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h(concen) n(trate) e(on) h (problems) g(in) n(v) n(olving) f(small) h(denominators) 515 2899 y(and) f(only) g(brie\015y) h(rep) r(ort) e(on) i(some) f(imp) r (ortan) n(t) g(results) g(of) h(a) f(di\013eren) n(t) h(kind.) p Fn 515 3171 a(2) 134 b(An) 45 b(abstract) g(theorem) h(for) f (nonresonan) l(t) h(PDEs) p Fm 515 3353 a(Let) p Fl 23 w(f) p Fk(X) p Fj 770 3365 a(s) p Fl 805 3353 a(g) p Fm 23 w(b) r(e) 24 b(a) f(scale) f(of) i(Hilb) r(ert) g(spaces) e(with) i(norms) p Fl 23 w(k) 10 b(\001) g(k) p Fj 2419 3365 a(s) p Fm 2477 3353 a(and) 23 b(scalar) f(pro) r(duct) p Fl 23 w(h\001) p Fm(;) p Fl 14 w(\001i) p Fj 3320 3365 a(s) p Fm 3356 3353 a(.) 515 3452 y(Let) p Fk 21 w(A) p Fm 21 w(b) r(e) g(a) e(\(linear\)) h(morphism) g(of) g(the) g(scale,) h (and) f(assume) f(that) h(there) g(exists) g(a) f(Hilb) r(ert) 515 3552 y(basis) p Fl 27 w(f) p Fk(') p Fj 815 3564 a(j) p Fl 849 3552 a(g) p Fi 891 3522 a(1) p Fj 891 3574 a(j) p Fh 3 w(=1) p Fm 1038 3552 a(suc) n(h) 27 b(that) p Fk 1532 3735 a(A') p Fj 1648 3747 a(j) p Fm 1707 3735 a(=) p Fk 22 w(!) p Fh 1849 3701 a(2) p Fj 1846 3755 a(j) p Fk 1886 3735 a(') p Fj 1940 3747 a(j) p Fk 2003 3735 a(;) 97 b(!) p Fj 2175 3747 a(j) p Fk 2233 3735 a(>) p Fm 22 w(0) 515 3904 y(Fix) p Fk 30 w(s) p Fm 31 w(and) 30 b(let) p Fk 31 w(g) p Fm 30 w(:) p Fk 28 w(X) p Fj 1213 3916 a(s) p Fl 1276 3904 a(\033) e(U) 36 b(!) p Fk 27 w(X) p Fj 1636 3916 a(s) p Fm 1672 3904 a(,) 31 b(where) p Fl 30 w(U) p Fm 39 w(is) f(a) g(neigh) n(b) r(ourho) r(o) r(d) f(of) i (the) g(origin,) f(b) r(e) 515 4004 y(a) p Fk 32 w(C) p Fi 654 3974 a(1) p Fm 756 4004 a(nonlinear) i(op) r(erator) e(ha) n (ving) h(at) i(the) f(origin) f(a) h(zero) f(of) h(second) g(order.) 49 b(W) -7 b(e) 33 b(are) 515 4104 y(in) n(terested) 27 b(in) h(families) f(of) h(small) f(amplitude) h(p) r(erio) r(dic) g (solutions) f(of) g(the) h(equation) 1665 4273 y(\177) p Fk -47 w(x) p Fm 18 w(+) p Fk 18 w(Ax) p Fm 24 w(=) p Fk 23 w(g) p Fm 3 w(\() p Fk(x) p Fm(\)) p Fk 28 w(:) p Fm 974 w(\(2.1\)) p Fg 515 4443 a(Example) p Fm 36 w(2.1) p Fg(.) p Fm 41 w(The) f(nonlinear) g(w) n(a) n(v) n(e) e(equation) i (with) h(p) r(erio) r(dic) f(b) r(oundary) f(conditions:) p Fk 1388 4612 a(w) p Fj 1447 4624 a(tt) p Fl 1520 4612 a(\000) p Fk 18 w(w) p Fj 1662 4624 a(xx) p Fm 1761 4612 a(+) p Fk 18 w(V) p Fm 18 w(\() p Fk(x) p Fm(\)) p Fk(w) p Fm 27 w(=) p Fk 23 w(f) p Fm 9 w(\() p Fk(x;) 14 b(w) p Fm 2 w(\)) p Fk 29 w(;) p Fm 702 w(\(2.2\)) p Fk 1023 4737 a(w) p Fm 2 w(\() p Fk(x;) g(t) p Fm(\)) 25 b(=) p Fk 22 w(w) p Fm 2 w(\() p Fk(x) p Fm 20 w(+) 18 b(2) p Fk(\031) s(;) c(t) p Fm(\)) p Fk 28 w(;) 97 b(w) p Fj 2015 4749 a(x) p Fm 2057 4737 a(\() p Fk(x;) 14 b(t) p Fm(\)) 24 b(=) p Fk 23 w(w) p Fj 2406 4749 a(x) p Fm 2448 4737 a(\() p Fk(x) p Fm 19 w(+) 18 b(2) p Fk(\031) s(;) c(t) p Fm(\)) p Fk 28 w(;) p Fm 337 w(\(2.3\)) 515 4907 y(where) 24 b(the) h(p) r(oten) n(tial) p Fk 25 w(V) p Fm 44 w(and) f(the) h (nonlinearit) n(y) p Fk 24 w(f) p Fm 33 w(are) f(p) r(erio) r(dic) h (of) g(p) r(erio) r(d) f(2) p Fk(\031) p Fm 28 w(in) p Fk 25 w(x) p Fm 25 w(and) p Fk 515 5006 a(f) p Fm 9 w(\() p Fk(x;) 14 b(w) p Fm 2 w(\)) 29 b(=) p Fk 28 w(O) p Fm 2 w(\() p Fl(j) p Fk(w) p Fl 2 w(j) p Fh 1100 4976 a(2) p Fm 1139 5006 a(\).) 46 b(The) 31 b(frequencies) p Fk 30 w(!) p Fj 1894 5018 a(j) p Fm 1959 5006 a(of) g(the) g(linearized) e (system) i(are) e(the) i(square) 1926 5255 y(1) p 90 rotate dyy eop %%Page: 2 2 2 1 bop Fm 515 523 a(ro) r(ots) 27 b(of) i(the) g(p) r(erio) r(dic) f (eigen) n(v) -5 b(alues) 27 b(of) i(the) g(Sturm) f(Liouville) g(op) r (erator) p Fl 27 w(\000) p Fk(@) p Fj 2995 535 a(xx) p Fm 3093 523 a(+) p Fk 19 w(V) p Fm 19 w(\() p Fk(x) p Fm(\),) 515 623 y(that) i(w) n(e) g(assume) f(to) h(b) r(e) g(p) r (ositiv) n(e.) 44 b(In) 30 b(this) h(case) p Fk 29 w(X) p Fj 2195 635 a(s) p Fm 2257 623 a(=) p Fk 27 w(H) p Fj 2425 593 a(s) p Fm 2460 623 a(\() p Ff(T) p Fm(\).) 44 b(Pro) n(vided) p Fk 29 w(s) 27 b(>) p Fm 26 w(1) p Fk(=) p Fm(2,) i(a) 515 722 y(smo) r(oth) p Fk 27 w(f) p Fm 36 w(induces) f(a) f(smo) r(oth) h(op) r(erator) d(from) p Fk 28 w(X) p Fj 2144 734 a(s) p Fm 2207 722 a(to) i(itself.) p 3318 722 4 57 v 3322 670 50 4 v 3322 722 V 3372 722 4 57 v Fg 515 844 a(Example) p Fm 37 w(2.2) p Fg(.) p Fm 41 w(The) g(nonlinear) g(plate) h(equation) f(in) g(the) p Fk 28 w(d) p Fm 28 w(dimensional) g(cub) r(e:) p Fk 1310 997 a(w) p Fj 1369 1009 a(tt) p Fm 1443 997 a(+) 18 b(\001\001) p Fk(w) p Fm 21 w(+) p Fk 18 w(aw) p Fm 26 w(=) p Fk 22 w(f) p Fm 9 w(\() p Fk(w) p Fm 2 w(\)) p Fk 29 w(;) 97 b(x) p Fl 23 w(2) 24 b(Q) p Fm 624 w(\(2.4\)) p Fk 1571 1131 a(w) p Fe 1632 1060 a(\014) 1632 1110 y(\014) p Fj 1660 1164 a(@) p Fi 4 w(Q) p Fm 1781 1131 a(=) e(\001) p Fk(w) p Fl 2 w(j) p Fj 2021 1143 a(@) p Fi 4 w(Q) p Fm 2143 1131 a(=) h(0) p Fk 27 w(;) p Fm 885 w(\(2.5\)) 515 1284 y(where) p Fk 27 w(a) g(>) p Fm 22 w(0) 28 b(and) p Fl 1173 1438 a(Q) p Fm 23 w(:=) p Fe 1374 1371 a(\010) p Fk 1423 1438 a(x) p Fm 23 w(=) 23 b(\() p Fk(x) p Fh 1660 1450 a(1) p Fk 1698 1438 a(;) 14 b(:::;) g(x) p Fj 1888 1450 a(d) p Fm 1927 1438 a(\)) p Fl 23 w(2) p Ff 24 w(R) p Fj 2115 1404 a(d) p Fm 2210 1438 a(:) 51 b(0) p Fk 23 w(<) 22 b(x) p Fj 2483 1450 a(i) p Fk 2534 1438 a(<) h(\031) p Fe 2672 1371 a(\011) p Fm 515 1592 a(Assume) p Fk 33 w(f) p Fm 9 w(\() p Fk(w) p Fm 2 w(\)) 34 b(=) p Fk 32 w(O) p Fm 2 w(\() p Fl(j) p Fk(w) p Fl 2 w(j) p Fh 1338 1561 a(3) p Fm 1377 1592 a(\),) h(then) f(the) f (eigenfunctions) g(of) h(the) f(linearized) g(system) g(are) 515 1691 y(giv) n(en) e(b) n(y) p Fk 33 w(') p Fj 911 1703 a(n) p Fm 988 1691 a(=) f(sin\() p Fk(n) p Fh 1267 1703 a(1) p Fk 1305 1691 a(x) p Fh 1352 1703 a(1) p Fm 1389 1691 a(\)) 14 b(sin\() p Fk(n) p Fh 1619 1703 a(2) p Fk 1657 1691 a(x) p Fh 1704 1703 a(2) p Fm 1742 1691 a(\)) p Fk(:::) p Fm 14 w(sin\() p Fk(n) p Fj 2041 1703 a(d) p Fk 2080 1691 a(x) p Fj 2127 1703 a(d) p Fm 2166 1691 a(\)) 33 b(and) f(the) h(corresp) r(onding) e(frequen-) 515 1791 y(cies) 39 b(are) p Fk 38 w(!) p Fj 886 1803 a(n) p Fm 974 1791 a(=) p Fe 1081 1720 a(p) p 1164 1720 695 4 v Fm 1164 1791 a(\() p Fk(n) p Fh 1246 1762 a(2) 1246 1813 y(1) p Fm 1302 1791 a(+) p Fk 18 w(:::) p Fm 19 w(+) p Fk 18 w(n) p Fh 1606 1762 a(2) p Fj 1606 1816 a(d) p Fm 1644 1791 a(\)) p Fh 1676 1767 a(2) p Fm 1732 1791 a(+) p Fk 18 w(a) p Fm 39 w(where) p Fk 39 w(n) p Fl 43 w(2) p Ff 43 w(Z) p Fj 2402 1761 a(d) p Fm 2474 1791 a(and) p Fk 40 w(n) p Fj 2698 1803 a(i) p Fl 2768 1791 a(\025) p Fm 42 w(1.) 72 b(T) -7 b(o) 39 b(\014t) h(the) 515 1890 y(ab) r(o) n(v) n(e) 25 b(sc) n(heme) h(w) n(e) g(order) f(the) i (basis) e(in) i(lexicographic) d(order.) 