NONLINEAR OSCILLATIONS OF HAMILTONIAN PDES
CONTENTS FINITE DIMENSION THE LYAPUNOV CENTER THEOREM THE WEINSTEIN–MOSER AND FADELL–RABINOWITZ RESONANT CENTER THEOREMS THE VARIATIONAL LYAPUNOV–SCHMIDT REDUCTION SOLUTION OF THE RANGE EQUATION SOLUTION OF THE BIFURCATION EQUATION PROOF OF THE WEINSTEIN–MOSER THEOREM PROOF OF THE FADELL–RABINOWITZ THEOREM IN?NITE DIMENSION THE LYAPUNOV CENTER THEOREM FOR PDES COMPLETELY RESONANT WAVE EQUATIONS THE CASE P ODD THE RANGE EQUATION THE BIFURCATION EQUATION THE MOUNTAIN PASS ARGUMENT THE CASE P EVEN MULTIPLICITY THE SMALL-DIVISOR PROBLEM A TUTORIAL IN NASH–MOSER THEORY INTRODUCTION AN ANALYTIC NASH–MOSER THEOREM A DIFFERENTIABLE NASH–MOSER THEOREM APPLICATION TO THE NONLINEAR WAVE EQUATION THE ZEROTH-ORDER BIFURCATION EQUATION THE FINITE-DIMENSIONAL REDUCTION SOLUTION OF THE (Q2)-EQUATION THE NASH–MOSER SCHEME SOLUTION OF THE (Q1)-EQUATION THE LINEARIZED OPERATOR DECOMPOSITION OF LN STEP 1: INVERSION OF D STEP 2: INVERSION OF LN FORCED VIBRATIONS THE FORCING FREQUENCY ? ?Q MONOTONE F STEP 1: THE L?ESTIMATE STEP 2: THE H1 ESTIMATE NONMONOTONE F STEP 1: THE L2K ESTIMATE STEP 2: THE L?ESTIMATE STEP 3: THE H1 ESTIMATE THE “MAXIMUM PRINCIPLE” FINITE DIMENSION LET US CONSIDER A ?NITE-DIMENSIONAL DYNAMICAL SYSTEM ?X = F (X) , X ?RN , (1 1) WHERE THE VECTOR ?ELD F ?C1(RN, RN) POSSESSES AN EQUILIBRIUM AT X = 0, I E , F (0) = 0 QUESTION: DO THERE EXIST PERIODIC SOLUTIONS OF (1 1) CLOSE TO X = 0? THE ?RST THING IS TO STUDY THE LINEARIZED SYSTEM ?X = AX, A = (DX F )(0) (1 2) A NECESSARY CONDITION TO ?ND PERIODIC SOLUTIONS OF THE NONLINEAR SYSTEM (1 1) CLOSE TO THE EQUILIBRIUM X = 0 IS THE PRESENCE OF PURELY IMAGINARY EIGENVALUES OF A INDEED, IF ALL THE EIGENVALUES OF A HAVE NONZERO REAL PART (I E , THE MATRIX A IS HYPERBOLIC), THEN THE HARTMANN–GROBMANN THEOREM (SEE, E G , [68]) ENSURES A TOPO- LOGICAL CONJUGACY BETWEEN THE LINEAR AND THE NONLINEAR ?OWS CLOSE TO THE EQUILIBRIUM; SEE FIGURE 1 1 IN THIS CASE, SINCE THE LINEAR SYSTEM (1 2) DOES NOT POSSESS PERIODIC SOLUTIONS, NEITHER DOES THE NONLINEAR SYSTEM (1 1) THEREFORE, IF A HAS THE EIGENVALUES ?1, , ?N , NOT NECESSARILY DISTINCT, WE SHALL ASSUME (RECALLING A IS A REAL MATRIX) THAT ?1 = I? , ?2 = ?I? (1 3) IS A PAIR OF PURELY IMAGINARY EIGENVALUES WITH COMPLEX EIGENVECTORS ? = E1 + IE2 , ? = E1 ?IE2 , E1, E2 ?RN THEN AE1 = ??E2, AE2 = ?E1 AND THE REAL PLANE 2(T))2 , 1 FINITE DIMENSION E ? FIG 1 2 THE INVARIANT PLANE E OF THE LINEAR SYSTEM (1 2) ?LLED BY 2?