v 0 ) ∈ R × E 0 , = o ( || v 0 || p ) . ρ ( v 0 , µ ) = NB. In case of analytical vector fields, there are results optimizing the degree p , giving a rest ρ exponentially small (G.I., E.Lombardi 2005) G. Iooss (IUF, Univ. Nice) water waves 14 / 34