36 b(Here) p Fk 26 w(X) p Fh 2822 1902 a(0) p Fm 2882 1890 a(=) p Fk 22 w(L) p Fh 3026 1860 a(2) p Fm 3063 1890 a(\() p Fl(Q) p Fm(\),) 27 b(and) p Fk 515 1990 a(X) p Fj 584 2002 a(s) p Fm 650 1990 a(=) p Fk 31 w(D) p Fm 2 w(\(\(\001\001\)) p Fj 1051 1960 a(s) p Fm 1088 1990 a(\)) p Fl 32 w(\032) p Fk 30 w(H) p Fh 1323 1960 a(4) p Fj(s) p Fm 1424 1990 a(endo) n(w) n(ed) 32 b(b) n(y) g(the) h(graph) f(norm.) 51 b(If) 33 b(the) g(nonlinearit) n(y) p Fk 31 w(f) p Fm 41 w(is) 515 2090 y(smo) r(oth) c(and) g(o) r(dd) g(\(i.e.) p Fk 42 w(f) p Fm 9 w(\() p Fl(\000) p Fk(w) p Fm 2 w(\)) d(=) p Fl 25 w(\000) p Fk(f) p Fm 9 w(\() p Fk(w) p Fm 2 w(\)\)) k(then) g(it) f(de\014nes) g(a) g(smo) r(oth) g(map) g(from) p Fk 29 w(X) p Fj 3344 2102 a(s) p Fm 515 2189 a(to) e(itself) h(for) f(an) n (y) p Fk 27 w(s) c(>) p Fm 23 w([) p Fk(d=) p Fm(2]) p Fk(=) p Fm(4.) p 3318 2189 4 57 v 3322 2137 50 4 v 3322 2189 V 3372 2189 4 57 v 639 2311 a(In) 41 b(the) g(linear) f(appro) n (ximation) f(\() p Fk(g) p Fl 47 w(\021) p Fm 45 w(0\)) h(the) h (general) e(solution) h(of) h(\(2.1\)) f(is) h(the) 515 2410 y(sup) r(erp) r(osition) 18 b(of) i(the) f(linear) g(normal) f(mo) r(des,) j(i.e.) 34 b(of) 19 b(the) g(families) h(of) f(p) r(erio) r (dic) g(solutions) p Fk 1248 2564 a(x) p Fh 1295 2530 a(\() p Fj(j) p Fh 3 w(\)) p Fm 1382 2564 a(\() p Fk(t) p Fm(\)) 24 b(=) e(\() p Fk(a) p Fj 1663 2576 a(j) p Fm 1712 2564 a(cos) o(\() p Fk(!) p Fj 1907 2576 a(j) p Fk 1942 2564 a(t) p Fm(\)) d(+) p Fk 18 w(b) p Fj 2142 2576 a(j) p Fm 2191 2564 a(sin) o(\() p Fk(!) p Fj 2376 2576 a(j) p Fk 2411 2564 a(t) p Fm(\)\)) p Fk(') p Fj 2559 2576 a(j) p Fk 2623 2564 a(:) p Fm 562 w(\(2.6\)) 515 2717 y(Fix) 25 b(one) g(of) h(the) g(families,) g(sa) n(y) p Fk 24 w(x) p Fh 1557 2687 a(\(1\)) p Fm 1647 2717 a(;) g(to) f(ensure) g (its) h(p) r(ersistence) f(in) h(the) g(nonlinear) e(prob-) 515 2817 y(lem) k(w) n(e) f(mak) n(e) g(the) h(follo) n(wing) e (assumptions:) 545 2971 y(H1\)) 41 b(\(Nonresonance\)) 22 b(F) -7 b(or) 22 b(small) g(enough) p Fk 22 w(\015) 28 b(>) p Fm 23 w(0) 22 b(there) h(exists) f(a) g(closed) g(set) p Fk 23 w(W) p Fj 3110 2983 a(\015) p Fl 3176 2971 a(\032) p Ff 23 w(R) p Fh 3318 2940 a(+) p Fm 722 3070 a(ha) n(ving) p Fk 25 w(!) p Fh 1040 3082 a(1) p Fm 1102 3070 a(as) j(an) h(accum) n (ulation) e(p) r(oin) n(t) i(b) r(oth) g(from) g(the) g(righ) n(t) f (and) g(from) g(the) h(left) 722 3170 y(suc) n(h) i(that,) f(for) h(an) n(y) p Fk 26 w(!) p Fl 26 w(2) p Fk 24 w(W) p Fj 1631 3182 a(\015) p Fm 1701 3170 a(one) f(has) p Fl 1379 3348 a(j) p Fk(!) s(l) p Fl 20 w(\000) p Fk 18 w(!) p Fj 1637 3360 a(j) p Fl 1671 3348 a(j) d(\025) p Fk 1815 3292 a(\015) p 1815 3329 48 4 v 1826 3405 a(l) 1900 3348 y(;) p Fl 97 w(8) p Fk(l) p Fl 24 w(\025) p Fm 23 w(1) p Fk 27 w(;) p Fl 41 w(8) p Fk(j) p Fl 27 w(\025) p Fm 23 w(2) p 2662 3348 4 57 v 2666 3295 50 4 v 2666 3348 V 2715 3348 4 57 v 633 w(\(2.7\)) 545 3557 y(H2\)) 41 b (\(Nondegeneracy\)) c(Let) p Fk 39 w(g) p Fj 1574 3569 a(r) p Fm 1610 3557 a(\() p Fk(x) p Fm(\)) j(b) r(e) f(the) f(\014rst) h (non) f(v) -5 b(anishing) 38 b(T) -7 b(a) n(ylor) 36 b(\(homoge-) 722 3656 y(neous\)) 28 b(p) r(olynomial) f(of) p Fk 27 w(g) p Fm 3 w(;) g(assume) p Fk 27 w(r) p Fl 26 w(\025) p Fm 23 w(3) g(and) g(de\014ne) p Fk 1310 3860 a(\014) p Fh 1357 3872 a(0) p Fm 1417 3860 a(:=) p Fe 1528 3743 a(\032) p Fl 1632 3809 a(h) p Fk(g) p Fj 1704 3821 a(r) p Fm 1741 3809 a(\() p Fk(') p Fh 1827 3821 a(1) p Fm 1865 3809 a(\)) p Fk(;) 14 b(') p Fh 1988 3821 a(1) p Fl 2026 3809 a(i) p Fh 2058 3821 a(0) p Fm 2220 3809 a(if) p Fk 90 w(r) p Fm 86 w(is) 95 b(o) r(dd) p Fl 1590 3909 a(h) p Fk(g) p Fj 1662 3921 a(r) p Fh 2 w(+1) p Fm 1783 3909 a(\() p Fk(') p Fh 1869 3921 a(1) p Fm 1907 3909 a(\)) p Fk(;) 14 b(') p Fh 2030 3921 a(1) p Fl 2068 3909 a(i) p Fh 2100 3921 a(0) p Fm 2220 3909 a(if) p Fk 90 w(r) p Fm 86 w(is) 83 b(ev) n(en) 3208 3860 y(\(2.8\)) 722 4059 y(Assume) p Fk 28 w(\014) p Fh 1077 4071 a(0) p Fl 1137 4059 a(6) p Fm(=) 23 b(0.) p 3318 4059 V 3322 4006 50 4 v 3322 4059 V 3372 4059 4 57 v 639 4212 a(Denoting) p Fk 28 w(\030) p Fh 1034 4224 a(1) p Fm 1071 4212 a(\() p Fk(!) p Fh 1155 4224 a(1) p Fk 1193 4212 a(t) p Fm(\)) g(=) g(cos) o(\() p Fk(!) p Fh 1561 4224 a(1) p Fk 1598 4212 a(t) p Fm(\)) p Fk(') p Fh 1714 4224 a(1) p Fm 1779 4212 a(one) 28 b(has) p Fd 515 4355 a(Theorem) 40 b(2.3.) p Fg 45 w(Supp) l(ose) e(the) f (assumptions) h(H1,H2) g(hold,) k(then) 37 b(ther) l(e) g(exists) g(a) h (set) p Fl 515 4455 a(E) 45 b(\032) p Ff 37 w(R) p Fg 43 w(having) 39 b(zer) l(o) f(as) h(an) e(ac) l(cumulation) h(p) l (oint,) j(a) d(p) l(ositive) p Fk 39 w(!) p Fi 2698 4467 a(\003) p Fg 2736 4455 a(,) i(and) e(a) h(family) g(of) 515 4554 y(p) l(erio) l(dic) 32 b(solutions) p Fl 30 w(f) p Fk(x) p Fj 1263 4566 a(\017) p Fm 1294 4554 a(\() p Fk(t) p Fm(\)) p Fl(g) p Fj 1430 4566 a(\017) p Fi(2E) p Fg 1577 4554 a(of) f(\(2.1\)) g(with) f(fr) l(e) l(quencies) p Fl 30 w(f) p Fk(!) p Fj 2583 4524 a(\017) p Fl 2614 4554 a(g) p Fj 2656 4566 a(\017) p Fi(2E) p Fg 2803 4554 a(ful\014l) t(ling) p Fm 984 4708 a(sup) p Fj 1034 4774 a(t) p Fl 1123 4708 a(k) p Fk(x) p Fj 1212 4720 a(\017) p Fm 1243 4708 a(\() p Fk(t) p Fm(\)) p Fl 19 w(\000) p Fk 18 w(\017\030) p Fh 1509 4720 a(1) p Fm 1547 4708 a(\() p Fk(t!) p Fj 1664 4674 a(\017) p Fm 1696 4708 a(\)) p Fl(k) p Fj 1770 4720 a(s) p Fl 1828 4708 a(\024) p Fk 22 w(C) 6 b(\017) p Fj 2014 4674 a(r) p Fk 2081 4708 a(;) p Fl 99 w(j) p Fk(!) p Fj 2281 4674 a(\017) p Fl 2331 4708 a(\000) p Fk 18 w(!) p Fh 2466 4720 a(1) p Fl 2503 4708 a(j) 23 b(\024) p Fk 22 w(C) 6 b(\017) p Fj 2735 4674 a(r) p Fi 2 w(\000) p Fh(1) p Fk 2887 4708 a(:) p Fm 298 w(\(2.9\)) p Fg 515 4907 a(Mor) l(e) l(over) 24 b(the) f(set) p Fl 23 w(E) p Fg 30 w(is) g(in) g(one) g(to) g(one) g(c) l(orr) l(esp) l (ondenc) l(e) h(either) g(with) p Fk 23 w(W) p Fj 2816 4919 a(\015) p Fl 2863 4907 a(\\) p Fm 4 w([) p Fk(!) p Fh 2997 4919 a(1) p Fk 3034 4907 a(;) 14 b(!) p Fh 3123 4919 a(1) p Fm 3163 4907 a(+) p Fk 4 w(!) p Fi 3284 4919 a(\003) p Fm 3321 4907 a(\)) p Fg(,) 515 5006 y(if) p Fk 30 w(\014) p Fh 642 5018 a(0) p Fk 703 5006 a(<) p Fm 22 w(0) p Fg(,) 30 b(or) g(with) p Fk 30 w(W) p Fj 1252 5018 a(\015) p Fl 1314 5006 a(\\) p Fm 19 w(\() p Fk(!) p Fh 1472 5018 a(1) p Fl 1527 5006 a(\000) p Fk 18 w(!) p Fi 1662 5018 a(\003) p Fk 1700 5006 a(;) 14 b(!) p Fh 1789 5018 a(1) p Fm 1826 5006 a(]) p Fg(,) 30 b(if) p Fk 31 w(\014) p Fh 2032 5018 a(0) p Fk 2092 5006 a(>) p Fm 23 w(0) p Fg(.) p Fm 1926 5255 a(2) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fd 515 523 a(Pro) s(of.) p Fm 49 w(W) -7 b(e) 32 b(consider) f(only) g(the) h(case) f(of) h(o) r(dd) p Fk 32 w(r) p Fm 2 w(,) i(the) e(general) f(case) g(can) g(b) r(e) h (obtained) 515 623 y(b) n(y) i(a) g(sligh) n(tly) g(di\013eren) n(t) h (treatmen) n(t) f(of) h(the) g(forthcoming) e(equation) p Fk 34 w(!) p Fm 3 w(.) 58 b(Lo) r(ok) 33 b(for) h(an) p Fk 515 722 a(X) p Fj 584 734 a(s) p Fm 644 722 a(v) -5 b(alued) 25 b(function) p Fk 26 w(q) p Fm 3 w(\() p Fk(t) p Fm(\)) g(p) r(erio) r(dic) g(of) g(p) r(erio) r(d) g(2) p Fk(\031) p Fm 28 w(and) g(rev) n(ersible) f(\(namely) h(s.t.) p Fk 36 w(q) p Fm 3 w(\() p Fk(t) p Fm(\)) e(=) p Fk 515 822 a(q) p Fm 3 w(\() p Fl(\000) p Fk(t) p Fm(\)\)) 34 b(and) g(for) f(a) h(p) r(ositiv) n(e) p Fk 34 w(!) p Fm 36 w(close) f(to) p Fk 34 w(!) p Fh 1924 834 a(1) p Fm 1995 822 a(suc) n(h) h(that) p Fk 34 w(q) p Fm 3 w(\() p Fk(!) s(t) p Fm(\)) g(is) g(a) g(solution) f(of) h(\(2.1\).) 