/?-PERIODIC SOLUTIONS SO THAT IF A PERIODIC SOLUTION X(T) EXISTS, WHATEVER ITS PERIOD T IS, 0 = = T 2(T)) DT = = T 2(T))2 DT , AND THEN X(T) ?0 1 1 EXERCISE: MAKE THE PHASE PORTRAIT OF (1 5) THE PREVIOUS EXAMPLE SHOWS THAT IN ORDER TO ?ND PERIODIC SOLUTIONS OF (1 1), THE CLASS OF THE VECTOR ?ELDS F (X) MUST BE RESTRICTED TWO TYPICAL CONDITIONS ARE ASSUMED: (I) SYSTEMS WITH A PRIME INTEGRAL (FOR EXAMPLE HAMILTONIAN SYSTEMS); (II) REVERSIBLE SYSTEMS IN THIS BOOK WE SHALL CONSIDER ONLY THE CLASS (I) DE?NITION 1 3 A SMOOTH FUNCTION G RN ?R IS A PRIME INTEGRAL OF (1 1) IFF G(X(T)) = CONSTANT ALONG ANY SOLUTION X(T) OF (1 1) EQUIVALENTLY, ?G(X) F (X) = 0 , ?X ?RN (1 6) IN OTHER WORDS, EVERY ORBIT EVOLVES WITHIN THE LEVEL SET {G(X) = G(X(0))} OF THE PRIME INTEGRAL G; SEE FIGURE 1 3 THE MAIN EXAMPLE OF SYSTEMS POSSESSING A ?RST INTEGRAL IS THAT OF THE AUTONOMOUS HAMILTONIAN SYSTEMS (HS) ?X = J?H(X) , X ?R2N , WHERE J = T 0 I ?I U ?MAT(2N × 2N) IS THE STANDARD SYMPLECTIC MATRIX, I IS THE IDENTITY MATRIX IN RN, AND THE FUNCTION H R2N ?R 1 FINITE DIMENSION × × X(0) X(T) G(X) = G(X(0)) FIG 1 3 EVERY ORBIT EVOLVES WITHIN THE LEVEL SET {G(X) = G(X(0))} OF THE PRIME INTEGRAL G IS CALLED THE HAMILTONIAN OF (HS) THE HAMILTONIAN H ITSELF IS A PRIME INTEGRAL OF (HS) BECAUSE D DT H(X(T)) = ?H(X(T)) J?H(X(T)) = DQI ?DPI , THE HAMILTONIAN VECTOR ?ELD X H(Q, P) = (?PH(Q, P), ??Q H(Q, P)) ?R2N IS UNIQUELY DE?NED, BY THE NONDEGENERACY OF ?, BY DH(Q, P) [H] = ?(X H(Q, P), H ) = X H(Q, P) JH , ?H ?R2N 1 1 THE LYAPUNOV CENTER THEOREM WE ?RST ANALYZE THE FORM OF A PRIME INTEGRAL LEMMA 1 5 SUPPOSE G IS A C2 PRIME INTEGRAL OF (1 1), F (0) = 0, A = (DX F )(0) POSSESSES A PAIR OF PURELY IMAGINARY EIGENVALUES (CF (1 3)) AND DETA /= 0 THEN 1 1 THE LYAPUNOV CENTER THEOREM /?Z (1 9) AND IF G IS A C2 PRIME INTEGRAL OF (1 1) WITH (D2G)|E(0) /?0, THEN FOR EVERY SMALL ?, THERE EXISTS A UNIQUE T(?)-PERIODIC SOLUTION P(?, T) OF (1 1) NEAR E, ON THE LEVEL SET1 {G(X) = G(0) + ?2}, WITH PERIOD T(?) NEAR 2?/? AS ? ?0 THE SOLUTION P(?, T) L? ?0 AND T(?) ?2?/? 