515 922 y(They) 27 b(m) n(ust) h(satisfy) f(the) h(equation) p Fk 1362 1153 a(L) p Fj 1419 1165 a(!) p Fk 1467 1153 a(q) p Fm 26 w(=) p Fk 22 w(g) p Fm 3 w(\() p Fk(q) p Fm 3 w(\)) p Fk 28 w(;) 42 b(L) p Fj 1914 1165 a(!) p Fm 1984 1153 a(:=) p Fk 23 w(!) p Fh 2150 1118 a(2) p Fk 2212 1097 a(d) p Fh 2255 1066 a(2) p 2197 1134 111 4 v Fk 2197 1210 a(dt) p Fh 2270 1186 a(2) p Fm 2336 1153 a(+) p Fk 18 w(A) 28 b(:) p Fm 635 w(\(2.10\)) 515 1359 y(whic) n(h) 34 b(will) i(b) r(e) f(considered) f(as) g(an) p Fk 34 w(!) p Fm 38 w(dep) r(enden) n(t) h(functional) g(equation) g(in) g(the) g(space) p Fl 515 1458 a(H) 24 b(\032) p Fk 23 w(H) p Fh 773 1428 a(1) p Fm 810 1458 a(\() p Ff(T) p Fk(;) 14 b(X) p Fj 1004 1470 a(s) p Fm 1039 1458 a(\)) 28 b(con) n(taining) e(the) i(rev) n(ersible) e(p) r(erio) r(dic) i (functions.) 38 b(Equation) 26 b(\(2.10\)) h(is) 515 1558 y(studied) d(using) f(the) h(Ly) n(apuno) n(v{Sc) n(hmidt) e (decomp) r(osition,) i(namely) g(b) n(y) f(decomp) r(osing) g(it) 515 1658 y(in) n(to) 30 b(an) h(equation) f(on) g(Ker) p Fk(L) p Fj 1457 1670 a(!) p Fc 1499 1678 a(1) p Fl 1563 1658 a(\021) p Fm(span\() p Fk(\030) p Fh 1863 1670 a(1) p Fm 1900 1658 a(\)) h(and) f(an) h(equation) f(on) h(its) g(complemen) n (t) p Fk 30 w(R) p Fm 1 w(.) 515 1757 y(Precisely) -7 b(,) 27 b(denote) i(b) n(y) p Fk 28 w(Q) p Fm 28 w(the) g(pro) 5 b(jector) 27 b(on) p Fk 29 w(\030) p Fh 2017 1769 a(1) p Fm 2083 1757 a(and) h(b) n(y) p Fk 29 w(P) p Fm 40 w(the) h(pro) 5 b(jector) 27 b(on) p Fk 28 w(R) p Fm 1 w(,) i(mak) n(e) 515 1857 y(the) f(Ansatz) p Fk 27 w(q) p Fm 26 w(=) p Fk 23 w(\017\030) p Fh 1158 1869 a(1) p Fm 1214 1857 a(+) p Fk 18 w(\017) p Fj 1331 1827 a(r) p Fk 1367 1857 a(u) p Fm(,) f(with) p Fk 29 w(u) p Fl 22 w(2) p Fk 24 w(R) p Fm 1 w(:) 36 b(Then) 28 b(\(2.10\)) f(is) g(equiv) -5 b(alen) n(t) 28 b(to) f(the) h(system) p Fk 1645 2039 a(!) p Fh 1700 2005 a(2) p Fm 1760 2039 a(=) p Fk 23 w(!) p Fh 1903 2005 a(2) 1900 2060 y(1) p Fm 1958 2039 a(+) p Fk 18 w(\014) t(\017) p Fj 2126 2005 a(r) p Fi 2 w(\000) p Fh(1) p Fm 3167 2039 a(\(2.11\)) p Fk 1469 2164 a(L) p Fj 1526 2176 a(!) p Fk 1573 2164 a(u) p Fm 23 w(=) p Fk 22 w(P) 12 b(g) p Fj 1836 2176 a(r) p Fm 1873 2164 a(\() p Fk(\030) p Fh 1941 2176 a(1) p Fm 1979 2164 a(\)) 18 b(+) p Fk 18 w(P) 12 b(G) p Fm(\() p Fk(\017;) i(u) p Fm(\)) 742 b(\(2.12\)) p Fl 1449 2289 a(\000) p Fk(\014) t(\030) p Fh 1601 2301 a(1) p Fm 1662 2289 a(=) p Fk 22 w(Qg) p Fj 1855 2301 a(r) p Fm 1891 2289 a(\() p Fk(\030) p Fh 1959 2301 a(1) p Fm 1997 2289 a(\)) 19 b(+) p Fk 18 w(QG) p Fm(\() p Fk(\017;) 14 b(u) p Fm(\)) 722 b(\(2.13\)) 515 2471 y(for) 37 b(the) h(unkno) n(wns) f(\() p Fk(\017;) 14 b(u;) g(\014) p Fm 4 w(\):) 56 b(Here) p Fk 38 w(G) p Fm 38 w(con) n(tains) 36 b(all) i(higher) e(order) h(corrections) e (and) p Fk 515 2571 a(!) p Fl 29 w(2) p Fk 28 w(W) p Fj 757 2583 a(\015) p Fm 830 2571 a(is) 30 b(a) f(parameter.) 43 b(The) 30 b(equations) f(\(2.11\),) h(\(2.12\)) f(and) h(\(2.13\)) f (are) g(called) h(the) p Fk 515 2670 a(!) p Fm 3 w(,) d(the) h(P) f (and) h(the) g(Q) f(equation) g(resp) r(ectiv) n(ely) -7 b(.) 639 2770 y(First) 20 b(one) f(solv) n(es) f(the) i(P) f(equation) g (\(2.12\).) 33 b(T) -7 b(o) 19 b(this) h(end) g(one) f(has) g(to) g(in) n(v) n(ert) g(the) h(linear) 515 2870 y(op) r(erator) p Fk 22 w(L) p Fj 903 2882 a(!) p Fe 950 2799 a(\014) 950 2849 y(\014) p Fj 978 2903 a(R) p Fm 1032 2870 a(.) 36 b(Its) 24 b(eigenfunctions) f(are) g(giv) n(en) g(b) n(y) g(cos\() p Fk(l) r(t) p Fm(\)) p Fk(') p Fj 2494 2882 a(j) p Fm 2529 2870 a(,) h(and) g(the) g(corresp) r(onding) 515 2969 y(eigen) n(v) -5 b(alues) 26 b(are) p Fk 966 3152 a(\025) p Fj 1014 3164 a(j) s(l) p Fm 1094 3152 a(=) p Fl 23 w(\000) p Fk(l) p Fh 1274 3118 a(2) p Fk 1310 3152 a(!) p Fh 1365 3118 a(2) p Fm 1420 3152 a(+) p Fk 18 w(!) p Fh 1558 3118 a(2) p Fj 1555 3172 a(j) p Fm 1618 3152 a(=) d(\() p Fk(l) r(!) p Fm 21 w(+) p Fk 18 w(!) p Fj 1973 3164 a(j) p Fm 2007 3152 a(\)\() p Fk(!) p Fj 2123 3164 a(j) p Fl 2177 3152 a(\000) p Fk 18 w(l) r(!) p Fm 3 w(\)) p Fk 27 w(;) 42 b(j) p Fl 28 w(\025) p Fm 22 w(2) p Fk 27 w(;) g(l) p Fl 24 w(\025) p Fm 23 w(1) 515 3346 y(By) 19 b(\(2.7\),) p Fl 20 w(j) p Fk(\025) p Fj 922 3358 a(j) s(l) p Fl 979 3346 a(j) p Fk 23 w(>) k(C) 6 b(\015) p Fm 5 w(.) 34 b(So) 19 b(\() p Fk(L) p Fj 1479 3358 a(!) p Fe 1527 3275 a(\014) 1527 3325 y(\014) p Fj 1554 3379 a(R) p Fm 1609 3346 a(\)) p Fi 1641 3316 a(\000) p Fh(1) p Fm 1750 3346 a(exists) f(and) h(is) g(b) r(ounded.) 35 b(Applying) 19 b(this) h(op) r(erator) 515 3445 y(to) j(the) h(P) e (equation) h(and) g(using) g(the) h(implicit) g(function) g(theorem) f (one) g(obtains) f(a) h(smo) r(oth) 515 3545 y(function) p Fk 22 w(u) p Fm(\() p Fk(\017) p Fm(\)) e(that) h(dep) r(ends) g (parametrically) e(on) p Fk 21 w(!) p Fl 26 w(2) p Fk 23 w(W) p Fj 2379 3557 a(\015) p Fm 2444 3545 a(and) h(solv) n(es) f (the) i(P) g(equation.) 639 3645 y(Inserting) p Fk 18 w(u) p Fm(\() p Fk(\017) p Fm(\)) d(in) g(the) p Fk 19 w(Q) p Fm 19 w(equation) f(one) g(determines) h(the) g(parameter) p Fk 17 w(\014) p Fm 23 w(as) f(a) g(function) 515 3744 y(of) p Fk 28 w(\017) p Fm(:) 39 b(In) 29 b(particular) e(one) i(has) p Fk 28 w(\014) p Fm 4 w(\() p Fk(\017) p Fm(\)) c(=) p Fk 24 w(C) 6 b(\014) p Fh 1873 3756 a(0) p Fm 1911 3744 a(+higher) 27 b(order) h(corrections,) f(where) p Fk 28 w(C) k(>) p Fm 24 w(0.) 515 3844 y(Inserting) p Fk 31 w(\014) p Fm 4 w(\() p Fk(\017) p Fm(\)) h(in) g(the) p Fk 32 w(!) p Fm 34 w(equation) f(one) g(gets) g(an) h(equation) f(for) p Fk 31 w(\017) p Fm 31 w(\(remem) n(b) r(er) g(that) p Fk 32 w(!) p Fm 515 3943 a(is) h(\014xed\)) h(whic) n(h) g(is) f(a) g (p) r(erturbation) g(of) h(the) g(equation) p Fk 32 w(!) p Fh 2385 3913 a(2) p Fl 2443 3943 a(\000) p Fk 22 w(!) p Fh 2585 3913 a(2) 2582 3964 y(1) p Fm 2653 3943 a(=) p Fk 31 w(C) 6 b(\014) p Fh 2861 3955 a(0) p Fk 2898 3943 a(\017) p Fj 2932 3913 a(r) p Fi 2 w(\000) p Fh(1) p Fm 3054 3943 a(.) 51 b(By) 33 b(the) 515 4043 y(nondegeneracy) 28 b(this) j(can) f(b) r(e) h(reduced) f(to) g(a) g(\014xed) h(p) r(oin) n (t) f(equation) g(for) p Fk 30 w(\017) p Fj 2931 4013 a(r) p Fi 2 w(\000) p Fh(1) p Fm 3083 4043 a(whic) n(h) g(is) 515 4143 y(solv) n(ed) c(b) n(y) i(the) g(con) n(traction) e(mapping) h (principle.) p 3318 4143 4 57 v 3322 4090 50 4 v 3322 4143 V 3372 4143 4 57 v Fg 515 4276 a(R) l(emark) p Fm 47 w(2.4) p Fg(.) p Fm 47 w(The) 37 b(theorem) g(holds) f(also) g(in) i (the) f(case) p Fk 36 w(r) p Fm 42 w(=) h(2) f(but) h(in) f(this) g (case) g(the) 515 4375 y(nondegeneracy) 25 b(condition) j(tak) n(es) e (a) i(more) e(complicated) i(form.) p 3318 4375 V 3322 4322 50 4 v 3322 4375 V 3372 4375 4 57 v 639 4508 a(Theorem) i(2.3) g (w) n(as) g(pro) n(v) n(ed) g(in) h([Bam00) n(].) 47 b(The) 31 b(tec) n(hnique) g(of) g(Ly) n(apuno) n(v{Sc) n(hmidt) 515 4608 y(decomp) r(osition) 25 b(w) n(as) h(used) g(for) g(the) g (\014rst) g(time) h(to) f(construct) g(families) g(of) h(p) r(erio) r (dic) f(solu-) 515 4707 y(tions) g(in) g(PDEs) g(b) n(y) f(Craig) g (and) h(W) -7 b(a) n(yne) 26 b([CW93) o(]) h(who) f(considered) f(the) h (mo) r(del) h(problem) 515 4807 y(of) j(the) h(w) n(a) n(v) n(e) d (equation) i(with) h(p) r(erio) r(dic) f(b) r(oundary) f(conditions) h (\(see) g(example) g(2.1\);) h(w) n(e) 515 4906 y(will) d(rep) r(ort) e (on) i(this) g(w) n(ork) e(in) i(Section) f(4.) 1926 5255 y(3) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fg 515 523 a(Example) p Fm 47 w(2.5) p Fg(.) p Fm 48 w(Consider) 38 b(the) i(nonlinear) e(w) n(a) n(v) n(e) g (equation) g(with) i(p) r(erio) r(dic) f(b) r(oundary) 515 623 y(conditions) e(\(see) i(example) e(2.1\).) 69 b(Let) p Fk 38 w(!) p Fh 1877 635 a(1) p Fm 1952 623 a(b) r(e) 39 b(suc) n(h) f(that) p Fk 38 w(!) p Fh 2516 635 a(1) p Fl 2594 623 a(6) p Fm(=) p Fk 40 w(!) p Fj 2751 635 a(j) p Fm 2824 623 a(for) g(an) n(y) p Fk 37 w(j) p Fl 46 w(6) p Fm(=) i(1.) 515 731 y(Decomp) r(ose) p Fk 22 w(V) p Fm 42 w(in) n(to) 22 b(its) h(a) n(v) n(erage) p Fk 20 w(a) p Fm 23 w(and) f(a) g(part) 2077 710 y(~) p Fk 2065 731 a(V) p Fm 41 w(of) h(zero) f(a) n(v) n(erage,) f(then) i(condition) f (H1) 515 831 y(is) i(satis\014ed) f(if) p Fk 24 w(a) p Fm 24 w(b) r(elongs) g(to) h(an) g(uncoun) n(table) f(set) h(whic) n(h) g(is) g(dense) f(in) i(a) e(neigh) n(b) r(ourho) r(o) r(d) 515 930 y(of) 37 b(the) g(origin) f(\(for) g(the) i(pro) r(of) e(see) g (lemma) h(3.1) f(of) h([BP02) n(]\).) 66 b(Condition) 36 b(H2) h(can) g(b) r(e) 515 1030 y(expressed) f(in) i(terms) f(of) h (the) g(eigenfunctions) f(of) h(the) g(Sturm) f(Liouville) h(op) r (erator.) 65 b(If) 515 1130 y(it) 30 b(holds) f(then) h(theorem) f(2.3) f(applies) i(and) f(ensures) g(p) r(ersistence) g(of) g(the) h(corresp) r(onding) 515 1229 y(family) e(of) g(p) r(erio) r(dic) f(orbits.) 38 b(Remark) 27 b(that,) h(as) f(a) h(di\013erence) g(with) g(resp) r(ect) g(to) g(the) g(case) 515 1329 y(of) k(Diric) n(hlet) g(b) r(oundary) f (conditions) h(the) h(nonlinearit) n(y) e(do) r(es) g(not) h(need) h (to) f(ha) n(v) n(e) f(some) 515 1429 y(particular) 26 b(parit) n(y) -7 b(.) p 3318 1429 4 57 v 3322 1376 50 4 v 3322 1429 V 3372 1429 4 57 v Fg 515 1556 a(Example) p Fm 39 w(2.6) p Fg(.) p Fm 43 w(Consider) 30 b(the) h(nonlinear) f (plate) g(equation) g(\(see) h(example) f(2.2\).) 46 b(In) 31 b(the) 515 1656 y(case) p Fk 30 w(d) p Fm 29 w(=) e(1) i(all) g(the) g(frequencies) g(are) f(simple) h(and) g(the) h (assumption) f(H1) g(is) g(satis\014ed) g(if) p Fk 515 1756 a(a) p Fm 31 w(is) g(c) n(ho) r(osen) f(in) h(a) f(subset) h(of) p Ff 31 w(R) p Fh 1572 1725 a(+) p Fm 1665 1756 a(ha) n(ving) f(full) h (measure.) 46 b(In) 32 b(the) f(case) p Fk 30 w(d) e(>) p Fm 28 w(1,) j(all) f(the) 515 1855 y(frequencies) c(are) f(m) n (ultiple) i(except) g(the) g(smallest) f(one.) 37 b(T) -7 b(aking) 26 b(for) p Fk 27 w(!) p Fh 2766 1867 a(1) p Fm 2831 1855 a(suc) n(h) h(a) g(smallest) 515 1955 y(frequency) -7 b(,) 31 b(H1) f(is) h(ful\014lled) h(if) p Fk 31 w(a) p Fm 31 w(b) r(elongs) e(to) g(a) g(dense) h(uncoun) n(table) f(subset) h (of) g([0,1/4].) 515 2054 y(H2) 20 b(is) h(automatic) f(pro) n(vided) g (the) h(T) -7 b(a) n(ylor) 19 b(expansion) h(of) p Fk 20 w(f) p Fm 30 w(at) g(zero) g(do) r(es) g(not) h(v) -5 b(anish) 21 b(at) f(all) 515 2154 y(orders) 28 b(\(remem) n(b) r(er) i (that) p Fk 30 w(f) p Fm 9 w(\() p Fl(\000) p Fk(w) p Fm 2 w(\)) e(=) p Fk 27 w(f) p Fm 9 w(\() p Fk(w) p Fm 2 w(\)\).) 46 b(Then) 30 b(theorem) g(2.3) f(ensures) g(p) r (ersistence) 515 2254 y(of) e(the) h(corresp) r(onding) e(family) i(of) f(p) r(erio) r(dic) h(orbits) f(\(for) g(details) g(see) g([BP02) o (]\).) p 3318 2254 V 3322 2201 50 4 v 3322 2254 V 3372 2254 4 57 v Fn 515 2526 a(3) 134 b(The) 45 b(resonan) l(t) h(case) p Fm 515 2708 a(It) 27 b(is) g(p) r(ossible) g(to) f(generalize) g(the) h (ab) r(o) n(v) n(e) f(theorem) g(to) h(the) h(case) e(where) g(the) i (frequencies) 515 2807 y(ful\014ll) 21 b(some) e(resonance) f (relations.) 34 b(W) -7 b(e) 20 b(will) g(consider) f(only) h(the) g (Lagrangian) d(case) j(where) p Fk 515 2907 a(g) p Fm 25 w(=) p Fl 23 w(\000r) p Fk(H) p Fm 7 w(.) 639 3007 y(Fix) 41 b(a) f(frequency) p Fk 41 w(!) p Fh 1325 3019 a(1) p Fm 1402 3007 a(of) h(the) g(linearized) f(system,) k(then) d (the) g(assumption) g(H1) f(is) 515 3106 y(substituted) 28 b(b) n(y) 484 3262 y(H1R\)) 41 b(F) -7 b(or) 23 b(an) n(y) f(small) h (enough) p Fk 23 w(\015) p Fm 28 w(there) g(exists) g(a) g(closed) f (set) p Fk 23 w(W) p Fj 2526 3274 a(\015) p Fl 2592 3262 a(\032) p Ff 23 w(R) p Fh 2734 3232 a(+) p Fm 2818 3262 a(ha) n(ving) p Fk 23 w(!) p Fh 3134 3274 a(1) p Fm 3194 3262 a(as) g(an) 722 3362 y(accum) n(ulation) 31 b(p) r(oin) n(t) h(b) r (oth) g(from) f(the) h(righ) n(t) f(and) g(from) g(the) h(left) g(and) g (suc) n(h) f(that) 722 3461 y(for) c(an) n(y) p Fk 27 w(!) p Fl 26 w(2) p Fk 23 w(W) p Fj 1240 3473 a(\015) p Fm 1311 3461 a(one) g(has) 1154 3656 y(either) p Fl 166 w(j) p Fk(!) s(l) p Fl 19 w(\000) p Fk 18 w(!) p Fj 1785 3668 a(j) p Fl 1820 3656 a(j) c(\025) p Fk 1964 3599 a(\015) p 1964 3637 48 4 v 1975 3713 a(l) 2049 3656 y(;) p Fm 180 w(or) p Fk 165 w(l) r(!) p Fh 2571 3668 a(1) p Fl 2626 3656 a(\000) p Fk 18 w(!) p Fj 2761 3668 a(j) p Fm 2818 3656 a(=) g(0) 260 b(\(3.1\)) p 3318 3847 4 57 v 3322 3794 50 4 v 3322 3847 V 3372 3847 4 57 v 639 4003 a(T) -7 b(o) 27 b(come) h(to) f(the) h(nondegeneracy) e(assumption) h (de\014ne) h(the) g(resonan) n(t) e(set) h(b) n(y) p Fl 1222 4173 a(I) p Fj 1267 4185 a(R) p Fm 1345 4173 a(:=) p Fl 22 w(f) p Fk(k) p Fl 26 w(\025) p Fm 22 w(1) 51 b(:) p Fl 50 w(9) p Fk(l) p Fl 25 w(\025) p Fm 23 w(1) f(:) p Fk 51 w(l) r(!) p Fh 2248 4185 a(1) p Fl 2303 4173 a(\000) p Fk 18 w(!) p Fj 2438 4185 a(k) p Fm 2501 4173 a(=) 23 b(0) p Fl(g) p Fm 535 w(\(3.2\)) 515 4343 y(and) i(let) p Fl 25 w(N) p Fm 37 w(b) r(e) h(the) f(space) g(obtained) f(b) n(y) h (closing) f(the) h(space) g(generated) f(b) n(y) p Fl 24 w(f) p Fk(') p Fj 3032 4355 a(k) p Fl 3073 4343 a(g) p Fj 3115 4355 a(k) p Fi 1 w(2I) p Fb 3234 4363 a(R) p Fm 3310 4343 a(in) 515 4442 y(the) d(graph) e(norm) h(of) p Fk 21 w(D) p Fm 2 w(\() p Fk(A) p Fm(\).) 35 b(Remark) 20 b(that) h(all) f(solutions) g(of) g(the) h(linearized) f(system) h (with) 515 4542 y(initial) 30 b(datum) g(in) p Fl 30 w(N) p Fm 42 w(and) g(v) -5 b(anishing) 29 b(initial) h(v) n(elo) r (cit) n(y) f(are) g(p) r(erio) r(dic) g(of) h(p) r(erio) r(d) g(2) p Fk(\031) s(=!) p Fh 3320 4554 a(1) p Fm 3356 4542 a(.) 515 4642 y(Let) p Fk 26 w(H) p Fj 731 4654 a(r) p Fm 793 4642 a(b) r(e) d(the) f(\014rst) g(non) f(v) -5 b(anishing) 26 b(T) -7 b(a) n(ylor) 24 b(co) r(e\016cien) n(t) h(of) p Fk 26 w(H) p Fm 33 w(and,) h(for) p Fk 25 w(x) p Fl 24 w(2) d(N) p Fm 12 w(,) k(de\014ne) 515 4741 y(the) h(a) n(v) n(erage) c (of) p Fk 28 w(H) p Fj 1119 4753 a(r) p Fm 1183 4741 a(b) n(y) p Fl 1230 4972 a(h) p Fk(H) p Fj 1331 4984 a(r) p Fl 1368 4972 a(i) p Fm(\() p Fk(x) p Fm(\)) g(:=) p Fk 1657 4916 a(!) p Fh 1709 4928 a(1) p 1656 4953 92 4 v Fm 1656 5029 a(2) p Fk(\031) p Fe 1771 4859 a(Z) p Fh 1854 4880 a(2) p Fj(\031) r(=!) p Fc 2004 4888 a(1) p Fh 1817 5048 a(0) p Fk 2054 4972 a(H) p Fj 2123 4984 a(r) p Fm 2160 4972 a(\(cos\() p Fk(At) p Fm(\)) p Fk(x) p Fm(\)) p Fk(dt) p Fm 29 w(;) 1926 5255 y(4) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Fm 515 523 a(consider) 26 b(the) i(h) n(yp) r(ersurface) p Fl 26 w(S) i(\032) 23 b(N) p Fm 40 w(of) k(the) h(p) r(oin) n(ts) p Fk 28 w(x) p Fl 23 w(2) c(N) p Fm 40 w(suc) n(h) j(that) p Fl 28 w(h) p Fk(x) p Fm(;) p Fk 14 w(Ax) p Fl(i) p Fh 3110 535 a(0) p Fm 3172 523 a(=) 22 b(1.) 