1 BY LEMMA 1 5, (D2G)|E(0) IS EITHER POSITIVE OR NEGATIVE DE?NITE; TO ?X NOTATION WE AS- SUME (D2G)|E(0) > 0 IN THE CASE (D2G)|E(0) < 0 THE PERIODIC SOLUTION P(?, T) LIES IN THE LEVEL SET {G(X) = G(0) ??2} 1 FINITE DIMENSION E X(T) = P(?, T) FIG 1 4 THE LYAPUNOV PERIODIC SOLUTIONS P(T, ?) ?LL A 2-DIMENSIONAL EMBEDDED DISK THAT IS TANGENT TO THE PLANE E AT X = 0 THE LYAPUNOV PERIODIC ORBITS P(T, ?) OF THEOREM 1 6 ARE PARAMETRIZED BY THE VALUE {G(P(T, ?)) = G(0)+?2} OF THE ?RST INTEGRAL G; OTHER PARAMETRIZATIONS ARE POS- SIBLE; SEE [6] NOTE THAT IN GENERAL, THE LYAPUNOV PERIODIC ORBITS ARE NOT PARAMETRIZED BY THEIR FREQUENCY: IT MAY HAPPEN THAT ALL THEIR FREQUENCIES ARE THE SAME, AS, FOR EX- AMPLE, IN LINEAR SYSTEMS REMARK 1 7 (REGULARITY) IF F ?CR+1, THE LYAPUNOV PERIODIC SOLUTIONS ?T TOGETHER INTO A TWO-DIMENSIONAL EMBEDDED INVARIANT MANIFOLD THAT IS CR AND TANGENT TO THE PLANE E AT X = 0; SEE FIGURE 1 4 IF F AND G ARE ANALYTIC, THEN THE EMBEDDING IS ANALYTIC; SEE [100] REMARK 1 8 (MULTIPLICITY) BY THEOREM 1 6, IF THERE ARE S PAIRS OF PURELY IMAGINARY EIGENVALUES ? J = ±I? J , J = 1, 2, , S , (1 10) WE OBTAIN S DISTINCT FAMILIES OF PERIODIC SOLUTIONS HAVING PERIODS CLOSE TO 2?/? J UNDER THE NONRESONANCE CONDITIONS ?K ? J /?Z ?K /= J , J = 1, 2, , S (1 11) (AND IF (D2G)(0) IS NONSINGULAR ON THE CORRESPONDING EIGENSPACES) REMARK 1 9 (EXISTENCE) IF THERE ARE S PAIRS OF PURELY IMAGINARY EIGENVALUES ? J = ±I? J WITH 0 < ?1 ? ??S?1 < ?S (NAMELY ?S IS SIMPLE), THEN THE LYAPUNOV CENTER THEOREM ENSURES THE PERSISTENCE OF THE PERIODIC ORBITS WITH THE SHORTEST PE- RIOD (HIGHEST FREQUENCY ?S) INDEED, THE NONRESONANCE CONDITION (1 11) IS SATIS?ED BECAUSE ?K ?S = I?K I?S < 1 2 THE WEINSTEIN–MOSER AND FADELL–RABINOWITZ RESONANT CENTER THEOREMS IS A PRIME INTEGRAL OF (1 12) WITH NONSINGULAR (D2G)(0) MOREOVER, D DT IM(Z1Z2) = (|Z2|2 + J2|Z1|2)|Z2|2 J?2 ?0 , SO THAT PERIODIC ORBITS HAVE Z2 = 0, AND THEREFORE Z1 = C EI JT , Z2 = 0 , GIVES THE SOLUTIONS OF SHORTEST PERIOD 1 2 EXERCISE: FIND EXPLICITLY THE LYAPUNOV PERIODIC SOLUTIONS OF THE HAMILTONIAN SYSTEM IN (R6, I=1 DQI ?DPI) WITH HAMILTONIAN H = 1) + RESONANT CENTER THEOREMS THE WEINSTEIN–MOSER AND FADELL–RABINOWITZ IF THE NONRESONANCE CONDITION (1 9) AMONG THE EIGENVALUES OF A IS VIOLATED (RESONANT CASE), NO PERIODIC SOLUTIONS EXCEPT THE EQUILIBRIUM POINT NEED EXIST AN EXAMPLE DUE TO MOSER [99] IS PROVIDED BY THE HAMILTONIAN SYSTEM IN (R4, , SO THAT THE UNIQUE PERIODIC SOLUTION IS Q = P = 0 , IS NEITHER POSITIVE NOR NEGATIVE DE?NITE, AND THAT ITS SIGNATURE SATIS?ES SIGN(H2) = (THE SIGNATURE IS THE