484 676 y(H2R\)) 41 b(There) 34 b(exists) g(a) f(nondegenerate) g(critical) h(p) r(oin) n (t) p Fk 34 w(x) p Fh 2390 688 a(0) p Fm 2462 676 a(of) g(the) g (functional) p Fl 35 w(h) p Fk(H) p Fj 3210 688 a(r) p Fl 3247 676 a(i) p Fe 3279 606 a(\014) 3279 656 y(\014) p Fi 3307 709 a(S) p Fm 3356 676 a(,) 722 776 y(and) 28 b(the) g(corresp) r(onding) d(Lagrange) h(m) n(ultiplier) h(do) r(es) h (not) f(v) -5 b(anish.) p 3318 776 4 57 v 3322 723 50 4 v 3322 776 V 3372 776 4 57 v 639 929 a(Denote) 32 b(b) n(y) p Fk 32 w(\030) p Fh 1084 941 a(0) p Fm 1121 929 a(\() p Fk(!) p Fh 1205 941 a(1) p Fk 1243 929 a(t) p Fm(\)) g(the) g(solution) g(of) f(the) i(linearized) e(system) g(with) i(initial) f(datum) p Fk 515 1029 a(x) p Fh 562 1041 a(0) p Fm 627 1029 a(and) c(v) -5 b(anishing) 27 b(initial) h(v) n(elo) r(cit) n(y) -7 b(.) p Fd 515 1182 a(Theorem) 46 b(3.1.) p Fg 48 w([BP01) r(]) c(Supp) l (ose) h(the) f(assumptions) h(H1R,H2R) f(hold,) 47 b(then) 42 b(ther) l(e) 515 1282 y(exists) 25 b(a) h(family) i(of) e(p) l(erio) l (dic) i(solutions) p Fl 26 w(f) p Fk(x) p Fj 1888 1294 a(\017) p Fm 1920 1282 a(\() p Fk(t) p Fm(\)) p Fl(g) p Fj 2056 1294 a(\017) p Fi(2E) p Fg 2199 1282 a(of) f(\(2.1\)) g(with) f (fr) l(e) l(quencies) p Fl 26 w(f) p Fk(!) p Fj 3189 1252 a(\017) p Fl 3220 1282 a(g) p Fj 3262 1294 a(\017) p Fi(2E) p Fg 515 1381 a(ful\014l) t(ling) p Fm 984 1481 a(sup) p Fj 1034 1547 a(t) p Fl 1123 1481 a(k) p Fk(x) p Fj 1212 1493 a(\017) p Fm 1243 1481 a(\() p Fk(t) p Fm(\)) p Fl 19 w(\000) p Fk 18 w(\017\030) p Fh 1509 1493 a(0) p Fm 1547 1481 a(\() p Fk(t!) p Fj 1664 1447 a(\017) p Fm 1696 1481 a(\)) p Fl(k) p Fj 1770 1493 a(s) p Fl 1828 1481 a(\024) p Fk 22 w(C) 6 b(\017) p Fj 2014 1447 a(r) p Fk 2081 1481 a(;) p Fl 99 w(j) p Fk(!) p Fj 2281 1447 a(\017) p Fl 2331 1481 a(\000) p Fk 18 w(!) p Fh 2466 1493 a(1) p Fl 2503 1481 a(j) 23 b(\024) p Fk 22 w(C) 6 b(\017) p Fj 2735 1447 a(r) p Fi 2 w(\000) p Fh(1) p Fk 2887 1481 a(:) p Fm 298 w(\(3.3\)) p Fg 515 1661 a(The) 30 b(set) p Fl 30 w(E) p Fg 37 w(has) g(the) g(same) g(pr) l(op) l(erties) h(as) f(in) g(the) g(nonr) l(esonant) f(c) l(ase.) p Fm 639 1815 a(The) 41 b(pro) r(of) e(is) i(obtained) f(pro) r(ceeding) f (as) h(in) g(the) h(nonresonan) n(t) e(case.) 74 b(The) 40 b(only) 515 1914 y(di\013erence) 26 b(is) h(that) g(in) g(this) g(case) f(the) h(k) n(ernel) f(of) p Fk 26 w(L) p Fj 2115 1926 a(!) p Fc 2157 1934 a(1) p Fm 2220 1914 a(is) g(no) h(more) f(one) g (dimensional,) g(but) 515 2014 y(is) 21 b(isomorphic) f(to) p Fl 21 w(N) p Fm 33 w(\(the) i(isomorphism) e(b) r(eing) h(giv) n(en) g (b) n(y) g(the) g(map) p Fk 21 w(x) p Fl 24 w(7!) p Fm 23 w(cos) o(\() p Fk(At=!) p Fh 3207 2026 a(1) p Fm 3244 2014 a(\)) p Fk(x) p Fm(\).) 515 2113 y(So) 33 b(the) g(Q) g(equation) g (can) g(b) r(e) g(transformed) f(in) n(to) h(an) g(equation) g(in) p Fl 33 w(N) p Fm 12 w(.) 54 b(This) 34 b(equation) 515 2213 y(turns) 28 b(out) h(to) f(b) r(e) h(a) f(p) r(erturbation) g(of) h (the) g(equation) f(for) g(the) h(critical) f(p) r(oin) n(ts) g(of) p Fl 29 w(h) p Fk(H) p Fj 3233 2225 a(r) p Fl 3270 2213 a(i) p Fe 3302 2143 a(\014) 3302 2192 y(\014) p Fi 3330 2246 a(S) p Fm 515 2313 a(and) 23 b(the) g(nondegeneracy) f(condition) g (H2R) i(allo) n(ws) d(to) i(solv) n(e) f(it) i(b) n(y) f(the) h (implicit) f(function) 515 2412 y(theorem.) 639 2512 y(Applying) 31 b(the) g(ab) r(o) n(v) n(e) f(theorem) g(one) g(can) g (construct) g(coun) n(tably) g(man) n(y) g(families) h(of) 515 2612 y(p) r(erio) r(dic) c(solutions) g(of) h(the) p Fk 28 w(\036) p Fh 1467 2581 a(4) p Fm 1532 2612 a(mo) r(del) p Fk 1245 2778 a(w) p Fj 1304 2790 a(tt) p Fl 1377 2778 a(\000) p Fk 18 w(w) p Fj 1519 2790 a(xx) p Fm 1622 2778 a(=) p Fl 22 w(\006) p Fk(w) p Fh 1835 2744 a(3) p Fm 1891 2778 a(+) 18 b(higher) 27 b(order) g(terms) 515 2945 y(with) 21 b(Diric) n(hlet) h(b) r(oundary) e(conditions,) i(and) f (also) f(higher) g(frequency) h(p) r(erio) r(dic) g(solutions) 515 3044 y(of) g(the) h(nonlinear) f(plate) h(equation) f(of) g(example) g (2.2) g(\(see) h([BP01) n(,) g(BP02) n(],) h(see) e(also) g([LS88) o(,) 515 3144 y(Bou99b) n(]\).) 639 3244 y(In) 29 b(general) d(it) j(is) f (di\016cult) h(to) f(c) n(hec) n(k) f(condition) h(H2R.) g(In) g(the) h (case) e(of) h(Hamiltonian) 515 3343 y(systems) 38 b(with) p Fk 39 w(n) p Fm 39 w(degrees) g(of) h(freedom,) p Fk 41 w(n) i(<) p Fl 42 w(1) p Fm(,) h(top) r(ological) 37 b(argumen) n(ts) g(allo) n(w) h(to) 515 3443 y(a) n(v) n(oid) 33 b(it.) 58 b(Indeed) 35 b(W) -7 b(einstein) 35 b(Moser) f(theorem) g (\(see) g([W) -7 b(ei73,) 35 b(Mos76) n(]\)) g(ensures) f(that) 515 3543 y(close) 29 b(to) i(a) f(minim) n(um) h(of) f(the) h(energy) e (there) h(exist) h(at) f(least) p Fk 30 w(n) p Fm 30 w(p) r(erio) r(dic) h(orbit) f(on) g(eac) n(h) 515 3642 y(surface) 36 b(of) h(constan) n(t) f(energy) -7 b(.) 64 b(In) 37 b(general) e(they) j(do) e(not) h(form) g(regular) e (families.) 65 b(A) 515 3742 y(corresp) r(onding) 27 b(result) i(for) f(PDEs) h(is) g(not) g(a) n(v) -5 b(ailable) 27 b(at) i(presen) n(t;) h(ho) n(w) n(ev) n(er) d(there) h(exists) 515 3842 y(an) p Fg 27 w(ad) j(ho) l(c) p Fm 28 w(v) -5 b(ariational) 26 b(result) h(for) h(the) f(w) n(a) n(v) n(e) f(equation) p Fk 1048 4008 a(w) p Fj 1107 4020 a(tt) p Fl 1180 4008 a(\000) p Fk 18 w(w) p Fj 1322 4020 a(xx) p Fm 1425 4008 a(=) p Fl 23 w(\006) p Fk(w) p Fj 1639 3974 a(p) p Fm 1696 4008 a(+) 18 b(higher) 27 b(order) f(terms) p Fk 27 w(;) 97 b(p) p Fl 23 w(\025) p Fm 23 w(2) p Fk 27 w(:) p Fm 362 w(\(3.4\)) 515 4175 y(whic) n(h) 23 b(ensures) f(that,) i (ha) n(ving) e(\014xed) p Fk 23 w(j) p Fl 28 w(\025) p Fm 22 w(1,) i(there) f(exists) f(a) h(sequence) f(of) h(p) r(erio) r (dic) g(orbits) 515 4274 y(accum) n(ulating) k(at) g(zero,) g(whose) g (frequencies) h(accum) n(ulate) f(at) p Fk 28 w(j) p Fm 32 w(\(whic) n(h) h(pla) n(ys) f(here) h(the) 515 4374 y(role) 38 b(of) h(the) p Fk 40 w(j) p Fl 5 w(\000) p Fm(th) g(linear) f(frequency\).) 72 b(Suc) n(h) 39 b(a) g(result) f(is) h(due) h(to) f(Berti) g(and) g(Bolle) 515 4474 y([BB02) n(].) 639 4573 y(P) n(erio) r(dic) 27 b(solutions) f(in) i(the) g(nonlinear) f(w) n(a) n(v) n(e) f(equation) p Fk 1083 4740 a(w) p Fj 1142 4752 a(tt) p Fl 1215 4740 a(\000) p Fk 18 w(w) p Fj 1357 4752 a(xx) p Fm 1455 4740 a(+) p Fk 18 w(f) p Fm 9 w(\() p Fk(x;) 14 b(w) p Fm 2 w(\)) 25 b(=) d(0) p Fk 27 w(;) 97 b(u) p Fm(\(0) p Fk(;) 14 b(t) p Fm(\)) 23 b(=) p Fk 22 w(u) p Fm(\() p Fk(\031) s(;) 14 b(t) p Fm(\)) 24 b(=) e(0) 397 b(\(3.5\)) 515 4907 y(where) 32 b(constructed) h(for) g (the) h(\014rst) e(time) i(b) n(y) f(Rabino) n(witz) g([Rab78) o(]) g (using) g(global) f(v) -5 b(ari-) 515 5006 y(ational) 31 b(metho) r(ds) h(and) g(a) g(Ly) n(apuno) n(v{Sc) n(hmidt) e(decomp) r (osition.) 50 b(Rabino) n(witz) 32 b(pro) n(v) n(ed) 1926 5255 y(5) p 90 rotate dyy eop %%Page: 6 6 6 5 bop Fm 515 523 a(that,) 28 b(under) g(suitable) g(assumptions) f (on) p Fk 28 w(f) p Fm 9 w(,) h(equation) f(\(3.5\)) h(has) f(at) h (least) g(one) f(p) r(erio) r(dic) 515 623 y(solution) e(with) h(p) r (erio) r(d) p Fk 25 w(T) p Fm 34 w(=) d(2) p Fk(\031) s(p=q) p Fm 3 w(,) i(for) g(an) n(y) g(c) n(hoice) f(of) i(the) g(in) n(tegers) p Fk 24 w(p) p Fm 25 w(and) p Fk 26 w(q) p Fm 3 w(.) 36 b(Remark) 515 722 y(that,) c(when) g(the) g(p) r(erio) r(d) p Fk 31 w(T) p Fm 42 w(is) g(in) f(rational) f(ratio) h(with) h(2) p Fk(\031) p Fm 3 w(,) g(the) g(op) r(erator) p Fk 29 w(L) p Fj 3029 734 a(!) p Fl 3077 722 a(j) p Fj 3100 734 a(R) p Fm 3186 722 a(has) e(a) 515 822 y(compact) g(in) n(v) n(erse,) g(i.e.) 46 b(there) 30 b(are) g(no) h(small) f(denominators.) 45 b(The) 30 b(w) n(ork) f([Rab78) o(]) i(w) n(as) 515 922 y(follo) n(w) n(ed) 18 b(b) n(y) h(a) g(series) g(of) g(pap) r(ers) g (simplifying) h(the) f(pro) r(of) g(and) h(sharp) r(ening) e(the) i (result) f(\(see) 515 1021 y([Bre83) n(]) 34 b(and) g(references) e (therein\).) 56 b(In) 34 b(particular) e(w) n(e) i(men) n(tion) f(the) h (pap) r(er) g([BCN80) o(]) 515 1121 y(b) n(y) 25 b(Brezis,) g(Coron) f (and) h(Niren) n(b) r(erg,) g(where) g(existence) g(of) g(p) r(erio) r (dic) h(orbits) e(is) i(pro) n(v) n(ed) d(b) n(y) 515 1220 y(a) 28 b(particularly) f(simple) i(metho) r(d:) 40 b(the) 29 b(authors) f(write) g(a) h(v) -5 b(ariational) 27 b(principle) i(dual) f(to) 515 1320 y(the) d(usual) g(one) g(and) g(lo) r(ok) f(for) h(its) g(critical) f(p) r(oin) n(ts) h(using) g(the) h (moun) n(tain) e(pass) h(lemma.) 36 b(It) 515 1420 y(is) 27 b(remark) -5 b(able) 26 b(that) i(in) g(this) g(approac) n(h) e(the) i (Q) f(equation) g(b) r(ecomes) g(trivial.) p Fn 515 1690 a(4) 134 b(W) -11 b(eak) l(ening) 46 b(the) g(nonresonance) f (condition) p Fm 515 1872 a(The) 35 b(main) g(limitation) g(of) g(the) g (results) f(presen) n(ted) h(in) g(Sections) g(2) f(and) h(3) f(rests) g (in) i(the) 515 1972 y(nonresonance) 25 b(conditions) h(H1) h(and) g (H1R.) g(Indeed) g(suc) n(h) g(conditions) g(are) f(ful\014lled) i (with) 515 2071 y(large) 19 b(probabilit) n(y) h(\(in) h(a) g(suitable) g(parameter) e(space\)) h(when) p Fk 21 w(!) p Fj 2516 2083 a(j) p Fl 2574 2071 a(\030) p Fk 23 w(j) p Fj 2701 2041 a(\027) p Fm 2763 2071 a(with) p Fk 21 w(\027) 29 b(>) p Fm 22 w(1;) 23 b(when) p Fk 515 2171 a(\027) p Fm 38 w(=) 32 b(1) h(the) g(nonresonance) f(conditions) h(are) f (satis\014ed) h(t) n(ypically) f(on) h(uncoun) n(table) g(sets) 515 2270 y(ha) n(ving) d(zero) g(measure,) i(but) g(when) p Fk 31 w(\027) j(<) p Fm 29 w(1) c(they) h(are) e(only) h(exceptionally) f(satify) n(ed) h(\(as) 515 2370 y(in) 25 b(the) h(plate) g (equation\).) 36 b(As) 25 b(a) g(consequence) g(the) h(results) f(of) g (Sections) g(2) g(and) h(3) f(are) f(not) 515 2470 y(applicable) 19 b(to) h(general) f(equations) g(in) h(more) f(than) h(one) g(space) f (dimensions.) 34 b(F) -7 b(urthermore) 515 2569 y(the) 32 b(metho) r(d) h(of) f(Ly) n(apuno) n(v{Sc) n(hmidt) e(decomp) r (osition) h(can) h(b) r(e) g(extended) h(to) f(the) g(case) 515 2669 y(of) d(rev) n(ersible) e(systems) h(of) h(\014rst) g(order) f(in) h(time,) h(but) g(the) f(approac) n(h) e(of) i(Section) g(2) f(is) h (no) 515 2769 y(more) e(applicable.) 639 2868 y(In) e(order) e(to) h(a) n(v) n(oid) e(suc) n(h) i(limitations) h(one) f(w) n(ould) f(lik) n(e) h (to) g(b) r(e) h(able) f(to) g(w) n(ork) f(with) i(the) 515 2968 y(w) n(eak) n(er) 19 b(nonresonance) h(condition) h(\\there) g (exists) g(a) p Fk 21 w(\034) 33 b(>) p Fm 22 w(0) 21 b(suc) n(h) g(that) p Fl 22 w(j) p Fk(l) r(!) p Fl 9 w(\000) p Fk 6 w(!) p Fj 2990 2980 a(j) p Fl 3024 2968 a(j) i(\025) p Fk 22 w(\015) 5 b(=l) p Fj 3274 2938 a(\034) p Fm 3314 2968 a(".) 515 3067 y(This) 26 b(w) n(as) f(done) g(b) n(y) h (Craig) f(and) g(W) -7 b(a) n(yne) 26 b([CW93) o(]) g(who) g(used) g (the) g(Nash) g(Moser) f(theorem) 515 3167 y(to) 32 b(solv) n(e) f(the) p Fk 33 w(P) p Fm 45 w(equation.) 51 b(It) 32 b(turns) h(out) f(that) h (the) g(application) f(of) g(the) h(Nash) f(Moser) 515 3267 y(theorem) i(requires) f(to) h(construct) g(and) g(estimate) g (the) h(in) n(v) n(erse) e(of) h(the) h(linear) f(op) r(erator) 515 3366 y(describing) 25 b(the) i(linearization) e(of) h(the) h(P) f (equation) g(at) g(an) g(appro) n(ximate) f(solution.) 36 b(This) 515 3466 y(is) 29 b(the) g(main) g(di\016cult) n(y) g(of) g (Craig{W) -7 b(a) n(yne's) 26 b(approac) n(h.) 39 b(T) -7 b(o) 29 b(o) n(v) n(ercome) d(it) k(they) f(use) g(the) 515 3566 y(tec) n(hniques) 35 b(b) n(y) h(F) -7 b(r\177) -42 b(olic) n(h) 34 b(and) i(Sp) r(encer) f([FS83],) j(p) r(erforming) d(a) g(v) n(ery) f(careful) h(analysis) 515 3665 y(of) k(small) g (denominators.) 72 b(The) 39 b(metho) r(d) h(b) n(y) g(Craig) e(and) h (W) -7 b(a) n(yne) 39 b(w) n(as) g(extended) g(b) n(y) 515 3765 y(Bourgain) 31 b(in) j(order) e(to) i(construct) e(p) r(erio) r (dic) i(\(and) f(also) g(quasi) f(p) r(erio) r(dic\)) i(solutions) f (in) 515 3864 y(higher) h(dimensional) g(equations.) 58 b(The) 35 b(resulting) g(metho) r(d) g(seems) f(v) n(ery) g(general,) h (but) 515 3964 y(at) 29 b(presen) n(t) g(a) g(theorem) g(\\ready) e (for) i(application") f(is) i(not) f(a) n(v) -5 b(ailable.) 41 b(W) -7 b(e) 30 b(presen) n(t) f(here) 515 4064 y(the) e(result) g (obtained) f(b) n(y) h(Bourgain) e(applying) i(suc) n(h) f(a) h(metho) r (d) h(to) e(the) i(nonlinear) e(w) n(a) n(v) n(e) 515 4163 y(equation) p Fk 1494 4263 a(w) p Fj 1553 4275 a(tt) p Fl 1626 4263 a(\000) p Fm 18 w(\001) p Fk(w) p Fm 22 w(+) p Fk 18 w(aw) p Fm 21 w(+) p Fk 18 w(w) p Fh 2210 4229 a(3) p Fm 2271 4263 a(=) c(0) 808 b(\(4.1\)) 515 4398 y(on) p Ff 27 w(T) p Fj 686 4368 a(d) p Fm 724 4398 a(.) 37 b(Fix) 28 b(a) f(m) n(ultiindex) p Fk 28 w(n) p Fl 23 w(2) p Ff 23 w(Z) p Fj 1630 4368 a(d) p Fm 1690 4398 a(di\013eren) n(t) h(from) f(zero,) g(and) g(let) p Fk 894 4588 a(\030) p Fj 930 4600 a(n) p Fm 975 4588 a(\() p Fk(!) p Fj 1059 4600 a(n) p Fk 1104 4588 a(t;) 14 b(x) p Fm(\)) 24 b(:=) f(cos) o(\() p Fk(n) p Fl 18 w(\001) p Fk 19 w(x) p Fm 19 w(+) p Fk 18 w(!) p Fj 1839 4600 a(n) p Fk 1884 4588 a(t) p Fm(\)) p Fk 28 w(;) 96 b(!) p Fj 2145 4600 a(n) p Fm 2213 4588 a(:=) p Fe 2324 4488 a(q) p 2407 4488 594 4 v Fk 2407 4588 a(n) p Fh 2457 4559 a(2) 2457 4610 y(1) p Fm 2513 4588 a(+) p Fk 18 w(:::) p Fm 18 w(+) p Fk 18 w(n) p Fh 2816 4559 a(2) p Fj 2816 4613 a(d) p Fm 2873 4588 a(+) p Fk 18 w(a) p Fm 515 4760 a(b) r(e) 28 b(the) g(corresp) r(onding) d(symmetric) j(rev) n(ersible) e(solution.) p Fd 515 4907 a(Theorem) 41 b(4.1.) p Fg 45 w([Bour) l(gain) e([Bou95) r (]].) 65 b(If) p Fk 38 w(a) p Fg 38 w(b) l(elongs) 39 b(to) f(a) g(subset) g(of) h(ful) t(l) f(me) l(asur) l(e) 515 5006 y(of) p Ff 37 w(R) p Fh 673 4976 a(+) p Fg 734 5006 a(,) h(then) e(ther) l(e) g(exists) f(a) i(Cantor) f(set) p Fl 36 w(E) p Fg 44 w(of) h(p) l(ositive) g(me) l(asur) l(e,) h(ac) l (cumulating) e(at) p Fm 1926 5255 a(6) p 90 rotate dyy eop %%Page: 7 7 7 6 bop Fg 515 523 a(zer) l(o,) 29 b(and) g(a) g(family) h(of) f(p) l (erio) l(dic) h(solutions) p Fl 29 w(f) p Fk(w) p Fj 2049 535 a(\017) p Fm 2080 523 a(\() p Fk(t;) 14 b(x) p Fm(\)) p Fl(g) p Fj 2300 535 a(\017) p Fi(2E) p Fg 2446 523 a(of) 30 b(\(4.1\)) f(with) g(fr) l(e) l(quency) p Fk 29 w(!) p Fj 3348 493 a(\017) p Fg 515 623 a(ful\014l) t(ling) p Fl 1033 722 a(j) p Fk(\017\030) p Fj 1126 734 a(n) p Fm 1171 722 a(\() p Fk(!) p Fj 1258 688 a(\017) p Fk 1290 722 a(t;) 14 b(x) p Fm(\)) p Fl 19 w(\000) p Fk 18 w(w) p Fj 1597 734 a(\017) p Fm 1630 722 a(\() p Fk(t;) g(x) p Fm(\)) p Fl(j) 24 b(\024) p Fk 22 w(C) 6 b(\017) p Fh 2041 688 a(3) p Fk 2108 722 a(;) p Fl 99 w(j) p Fk(!) p Fj 2305 734 a(n) p Fl 2368 722 a(\000) p Fk 18 w(!) p Fj 2506 688 a(\017) p Fl 2538 722 a(j) 23 b(\024) p Fk 23 w(C) 6 b(\017) p Fh 2771 688 a(2) p Fk 2837 722 a(:) p Fm 639 876 a(In) 27 b(the) g(case) p Fk 25 w(d) p Fm 24 w(=) 22 b(1) k(the) h(result) f(w) n(as) g(pro) n(v) n(ed) f(in) h ([CW93];) h(subsequen) n(tly) -7 b(,) 26 b(still) h(in) g(the) 515 976 y(case) p Fk 32 w(d) p Fm 33 w(=) 33 b(1) g(Kuksin) g(in) n(tro) r (duced) g(a) g(simpler) g(tec) n(hnique) h(in) g(order) e(to) h(\014nd) h(the) g(\\large) 515 1075 y(measure) 26 b(result") h(of) h(theorem) f (4.1) g(\(see) g([Bou99a) n(]) h(pp.90{94\).) 