DIFFERENCE BETWEEN THE DIMENSION OF THE MAXIMAL SUBSPACE WHERE H2 IS POSITIVE DE?NITE AND THE DIMENSION OF THE MAXIMAL SUBSPACE WHERE H2 IS NEGATIVE DE?NITE) IN CONTRAST, TWO REMARKABLE THEOREMS BY WEINSTEIN [129]–MOSER [98] AND FADELL– RABINOWITZ [61] PROVE THE EXISTENCE OF PERIODIC SOLUTIONS UNDER THE RESPECTIVE AS- SUMPTIONS (WM) (D2H)(0) > 0 (OR < 0 ) , (FR) SIGN(D2H)(0) /= 0 NOTE THAT (WM) IS A STRONGER CONDITION THAN (FR) AND THAT THE NONEXISTENCE EXAMPLE (1 13) VIOLATES THE CONDITION (FR) UNDER (WM), THE WEINSTEIN–MOSER THEOREM ?NDS BIFURCATION OF PERIODIC SOLU- TIONS OF THE HAMILTONIAN SYSTEM (HS) WITH ?XED ENERGY, WHILE ASSUMING (FR), THE FADELL–RABINOWITZ THEOREM PROVES THE EXISTENCE OF SOLUTIONS WITH ?XED PERIOD REMARK 1 11 ASSUMING JUST (FR), BIFURCATION OF PERIODIC SOLUTIONS WITH ?XED ENERGY WAS MORE RECENTLY PROVED BY BARTSCH [16] LET US GIVE THE PRECISE STATEMENTS OF THESE THEOREMS THEOREM 1 12 (WEINSTEIN 1973–MOSER 1976) LET H ?C2(R2N, R) BE SUCH THAT (?H)(0) = 0 AND (D2H)(0) > 0 FOR ALL ? SMALL ENOUGH THERE EXIST ON EACH ENERGY SURFACE {H(X) = H(0) + ?2} AT LEAST N GEOMETRICALLY DISTINCT PERIODIC SOLUTIONS OF THE HAMILTONIAN SYSTEM (HS); SEE FIGURE 1 5 REMARK 1 13 THEOREM 1 12 WAS ?RST PROVED BY WEINSTEIN [129] MOSER [98] PROVED THE EXTENSIONS DESCRIBED IN REMARKS 1 19 AND 1 16 BELOW 1 2 THE WEINSTEIN–MOSER AND FADELL–RABINOWITZ RESONANT CENTER THEOREMS D2H(0) > FIG 1 5 THE WEINSTEIN–MOSER THEOREM BY THE ASSUMPTIONS OF THE WEINSTEIN–MOSER THEOREM, THEOREM 1 12, THE POINT X = 0 IS AN ELLIPTIC EQUILIBRIUM INDEED, THE HAMILTONIAN OF THE LINEARIZED SYSTEM ?X = J(D2H)(0)X IS H2 = (1 14) THE NUMBER OF SOLUTIONS FOUND IN THEOREM 1 12 IS, IN GENERAL, OPTIMAL REMARK 1 15 (OPTIMALITY) A ?RST TRIVIAL EXAMPLE IS PROVIDED BY THE QUADRATIC HAMILTONIAN (1 14) WHERE THE FREQUENCY VECTOR ? = (?1, , ?N) OF THE DECOUPLED HARMONIC OSCILLATORS SATISFY THE NONRESONANCE CONDITION ? K /= 0 , ?K ?ZN \ 0 (1 15) IN THIS CASE THE ONLY PERIODIC SOLUTIONS OF (HS) HAVE THE FORM Q (Q J, P J) = A J(COS(? JT + ? J), SIN(? JT + ? J)) , QI = PI = 0 , ?I /= J R FOR J = 1, , N, AND A J IS UNIQUELY ?XED BY THE ENERGY LEVEL 2)(1 + H) = H (1 16) IN OTHER WORDS, THE ENERGY SURFACE {H = H} IS ALSO THE ENERGY SURFACE OF THE QUADRATIC HAMILTONIAN IN THE LEFT-HAND SIDE OF (1 16) BUT IF TWO HAMILTONIANS SHARE THE SAME ENERGY LEVEL, THEY POSSESS (ON THIS LEVEL) THE SAME TRAJECTORIES UP TO A REPARAMETRIZA- TION OF TIME 2 THEN ALL THE ORBITS OF THE HAMILTONIAN SYSTEM GENERATED BY H ON THE SURFACE {H = H} ARE HARMONIC OSCILLATIONS, AND IF 1 ?H 1 + H /?Q , THEN THE SYSTEM HAS EXACTLY ONLY N = 2 PERIODIC ORBITS REMARK 1 16 MOSER [98] PROVED ALSO AN ANALOGUE OF THEOREM 1 12 FOR NON-HAMIL- TONIAN SYSTEMS: IF (1 1) HAS A ?RST INTEGRAL G WITH (D2G)(0) > 0, THEN THERE EXISTS AT LEAST ONE PERIODIC SOLUTION ON EACH LEVEL {G(X) = G(0) + ?2} THE DIFFERENCE WITH RESPECT TO THE LYAPUNOV CENTER THEOREM, THEOREM 1 6, IS THAT THE NONRESONANCE CONDITION (1 9) IS NOT ASSUMED WE NOW DESCRIBE THE FADELL–RABINOWITZ RESONANT CENTER THEOREM THEOREM 1 17 (FADELL–RABINOWITZ 1978) LET H ?C2(R2N, R) WITH (?H)(0) = 0 ASSUME THAT ANY NONZERO SOLUTION OF THE LINEARIZED SYSTEM ?X = J(D2H)(0)X IS T-PERIODIC (AND NOT CONSTANT) IF SIGN(D2H)(0) = 2? /= 0 , THEN EITHER (I) X = 0 IS A NONISOLATED T -PERIODIC SOLUTION OF THE HAMILTONIAN SYSTEM (HS), OR 2 ASSUME M = {H(X) = H} = {F(X) = F } WITH ?H(X) /= 0, ?F(X) /= 0 ON M THEN ?F(X) = ?(X)?H(X) WITH ?(X) /= 0 ON M IF ?T DENOTES THE ?OW OF THE HAMILTONIAN VECTOR ?ELD J?H(X), AND ?S THE ?OW OF J?F(X), THEN WE HAVE, ON M, ?S(X) = ?T(X), WHERE T(S, X) SOLVES DT/DS = ?(?T(X)), T(0, X) = 0 1 2 THE WEINSTEIN–MOSER AND FADELL–RABINOWITZ RESONANT CENTER THEOREMS (II) THERE EXIST A PAIR OF INTEGERS K, M ?0 WITH K+M ?|?| AND A LEFT NEIGHBORHOOD, UL, RESP A RIGHT NEIGHBORHOOD, UR, OF T IN R SUCH THAT ?? ?UL \ {T}, RESP UR \{T }, THERE EXIST AT LEAST K, RESP M, DISTINCT, NONTRIVIAL, ?-PERIODIC SOLUTIONS OF THE HAMILTONIAN SYSTEM (HS) THE L?-NORM OF THE SOLUTIONS TENDS TO 0 AS ? ?T X X T T ? ? FIG 1 6 THE BIFURCATION DIAGRAM OF THE FADELL–RABINOWITZ THEOREM THE UPPER DIAGRAM REP- RESENTS CASE (I): THERE EXISTS A SEQUENCE OF NONTRIVIAL PERIODIC SOLUTIONS OF (HS) TENDING TO WITH PERIOD EQUAL TO T (THIS CASE HOLDS, FOR EXAMPLE, IF H ?H(0) + (1/2)(D2H)(0)X X IS QUADRATIC) IN CASE (II) THERE EXIST NONTRIVIAL PERIODIC SOLUTIONS WITH PERIOD ? NEAR T SUCH SOLUTIONS OF (HS) COULD NOT DEPEND CONTINUOUSLY WITH RESPECT TO THE PERIOD ? UNLIKE THE LYAPUNOV CENTER THEOREM (SEE REMARK 1 7), THE PERIODIC SOLUTIONS OF THEOREMS 1 12 AND 1 17 DO NOT, IN GENERAL, VARY SMOOTHLY WITH RESPECT TO THE PARAMETERS ? (ENERGY) AND ? (PERIOD) THIS WILL CAUSE A SERIOUS DIF?CULTY FOR IN?NITE-DIMENSIONAL PDES; SEE CHAPTER