639 1175 y(The) 20 b(Craig{W) -7 b(a) n(yne{Bourgain) 14 b(metho) r(d) 20 b(allo) n(ws) e(to) i(deal) f(also) f(with) i(\014rst) f(order) f (equa-) 515 1275 y(tions.) 49 b(F) -7 b(or) 31 b(example) h(it) g(w) n (as) f(applied) h(to) f(the) i(Sc) n(hr\177) -42 b(odinger) 30 b(equation) h(in) h(one) g([CW94) o(]) 515 1374 y(or) 26 b(t) n(w) n(o) h(dimensions) h([Bou98) n(].) p Fn 515 1646 a(5) 134 b(The) 45 b(w) l(ater) h(w) l(a) l(v) l(e) h(problem) p Fm 515 1828 a(A) 24 b(particular) f(problem) g(that) h(has) g (attracted) f(the) h(atten) n(tion) g(of) g(man) n(y) f(researc) n (hers) e(since) 515 1928 y(the) 27 b(v) n(ery) e(b) r(eginning) h(of) h (the) g(theory) e(of) i(PDEs) f(is) g(that) h(of) f(existence) h(of) f (standing) g(w) n(ater) 515 2027 y(w) n(a) n(v) n(es.) 34 b(The) 27 b(\014rst) f(rigorous) e(pro) r(of) i(of) g(their) h (existence) f(w) n(as) f(obtained) i(only) f(recen) n(tly) f(b) n(y) 515 2127 y(Plotnik) n(o) n(v) g(and) j(T) -7 b(oland) 27 b([PT01) o(];) g(w) n(e) h(presen) n(t) f(here) g(their) g(result.) 639 2226 y(Consider) j(a) h(p) r(erfect) h(\015uid) f(lying) g(ab) r(o) n (v) n(e) f(a) h(horizon) n(tal) e(b) r(ottom,) k(and) e(con\014ned) g (b) r(e-) 515 2326 y(t) n(w) n(een) 19 b(t) n(w) n(o) g(parallel) f(v) n (ertical) h(w) n(alls.) 33 b(The) 20 b(\015uid) g(is) g(sub) 5 b(ject) 20 b(to) f(gra) n(vit) n(y) -7 b(,) 20 b(and) f(atmospheric) 515 2426 y(pressure) f(acts) i(at) g(the) g(free) g(surface.) 34 b(This) 20 b(is) g(a) f(dynamical) h(system) g(go) n(v) n(erned) d(b) n (y) j(the) h(Eu-) 515 2525 y(ler) h(equations) g(supplemen) n(ted) h(b) n(y) g(appropriate) e(b) r(oundary) h(conditions.) 35 b(It) 23 b(w) n(as) f(p) r(oin) n(ted) 515 2625 y(out) f(b) n(y) h (Zakharo) n(v) d(that) j(this) g(system) f(is) h(Hamiltonian) f(\(see) h ([Zak68) n(]\).) 35 b(The) 22 b(corresp) r(ond-) 515 2725 y(ing) g(Hamiltonian) h(function) g(is) g(the) g(energy) f(of) g (the) h(\015uid,) i(and) d(conjugated) h(v) -5 b(ariables) 21 b(are) 515 2824 y(giv) n(en) 27 b(b) n(y) g(the) h(w) n(a) n(v) n(e) e (pro\014le) h(and) g(the) h(v) n(elo) r(cit) n(y) f(p) r(oten) n(tial) g (at) h(the) g(free) f(surface.) 639 2924 y(In) 21 b(the) g(linear) e (appro) n(ximation) g(the) i(general) e(solution) h(is) g(giv) n(en) g (b) n(y) g(the) h(sup) r(erp) r(osition) 515 3023 y(of) 26 b(the) h(normal) f(mo) r(des.) 36 b(The) 27 b(problem) f(is) h(to) f (con) n(tin) n(ue) g(the) h(normal) f(mo) r(des) h(to) f(families) 515 3123 y(of) 38 b(p) r(erio) r(dic) h(solutions) f(of) h(the) g (nonlinear) f(system) g(\(the) i(stending) f(w) n(a) n(v) n(es\).) 69 b(Fix) 38 b(one) 515 3223 y(of) d(the) g(normal) g(mo) r(des,) i(and) e (denote) g(b) n(y) p Fk 35 w(\021) p Fm 3 w(\() p Fk(t;) 14 b(x) p Fh 2103 3235 a(1) p Fm 2141 3223 a(\)) 35 b(the) h(corresp) r (onding) d(pro\014le) i(of) g(the) 515 3322 y(free) 30 b(surface) g(\() p Fk(x) p Fh 1039 3334 a(1) p Fm 1108 3322 a(b) r(eing) h(the) g(horizon) n(tal) e(v) -5 b(ariable\).) 45 b(Then) 31 b(it) g(is) g(p) r(ossible) f(to) g(tune) i(the) 515 3422 y(depth) p Fk 32 w(h) p Fm(,) g(the) g(width) p Fk 32 w(l) p Fm 33 w(of) f(the) h(region) e(o) r(ccupied) i(b) n(y) f (the) h(\015uid) f(and) h(the) g(gra) n(vitational) 515 3522 y(constan) n(t) p Fk 33 w(g) p Fm 38 w(in) i(suc) n(h) g(a) h(w) n (a) n(y) e(that) i(the) f(p) r(erio) r(d) h(of) f(the) h(solution) f (is) g(normalized) g(to) g(2) p Fk(\031) p Fm 515 3621 a(and) c(the) g(linear) f(frequencies) h(ful\014ll) h(a) e(suitable) h (nonresonance) e(condition.) 44 b(Denote) 30 b(b) n(y) 515 3721 y(\() p Fk(g) p Fh 587 3733 a(0) p Fk 624 3721 a(;) 14 b(l) p Fh 686 3733 a(0) p Fk 723 3721 a(;) g(h) p Fh 808 3733 a(0) p Fm 845 3721 a(\)) 28 b(a) f(c) n(hoice) g(of) g(the) h (parameters) e(realizing) g(suc) n(h) i(conditions,) f(then) h(one) f (has) p Fd 515 3875 a(Theorem) g(5.1.) p Fg 38 w([Plotnikov) j(&) d(T) -6 b(oland) 28 b([PT01) r(]]) h(Ther) l(e) f(exists) f(an) h (in\014nite) f(set) p Fl 27 w(E) j(\032) p Ff 23 w(R) p Fg 515 3974 a(having) h(zer) l(o) g(as) g(an) f(ac) l(cumulation) h(p) l (oint) f(and,) i(for) f(any) p Fk 30 w(\017) p Fl 24 w(2) 25 b(E) p Fg 7 w(,) 31 b(a) f(se) l(quenc) l(e) p Fk 30 w(g) p Fj 3073 3986 a(\017) p Fg 3105 3974 a(,) p Fk 31 w(l) p Fj 3186 3986 a(\017) p Fg 3247 3974 a(and) 515 4074 y(a) h(standing) h(wave) g(solution) g(of) g(the) f(water) h(wave) g(pr) l(oblem) g(with) g(gr) l(avity) p Fk 32 w(g) p Fj 2925 4086 a(\017) p Fg 2988 4074 a(in) f(a) h(b) l(ox) f(of) 515 4173 y(width) p Fk 27 w(l) p Fj 759 4185 a(\017) p Fg 790 4173 a(.) 38 b(Denote) 25 b(by) p Fk 26 w(\021) p Fj 1279 4185 a(\017) p Fg 1337 4173 a(the) h(c) l(orr) l(esp) l(onding) h(pr) l(o\014le) g(of) g(the) f(fr) l(e) l(e) g(surfac) l(e,) h(then) f (one) g(has) p Fl 901 4341 a(j) p Fk(\021) p Fj 965 4353 a(\017) p Fm 997 4341 a(\() p Fk(t;) 14 b(x) p Fh 1143 4353 a(1) p Fm 1181 4341 a(\)) p Fl 18 w(\000) p Fk 18 w(\017) p Fh 1348 4307 a(2) p Fk 1385 4341 a(\021) p Fm 3 w(\() p Fk(t;) g(x) p Fh 1575 4353 a(1) p Fm 1613 4341 a(\)) p Fl(j) p Fk 24 w(<) 22 b(C) 6 b(\017) p Fh 1878 4307 a(3) p Fk 1945 4341 a(;) p Fl 99 w(j) p Fk(g) p Fj 2130 4353 a(\017) p Fl 2180 4341 a(\000) p Fk 18 w(g) p Fh 2303 4353 a(0) p Fl 2340 4341 a(j) p Fm 18 w(+) p Fl 18 w(j) p Fk(l) p Fj 2512 4353 a(\017) p Fl 2563 4341 a(\000) p Fk 18 w(l) p Fh 2671 4353 a(0) p Fl 2708 4341 a(j) 23 b(\024) p Fk 22 w(C) 6 b(\017) 30 b(:) p Fm 639 4508 a(The) 25 b(main) f(di\016culties) h(in) g(pro) n(ving) e(this) i (result) f(are) f(as) h(follo) n(ws:) 35 b(\014rst) 24 b(the) h(linear) f(fre-) 515 4608 y(quencies) 31 b(b) r(eha) n(v) n(e) g (as) p Fk 31 w(!) p Fj 1288 4620 a(n) p Fl 1363 4608 a(\030) p Fk 29 w(n) p Fh 1507 4578 a(1) p Fj(=) p Fh(2) p Fm 1643 4608 a(and) h(therefore) f(the) h(nonresonance) e(conditions) h (that) 515 4707 y(can) 21 b(b) r(e) i(satis\014ed) e(are) h(quite) g(w) n(eak.) 34 b(Second,) 23 b(one) e(has) h(that) g(the) h(mathematical) e (form) n(ula-) 515 4807 y(tion) i(of) h(the) f(problem) g(in) n(v) n (olv) n(es) f(a) h(nonlinear) f(non) i(lo) r(cal) e(op) r(erator) g (whic) n(h) h(is) h(un) n(b) r(ounded.) 515 4907 y(T) -7 b(o) 30 b(o) n(v) n(ercome) e(suc) n(h) i(di\016culties,) h(Plotnik) n (o) n(v) e(and) h(T) -7 b(oland) 29 b(use) h(a) g(Lagrangian) e (descrip-) 515 5006 y(tion) 36 b(of) g(the) g(\015uid) h(motion) f(and) g(a) f(Ly) n(apuno) n(v{Sc) n(hmidt) g(approac) n(h) f(to) i(the) g (nonlinear) 1926 5255 y(7) p 90 rotate dyy eop %%Page: 8 8 8 7 bop Fm 515 523 a(problem) 28 b(whic) n(h) g(ensues.) 39 b(The) 29 b(P) f(equation) g(is) g(solv) n(ed) f(b) n(y) i(Nash) f (Moser) f(theorem.) 39 b(The) 515 623 y(required) 26 b(in) n(v) n(ertibilit) n(y) h(of) g(the) h(linearized) e(op) r(erator) g(is) h(obtained) g(in) g(t) n(w) n(o) g(steps:) 37 b(\014rst) 27 b(it) 515 722 y(is) i(reduced) g(to) g(a) g(suitable) g(canonical) f (form) h(and) g(then) g(suc) n(h) g(a) g(canonical) f(form) h(\(whic) n (h) 515 822 y(is) 37 b(essen) n(tially) f(a) g(p) r(erturbation) h(of) f (an) h(op) r(erator) e(in) n(v) n(olving) h(deriv) -5 b(ativ) n(es) 36 b(and) g(Hilb) r(ert) 515 922 y(transform\)) 26 b(is) i(studied) g(in) g(detail.) p Fn 515 1193 a(References) p Fm 515 1375 a([Bam00]) 63 b(D.) 33 b(Bam) n(busi,) p Fg 34 w(Lyapunov) i(c) l(enter) f(the) l(or) l(em) h(for) g(some) g (nonline) l(ar) g(PDEs:) 49 b(a) 878 1474 y(simple) 31 b(pr) l(o) l(of) p Fm(,) e(Ann.) f(S.N.S.) g(Pisa) f(Cl.) h(Sci) p Fd 27 w(29) p Fm 28 w(\(2000\),) e(823{837.) 515 1632 y([BB02]) 115 b(M.) 26 b(Berti) g(and) f(P) -7 b(.) 26 b(Bolle,) p Fg 25 w(Perio) l(dic) 31 b(solutions) d(of) h(nonline) l (ar) g(wave) g(e) l(quations) 878 1732 y(with) h(gener) l(al) h (nonline) l(arities) p Fm(,) e(Preprin) n(t) d(\(2002\).) 515 1890 y([BCN80]) 52 b(H.) 25 b(Brezis,) g(J.) f(Coron,) g(and) h(L.) g (Niren) n(b) r(erg,) p Fg 24 w(F) -6 b(r) l(e) l(e) 27 b(vibr) l(ations) h(for) g(a) f(nonline) l(ar) 878 1990 y(wave) h(e) l(quation) f(and) g(a) g(the) l(or) l(em) g(by) h(P.) f (Rabinowitz) p Fm(,) f(Comm) n(un.) f(Pure) e(Appl.) 878 2089 y(Math.) p Fd 28 w(33) p Fm 27 w(\(1980\),) j(667{689.) 515 2247 y([Bou95]) 86 b(J.) 44 b(Bourgain,) p Fg 46 w(Construction) h(of) h (p) l(erio) l(dic) h(solutions) e(of) h(nonline) l(ar) f(wave) 878 2347 y(e) l(quations) c(in) f(higher) j(dimension) p Fm(,) g(Geometric) c(and) g(F) -7 b(unctional) 40 b(Analysis) p Fd 878 2446 a(5) p Fm 27 w(\(1995\),) 27 b(629{639.) 515 2605 y([Bou98]) 86 b(J.) 24 b(Bourgain,) p Fg 24 w(Quasi-p) l(erio) l (dic) 29 b(solutions) e(of) h(Hamiltonian) g(p) l(erturb) l(ations) f (of) 878 2704 y(2D) j(line) l(ar) g(Sh\177) -42 b(odinger) 31 b(e) l(quation) p Fm(,) d(Ann.) h(Math.) p Fd 27 w(148) p Fm 27 w(\(1998\),) e(363{439.) 515 2862 y([Bou99a]) 44 b(J.) 31 b(Bourgain,) p Fg 31 w(Nonline) l(ar) j(Schr\177) -42 b(odinger) 35 b(e) l(quations) p Fm(,) e(Hyp) r(erb) r(olic) f (equations) 878 2962 y(and) 27 b(frequency) h(in) n(teractions,) e (American) h(Mathematical) g(So) r(ciet) n(y) -7 b(,) 28 b(1999.) 515 3120 y([Bou99b]) 40 b(J.) 19 b(Bourgain,) p Fg 19 w(Perio) l(dic) 24 b(solutions) e(of) h(nonline) l(ar) f(wave) h (e) l(quations) p Fm(,) e(Harmonic) 878 3220 y(analysis) 26 b(and) i(partial) f(di\013eren) n(tial) h(equations,) f(Chicago) f (Univ.Press,) h(1999,) 878 3319 y(pp.) h(69{97.) 515 3477 y([BP01]) 117 b(D.) 35 b(Bam) n(busi) f(and) g(S.) h(P) n(aleari,) p Fg 34 w(F) -6 b(amilies) 38 b(of) e(p) l(erio) l(dic) j(orbits) d (for) h(r) l(esonant) 878 3577 y(PDE's) p Fm(,) 28 b(J.) g(Nonlinear) f (Science) p Fd 27 w(11) p Fm 27 w(\(2001\),) g(69{87.) 515 3735 y([BP02]) 117 b(D.) 21 b(Bam) n(busi) f(and) h(S.) g(P) n(aleari,) p Fg 20 w(F) -6 b(amilies) 25 b(of) f(p) l(erio) l(dic) i(orbits) e (for) g(some) g(PDE's) 878 3835 y(in) 44 b(higher) j(dimensions) p Fm(,) h(Comm) n(un.) c(on) f(Pure) g(and) g(Applied) h(Analysis) p Fd 43 w(1) p Fm 878 3934 a(\(2002\),) 26 b(269{279.) 515 4092 y([Bre83]) 104 b(H.) 26 b(Brezis,) p Fg 26 w(Perio) l(dic) k (solutions) f(of) g(nonline) l(ar) g(vibr) l(ating) g(string) f(and) h (duality) 878 4192 y(principle) p Fm(,) h(Bull.) e(A.M.S.) p Fd 28 w(8) p Fm 27 w(\(1983\),) f(409{426.) 515 4350 y([CW93]) 88 b(W.) 32 b(Craig) d(and) i(C.) h(E.) e(W) -7 b(a) n(yne,) p Fg 32 w(Newton) f('s) 33 b(metho) l(d) g(and) h(p) l (erio) l(dic) h(solutions) 878 4450 y(of) 41 b(nonline) l(ar) f(wave) i (e) l(quations) p Fm(,) g(Comm.) c(Pure) g(Appl.) i(Math.) p Fd 39 w(46) p Fm 38 w(\(1993\),) 878 4549 y(1409{1498.) 515 4707 y([CW94]) 88 b(W.) 67 b(Craig) e(and) h(C.) g(E.) g(W) -7 b(a) n(yne,) p Fg 76 w(Perio) l(dic) 68 b(solutions) d(of) h(nonline) l (ar) 878 4807 y(Schr\177) -42 b(odinger) 30 b(e) l(quations) g(and) f (the) g(Nash-Moser) h(metho) l(d) p Fm(,) e(Hamiltonian) e(me-) 878 4907 y(c) n(hanics.) 37 b(In) n(tegrabilit) n(y) g(and) g(c) n(haotic) g (b) r(eha) n(vior,) i(NA) -7 b(TO) 38 b(ASI,) h(v) n(ol.) e(B331,) 878 5006 y(Plen) n(um) 27 b(Press,) f(1994,) g(pp.) i(103{122.) 1926 5255 y(8) p 90 rotate dyy eop %%Page: 9 9 9 8 bop Fm 515 523 a([FR78]) 118 b(E.) 32 b(R.) g(F) -7 b(adell) 32 b(and) g(P) -7 b(.) 32 b(H.) h(Rabino) n(witz,) p Fg 32 w(Gener) l(alize) l(d) j(c) l(ohomolo) l(gic) l(al) h(index) 878 623 y(the) l(ories) 32 b(for) g(lie) g(gr) l(oup) f(actions) h(with) f (an) g(applic) l(ation) j(to) d(bifur) l(c) l(ation) g(ques-) 878 722 y(tions) f(for) g(Hamiltonian) h(systems) p Fm(,) d(In) n(v) n(en) n (t.) f(Math.) p Fd 27 w(45) p Fm 28 w(\(19878\),) e(139{174.) 515 888 y([FS83]) 133 b(J.) 36 b(F) -7 b(r\177) -42 b(ohlic) n(h) 35 b(and) h(T.) g(Sp) r(encer,) p Fg 37 w(A) n(bsenc) l(e) i(of) g (di\013usion) g(in) f(Anderson) h(tight) 878 988 y(binding) 26 b(mo) l(del) g(for) f(lar) l(ge) h(disor) l(der) h(or) e(low) g(ener) l (gy) p Fm(,) f(Comm) n(un.) f(Math.) f(Ph) n(ys.) p Fd 878 1088 a(88) p Fm 27 w(\(1983\),) k(151{184.) 515 1254 y([LS88]) 135 b(B.) 34 b(V.) h(Lidskij) g(and) f(E.) g(I.) h(Sh) n (ulman,) p Fg 36 w(Perio) l(dic) j(solutions) f(of) f(the) h(e) l (quation) p Fk 878 1353 a(u) p Fj 926 1365 a(tt) p Fl 998 1353 a(\000) p Fk 18 w(u) p Fj 1129 1365 a(xx) p Fm 1227 1353 a(+) p Fk 18 w(u) p Fh 1358 1323 a(3) p Fm 1418 1353 a(=) 22 b(0,) 27 b(F) -7 b(unct.) 29 b(Anal.) f(Appl.) p Fd 28 w(22) p Fm 27 w(\(1988\),) e(332{333.) 515 1519 y([Mos76]) 82 b(J.) 29 b(Moser,) p Fg 28 w(Perio) l(dic) k(orbits) e (ne) l(ar) g(an) g(e) l(quilibrium) h(and) f(a) h(the) l(or) l(em) f (by) g(Alan) 878 1619 y(Weinstein) p Fm(,) d(Comm.) f(Pure) g(Appl.) h (Math.) p Fd 28 w(29) p Fm 27 w(\(1976\),) f(724{747.) 515 1785 y([PT01]) 116 b(P) -7 b(.) 37 b(Plotnik) n(o) n(v) f(and) i(J.) f (T) -7 b(oland,) p Fg 40 w(Nash-Moser) 40 b(the) l(ory) f(for) h (standing) g(water) 878 1885 y(waves) p Fm(,) 29 b(Arc) n(h.) e (Ration.) g(Mec) n(h.) h(Anal.) p Fd 27 w(159) p Fm 27 w(\(2001\),) f(1{83.) 515 2051 y([Rab78]) 84 b(P) -7 b(.) 30 b(Rabino) n(witz,) p Fg 30 w(F) -6 b(r) l(e) l(e) 31 b(vibr) l(ations) i(for) g(a) f(semiline) l(ar) h(wave) g(e) l(quation) p Fm(,) e(Com-) 878 2150 y(m) n(un.) d(Pure) f(Appl.) h(Math.) p Fd 28 w(31) p Fm 27 w(\(1978\),) e(31{68.) 515 2316 y([W) -7 b(ei73]) 95 b(A.) 30 b(W) -7 b(einstein,) p Fg 31 w(Normal) 33 b(mo) l(des) f(for) h(nonline) l(ar) f(Hamiltonian) h(systems) p Fm(,) d(In-) 878 2416 y(v) n(en) n(t.) d(Math.) p Fd 28 w(20) p Fm 27 w(\(1973\),) f(47{57.) 515 2582 y([Zak68]) 96 b(V.) 28 b(E.) f(Zakharo) n(v,) p Fg 25 w(Stability) k(of) f(p) l(erio) l(dic) i(waves) e(of) h(\014nite) e(amplitude) i(on) f(the) 878 2682 y(surfac) l(e) f(of) h(a) f(de) l(ep) g(\015uid) p Fm(,) e(Appl.) g(Mec) n(h.) g(T) -7 b(ec) n(h.) 26 b(Ph) n(ysics) p Fd 25 w(2) p Fm 26 w(\(1968\),) f(190{194.) p Fg 639 2964 a(Dip) l(artimento) 31 b(di) f(Mathematic) l(a) 639 3064 y(Via) h(Saldini) g(50,) g(20133) h(Milano,) f(Italy) 639 3188 y(E-mail:) p Fa 40 w(dario.bambusi@un) o(im) o(i.i) o(t) p Fg 639 3313 a(web-addr) l(ess) p Fm(:) p Fa 39 w(http://users.ma) o (t.u) o(ni) o(mi.) o(it) o(/u) o(ser) o(s/) o(bam) o(bu) o(si) o(/) p Fm 1926 5255 a(9) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF