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View File Name : yesiltas.ps

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4921 y(The)19 b(Mathieu)f(equation)g(is)h(an)g(e)o(xample)f(of)g(a)i
(dif)n(ferential)e(equation)g(with)g(periodic)g(coef)n
(\002cients[1,15,16,17].)1983 5753 y(3)p eop
%%Page: 4 4
4 3 bop 0 52 a Fl(3)143 b(Multiple)35 b(Scale)g(analysis)f(of)h
(Reducable)f(V)-13 b(an)35 b(der)g(P)m(ol)g(Equation)0
276 y Fo(It)28 b(is)g(possible)f(to)g(solv)o(e)g(\(5\),)j(disturbed)d
(VdP)h(oscillator)f(equation,)h(by)g(using)f(MSPT)-7
b(,)28 b(so)g(we)g(seek)h(the)f(pertur)n(-)0 396 y(bati)n(v)o(e)22
b(solution)f(to)i(\(5\))g(ha)n(ving)f(tw)o(o)h(v)n(ariables)f(the)h
(short-time)f(scale)h Fk(t)h Fo(and)f(the)g(long-time)e(scale)j
Fk(\034)39 b Fi(=)27 b Fk(")17 b(t)23 b Fo(when)0 517
y Fk(")29 b Fo(is)g(suf)n(\002ciently)g(small.)44 b(W)-8
b(e)30 b(w)o(ould)f(lik)o(e)g(to)g(in)l(v)o(estigate)e(the)j(solutions)
d(of)j(\(5\))g(by)f(disturbing)f(this)h(equation.)0 637
y(Because)e(to)f(terminate)53 b(more)26 b(secular)h(terms)e(and)i
(unstable)e(solutions)g(near)2803 598 y Fj(1)p 2803 614
36 4 v 2803 671 a(4)2848 637 y Fo([1])i(,)f(time)g(v)n(ariable)g(is)g
(changed)0 758 y(as)50 b Fk(t)28 b Fh(!)338 718 y Ff(t)p
333 734 V 333 792 a Fj(2)400 758 y Fi(+)508 718 y Ff(\031)p
508 734 43 4 v 512 792 a Fj(4)611 758 y Fo(and)d(the)f(\(5\))h(turns)g
(into)619 1160 y Fi(\177)-55 b Fk(u)o Fi(\()p Fk(t)p
Fi(\))22 b(+)899 989 y Fg(0)899 1139 y(@)982 1093 y Fi(1)p
982 1137 49 4 v 982 1228 a(4)1063 1160 y(+)1171 1093
y(1)p 1171 1137 V 1171 1228 a(8)1230 1160 y Fk("a)1327
1119 y Fj(2)1382 1160 y Fi(cos)c Fk(t)k Fh(\000)1721
1093 y Fi(3)p 1697 1137 98 4 v 1697 1228 a(16)1804 1160
y Fk(")1850 1119 y Fj(2)1906 1014 y Fg( )1972 1160 y
Fi(1)f Fh(\000)2152 1093 y Fk(a)2203 1057 y Fj(2)p 2152
1137 91 4 v 2173 1228 a Fi(2)2253 1160 y(\(1)g(+)h(sin)17
b Fk(t)p Fi(\))2669 1014 y Fg(!)2735 1037 y Fj(2)2774
989 y Fg(1)2774 1139 y(A)2863 1160 y Fk(u)p Fi(\()p Fk(t)p
Fi(\))28 b(=)f(0)p Fk(:)663 b Fo(\(7\))146 1461 y(Applying)21
b(the)h(multiple-scale)f(perturbation)g(theory)h(we)h(\002nd)f(the)g
(boundaries)g(between)g(the)g(re)o(gions)f(in)h(the)0
1581 y Fi(\(\001)p Fk(;)17 b(")p Fi(\))24 b Fo(plane)h(for)g(which)g
(all)f(solutions)f(to)h(general)h(equation)f(are)i(stable.)k(So)25
b(we)g(put)g Fi(\001)g Fo(parameter)g(in)f(\(7\))606
1984 y Fi(\177)-56 b Fk(u)p Fi(\()p Fk(t)p Fi(\))22 b(+)886
1813 y Fg(0)886 1962 y(@)969 1916 y Fi(\001)p 969 1960
82 4 v 985 2052 a(4)1082 1984 y(+)1190 1916 y(1)p 1190
1960 49 4 v 1190 2052 a(8)1249 1984 y Fk(\017a)1339 1942
y Fj(2)1396 1984 y Fi(cos)17 b Fk(t)22 b Fh(\000)1734
1916 y Fi(3)p 1710 1960 98 4 v 1710 2052 a(16)1817 1984
y Fk(")1863 1942 y Fj(2)1919 1837 y Fg( )1985 1984 y
Fi(1)g Fh(\000)2165 1916 y Fk(a)2216 1880 y Fj(2)p 2165
1960 91 4 v 2186 2052 a Fi(2)2266 1984 y(\(1)g(+)g(sin)16
b Fk(t)p Fi(\))2682 1837 y Fg(!)2748 1860 y Fj(2)2787
1813 y Fg(1)2787 1962 y(A)2877 1984 y Fk(u)p Fi(\()p
Fk(t)p Fi(\))27 b(=)g(0)p Fk(;)650 b Fo(\(8\))0 2295
y(and)33 b(e)o(xpand)g(it)g(as)h(a)f(po)n(wer)g(series)h(in)f([16])g
Fk(\017)p Fo(.)57 b(By)34 b(writing)66 b Fi(\001)44 b(=)f(1)29
b(+)f Fk(\017\016)2765 2310 y Fj(1)2834 2295 y Fi(+)g
Fk(\017)2977 2259 y Fj(2)3017 2295 y Fk(\016)3060 2310
y Fj(2)3128 2295 y Fi(+)g Fk(:::)34 b Fo(determine)f(the)g
Fi(\001)0 2415 y Fo(so)c(that)f(the)h(resulting)f(e)o(xpansion)g(is)g
(periodic.)44 b(Consequently)-6 b(,)28 b(the)h(resulting)f(e)o
(xpressions)g(for)h Fi(\001)g Fo(de\002ne)h(the)0 2536
y(transition)23 b(curv)o(es)i(separating)f(stability)f(from)i
(instability)326 2938 y Fi(\177)-55 b Fk(u)p Fi(\()p
Fk(t)p Fi(\))22 b(+)607 2767 y Fg(0)607 2917 y(@)690
2871 y Fi(1)p 690 2915 49 4 v 690 3007 a(4)770 2938 y(+)g
Fk(\017)p Fi(\()955 2871 y Fk(\016)998 2886 y Fj(1)p
955 2915 83 4 v 973 3007 a Fi(4)1071 2938 y(+)1179 2871
y(1)p 1179 2915 49 4 v 1179 3007 a(8)1237 2938 y Fk(a)1288
2897 y Fj(2)1344 2938 y Fi(cos)c Fk(t)p Fi(\))k(+)g Fk(\017)1724
2897 y Fj(2)1764 2938 y Fi(\()1812 2871 y Fk(\016)1855
2886 y Fj(2)p 1812 2915 83 4 v 1829 3007 a Fi(4)1926
2938 y Fh(\000)2060 2871 y Fi(3)p 2036 2915 98 4 v 2036
3007 a(16)2160 2792 y Fg( )2226 2938 y Fi(1)g Fh(\000)2406
2871 y Fk(a)2457 2835 y Fj(2)p 2406 2915 91 4 v 2427
3007 a Fi(2)2507 2938 y(\(1)g(+)g(sin)16 b Fk(t)p Fi(\))2923
2792 y Fg(!)2989 2815 y Fj(2)3029 2938 y Fi(\))3067 2767
y Fg(1)3067 2917 y(A)3156 2938 y Fk(u)p Fi(\()p Fk(t)p
Fi(\))27 b(=)h(0)p Fk(;)370 b Fo(\(9\))146 3239 y(is)25
b(the)f(general)h(equation.)30 b(An)25 b(uniform)f(e)o(xpansion)f(is)i
(in)f(the)h(form)f(:)940 3579 y Fk(u)p Fi(\()p Fk(t)p
Fi(\))k(=)f Fk(U)1304 3594 y Fj(0)1344 3579 y Fi(\()p
Fk(t;)17 b(\034)11 b Fi(\))22 b(+)g Fk("U)1784 3594 y
Fj(1)1824 3579 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))22
b(+)g Fk(")2198 3538 y Fj(2)2237 3579 y Fk(U)2303 3594
y Fj(2)2343 3579 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))22
b(+)g Fk(O)s Fi(\()p Fk(")2833 3538 y Fj(3)2872 3579
y Fi(\))940 b Fo(\(10\))146 3799 y(The)21 b(chain)f(rule)h(for)f(the)h
(partial)f(dif)n(ferentiation)f(to)h(compute)f(deri)n(v)n(ati)n(v)o(es)
f(of)j Fk(u)p Fi(\()p Fk(t)p Fi(\))f Fo(is,)g(with)g(using)3633
3760 y Ff(d\034)p 3633 3776 76 4 v 3640 3834 a(dt)3746
3799 y Fi(=)28 b Fk(")689 4177 y Fi(\177)-55 b Fk(u)o
Fi(\()p Fk(t)p Fi(\))84 b(=)1101 4110 y Fk(@)1157 4073
y Fj(2)1198 4110 y Fk(U)1264 4125 y Fj(0)1303 4110 y
Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))p 1101 4154 411 4
v 1241 4245 a Fk(@)5 b(t)1332 4217 y Fj(2)1544 4177 y
Fi(+)22 b Fk(")1721 4031 y Fg( )1786 4177 y Fi(2)1845
4110 y Fk(@)1901 4073 y Fj(2)1941 4110 y Fk(U)2007 4125
y Fj(0)2047 4110 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))p
1845 4154 V 1949 4245 a Fk(@)5 b(\034)11 b(@)5 b(t)2288
4177 y Fi(+)2396 4110 y Fk(@)2452 4073 y Fj(2)2492 4110
y Fk(U)2558 4125 y Fj(1)2598 4110 y Fi(\()p Fk(t;)17
b(\034)11 b Fi(\))p 2396 4154 V 2535 4245 a Fk(@)5 b(t)2626
4217 y Fj(2)2816 4031 y Fg(!)1091 4459 y Fi(+)598 b Fk(")1811
4418 y Fj(2)1867 4313 y Fg( )1942 4392 y Fk(@)1998 4356
y Fj(2)2039 4392 y Fk(U)2105 4407 y Fj(2)p 1942 4436
203 4 v 1978 4528 a Fk(@)5 b(t)2069 4499 y Fj(2)2177
4459 y Fi(+)22 b(2)2334 4392 y Fk(@)2390 4356 y Fj(2)2430
4392 y Fk(U)2496 4407 y Fj(1)p 2334 4436 V 2334 4528
a Fk(@)5 b(\034)11 b(@)5 b(t)2568 4459 y Fi(+)2676 4392
y Fk(@)2732 4356 y Fj(2)2772 4392 y Fk(U)2838 4407 y
Fj(0)p 2676 4436 V 2702 4528 a Fk(@)g(\034)2811 4499
y Fj(2)2888 4313 y Fg(!)2976 4459 y Fi(+)22 b Fk(O)s
Fi(\()p Fk(\017)3229 4418 y Fj(3)3268 4459 y Fi(\))p
Fk(:)517 b Fo(\(11\))146 4735 y(Substituting)19 b(\(10\))i(and)h
(\(11\))f(in)f(\(9\))i(,)g(partial)e(dif)n(ferential)h(equations)f(are)
i(obtained)e(for)h(the)g(dependent)g(v)n(ari-)0 4855
y(ables)g Fk(U)292 4870 y Fj(0)332 4855 y Fi(\()p Fk(t;)c(\034)11
b Fi(\))p Fk(;)17 b(U)650 4870 y Fj(1)690 4855 y Fi(\()p
Fk(t;)g(\034)11 b Fi(\))p Fk(;)17 b(::)k Fo(So,in)g(this)g(solution)e
(there)j(are)g(less)f(secular)g(terms)g(to)g(all)g(order)h(in)f(the)g
(perturbation)0 4976 y(series.)31 b(The)24 b(\002rst)h(three)g(terms)g
(of)g(the)f(series)h(are)h(gi)n(v)o(en)d(by)1365 5283
y Fk(@)1421 5247 y Fj(2)1461 5283 y Fk(U)1527 5298 y
Fj(0)1567 5283 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))p
1365 5327 411 4 v 1504 5419 a Fk(@)5 b(t)1595 5390 y
Fj(2)1807 5350 y Fi(+)1915 5283 y(1)p 1915 5327 49 4
v 1915 5419 a(4)1974 5350 y Fk(U)2040 5365 y Fj(0)2080
5350 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))28 b(=)f(0)p
Fk(;)1355 b Fo(\(12\))1983 5753 y(4)p eop
%%Page: 5 5
5 4 bop 564 239 a Fk(@)620 203 y Fj(2)661 239 y Fk(U)727
254 y Fj(1)767 239 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))p
564 284 411 4 v 704 375 a Fk(@)5 b(t)795 346 y Fj(2)1007
307 y Fi(+)1115 239 y(1)p 1115 284 49 4 v 1115 375 a(4)1174
307 y Fk(U)1240 322 y Fj(1)1279 307 y Fi(\()p Fk(t;)17
b(\034)11 b Fi(\))28 b(=)g Fh(\000)p Fi(2)1755 239 y
Fk(@)1811 203 y Fj(2)1851 239 y Fk(U)1917 254 y Fj(0)1957
239 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))p 1755 284 411
4 v 1859 375 a Fk(@)5 b(\034)11 b(@)5 b(t)2197 307 y
Fh(\000)23 b Fi(\()2345 239 y Fk(\016)2388 254 y Fj(1)p
2345 284 83 4 v 2362 375 a Fi(4)2460 307 y(+)2568 239
y(1)p 2568 284 49 4 v 2568 375 a(8)2626 307 y Fk(a)2677
266 y Fj(2)2734 307 y Fi(cos)17 b Fk(t)p Fi(\))p Fk(U)3020
322 y Fj(0)3060 307 y Fi(\()p Fk(t;)g(\034)11 b Fi(\))p
Fk(;)555 b Fo(\(13\))514 714 y Fk(@)570 677 y Fj(2)610
714 y Fk(U)676 729 y Fj(2)716 714 y Fi(\()p Fk(t;)17
b(\034)11 b Fi(\))p 514 758 411 4 v 653 849 a Fk(@)5
b(t)744 821 y Fj(2)956 781 y Fi(+)1064 714 y(1)p 1064
758 49 4 v 1064 849 a(4)1123 781 y Fk(U)1189 796 y Fj(2)1229
781 y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))28 b(=)83 b
Fh(\000)p Fi(2)1760 735 y Ff(@)1801 711 y Fd(2)1835 735
y Ff(U)1883 744 y Fd(1)1918 735 y Fj(\()p Ff(t;\034)8
b Fj(\))p 1760 758 297 4 v 1835 815 a Ff(@)t(\034)g(@)t(t)2089
781 y Fh(\000)22 b Fi(\()2236 742 y Ff(\016)2267 751
y Fd(1)p 2236 758 66 4 v 2252 815 a Fj(4)2334 781 y Fi(+)2442
742 y Fj(1)p 2442 758 36 4 v 2442 815 a(8)2488 781 y
Fk(a)2539 745 y Fj(2)2595 781 y Fi(cos)17 b Fk(t)p Fi(\))p
Fk(U)2881 796 y Fj(1)2921 781 y Fi(\()p Fk(t;)g(\034)11
b Fi(\))22 b Fh(\000)3261 742 y Ff(@)3302 718 y Fd(2)3337
742 y Ff(U)3385 751 y Fd(0)p 3261 758 159 4 v 3283 815
a Ff(@)t(\034)3363 796 y Fd(2)1687 1007 y Fh(\000)1781
886 y Fg(\022)1852 968 y Ff(\016)1883 977 y Fd(2)p 1852
984 66 4 v 1867 1041 a Fj(4)1950 1007 y Fh(\000)2077
968 y Fj(3)p 2059 984 71 4 v 2059 1041 a(16)2156 911
y Fg(\020)2206 1007 y Fi(1)g Fh(\000)2387 968 y Ff(a)2424
944 y Fd(2)p 2387 984 73 4 v 2405 1041 a Fj(2)2469 1007
y Fi(\(1)f(+)h(sin)17 b Fk(t)p Fi(\))2885 911 y Fg(\021)2935
934 y Fj(2)2974 886 y Fg(\023)3052 1007 y Fk(U)3118 1022
y Fj(0)3157 1007 y Fi(\()p Fk(t;)g(\034)11 b Fi(\))485
b Fo(\(14\))146 1258 y(The)25 b(form)g(of)g(general)g(solution)e(to)h
(\(12\))h(can)g(be)g(chosen)g(as)1203 1598 y Fk(U)1269
1613 y Fj(0)1309 1598 y Fi(\()p Fk(t;)17 b(\034)11 b
Fi(\))28 b(=)g Fk(A)1722 1613 y Fj(0)1761 1598 y Fi(\()p
Fk(\034)11 b Fi(\))p Fk(e)1935 1557 y Ff(it=)p Fj(2)2082
1598 y Fi(+)22 b Fk(A)2253 1557 y Ff(?)2253 1623 y Fj(0)2293
1598 y Fi(\()p Fk(\034)11 b Fi(\))p Fk(e)2467 1557 y
Fe(\000)p Ff(it=)p Fj(2)3850 1598 y Fo(\(15\))146 1818
y(In)30 b(order)h(to)e(determine)60 b Fk(\034)72 b Fi(\()p
Fk(\034)48 b Fi(=)38 b Fk("t)p Fi(\))30 b Fo(dependent)f(coef)n
(\002cients)h Fk(A)2570 1833 y Fj(0)2610 1818 y Fi(\()p
Fk(\034)11 b Fi(\))p Fo(,)61 b Fk(A)2898 1782 y Ff(?)2898
1843 y Fj(0)2938 1818 y Fi(\()p Fk(\034)11 b Fi(\))30
b Fo(substitute)e Fk(U)3575 1833 y Fj(0)3615 1818 y Fi(\()p
Fk(t;)17 b(\034)11 b Fi(\))38 b Fo(into)0 1939 y(the)29
b(right)g(hand)g(side)f(of)i(\(13\).)44 b(One)29 b(can)h(see)f(that)g
Fk(e)1872 1902 y Ff(it=)p Fj(2)2026 1939 y Fo(and)59
b Fk(e)2274 1902 y Fe(\000)p Ff(it=)p Fj(2)2482 1939
y Fo(are)30 b(the)f(solutions)e(of)j(the)f(left)g(-hand)g(side)0
2070 y(equation)23 b(i.e.)30 b(homogenous)22 b(equation)1428
2024 y Ff(@)1469 2000 y Fd(2)1504 2024 y Ff(U)1552 2033
y Fd(1)1586 2024 y Fj(\()p Ff(t;\034)8 b Fj(\))p 1428
2047 297 4 v 1526 2105 a Ff(@)t(t)1592 2086 y Fd(2)1753
2070 y Fi(+)1856 2031 y Fj(1)p 1856 2047 36 4 v 1856
2105 a(4)1901 2070 y Fk(U)1967 2085 y Fj(1)2007 2070
y Fi(\()p Fk(t;)17 b(\034)11 b Fi(\))28 b(=)f(0)d Fo(.)30
b(Therefore,)25 b(the)e(right-hand)g(side)g(contains)0
2191 y(terms)g(that)f(produce)h(secular)h(terms)e(in)h
Fk(U)1478 2206 y Fj(1)1518 2191 y Fo(.)30 b(F)o(or)23
b(a)h(uniform)e(e)o(xpansion,)g(these)h(terms)f(must)g(be)i
(eliminated,)e(this)0 2311 y(is)i(proceeding)h(by)f(setting)g(coef)n
(\002cients)h(of)50 b Fk(e)1634 2275 y Ff(it=)p Fj(2)1783
2311 y Fo(and)g Fk(e)2022 2275 y Fe(\000)p Ff(it=)p Fj(2)2226
2311 y Fo(equal)24 b(to)h(zero.)31 b(So,)50 b Fk(A)3049
2326 y Fj(0)3089 2311 y Fi(\()p Fk(\034)11 b Fi(\))25
b Fo(satisfy:)1327 2686 y Fh(\000)p Fk(i)1447 2618 y(@)5
b(A)1576 2633 y Fj(0)p 1447 2663 170 4 v 1477 2754 a
Fk(@)g(\034)1649 2686 y Fh(\000)1762 2618 y Fk(a)1813
2582 y Fj(2)p 1759 2663 98 4 v 1759 2754 a Fi(16)1866
2686 y Fk(A)1939 2645 y Ff(?)1939 2710 y Fj(0)2001 2686
y Fh(\000)2111 2618 y Fk(\016)2154 2633 y Fj(1)p 2111
2663 83 4 v 2128 2754 a Fi(4)2203 2686 y Fk(A)2276 2701
y Fj(0)2343 2686 y Fi(=)28 b(0)p Fk(;)1327 b Fo(\(16\))1366
3160 y Fk(i)1409 3093 y(@)5 b(A)1538 3057 y Ff(?)1538
3117 y Fj(0)p 1409 3137 170 4 v 1438 3228 a Fk(@)g(\034)1610
3160 y Fh(\000)1723 3093 y Fk(a)1774 3057 y Fj(2)p 1720
3137 98 4 v 1720 3228 a Fi(16)1828 3160 y Fk(A)1901 3175
y Fj(0)1962 3160 y Fh(\000)2072 3093 y Fk(\016)2115 3108
y Fj(1)p 2072 3137 83 4 v 2089 3228 a Fi(4)2164 3160
y Fk(A)2237 3119 y Ff(?)2237 3185 y Fj(0)2305 3160 y
Fi(=)27 b(0)p Fk(:)1366 b Fo(\(17\))146 3411 y(Setting)25
b Fk(A)527 3426 y Fj(0)566 3411 y Fi(\()p Fk(\034)11
b Fi(\))28 b(=)g Fk(w)897 3426 y Fj(0)936 3411 y Fi(\()p
Fk(\034)11 b Fi(\))22 b(+)g Fk(iv)1265 3426 y Fj(0)1305
3411 y Fi(\()p Fk(\034)11 b Fi(\))26 b Fo(we)f(\002nd:)1216
3786 y Fh(\000)p Fk(w)1366 3721 y Fc(0)1363 3810 y Fj(0)1403
3786 y Fi(\()p Fk(\034)11 b Fi(\))23 b Fh(\000)1664 3718
y Fk(\016)1707 3733 y Fj(1)p 1664 3763 V 1681 3854 a
Fi(4)1757 3786 y Fk(v)1804 3801 y Fj(0)1843 3786 y Fi(\()p
Fk(\034)11 b Fi(\))23 b(+)2106 3718 y Fk(a)2157 3682
y Fj(2)p 2103 3763 98 4 v 2103 3854 a Fi(16)2210 3786
y Fk(v)2257 3801 y Fj(0)2297 3786 y Fi(\()p Fk(\034)11
b Fi(\))28 b(=)f(0)p Fk(;)1217 b Fo(\(18\))1243 4260
y Fk(v)1294 4195 y Fc(0)1290 4285 y Fj(0)1330 4260 y
Fi(\()p Fk(\034)11 b Fi(\))22 b Fh(\000)1591 4193 y Fk(\016)1634
4208 y Fj(1)p 1591 4237 83 4 v 1608 4328 a Fi(4)1683
4260 y Fk(w)1753 4275 y Fj(0)1792 4260 y Fi(\()p Fk(\034)11
b Fi(\))23 b Fh(\000)2057 4193 y Fk(a)2108 4157 y Fj(2)p
2054 4237 98 4 v 2054 4328 a Fi(16)2161 4260 y Fk(w)2231
4275 y Fj(0)2270 4260 y Fi(\()p Fk(\034)11 b Fi(\))28
b(=)g(0)p Fk(;)1243 b Fo(\(19\))146 4504 y(from)25 b(\(18\))g(and)g
(\(19\))1370 4879 y Fk(v)1421 4814 y Fc(00)1417 4904
y Fj(0)1466 4879 y Fi(\()p Fk(\034)11 b Fi(\))22 b(+)1725
4812 y(1)p 1725 4856 49 4 v 1725 4947 a(4)1784 4879 y(\()p
Fk(\016)1869 4838 y Fj(2)1865 4904 y(1)1931 4879 y Fh(\000)2044
4812 y Fk(a)2095 4775 y Fj(4)p 2040 4856 98 4 v 2040
4947 a Fi(16)2148 4879 y(\))p Fk(v)2233 4894 y Fj(0)2300
4879 y Fi(=)27 b(0)p Fk(;)1371 b Fo(\(20\))146 5130 y(and)25
b(the)g(solutions)d(of)50 b Fk(w)1049 5145 y Fj(0)1088
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%%Page: 6 6
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%%Page: 7 7
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y(7)p eop
%%Page: 8 8
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y Fd(2)p 1531 2684 V 1560 2742 a Ff(@)t(t)1626 2723 y
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y Fd(2)2384 2668 y Ff(U)2432 2677 y Fd(0)p 2308 2684
V 2312 2742 a Ff(@)t(t@)t(\033)2476 2708 y Fi(\))g(+)g
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y Ff(@)2802 2645 y Fd(2)2837 2668 y Ff(U)2885 2677 y
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y Fk(U)2168 3730 y Fj(1)p 2102 3759 106 4 v 2131 3851
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y Fj(2)p 1293 4227 203 4 v 1328 4319 a Fk(@)5 b(t)1419
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y(=)27 b Fh(\000)1969 4183 y Fk(a)2020 4147 y Fj(2)p
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Fg(\021)2744 4251 y Fk(U)2810 4266 y Fj(0)2850 4251 y
Fk(;)973 b Fo(\(43\))766 4719 y Fk(\017)815 4650 y Fd(3)p
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y Fk(@)1158 4615 y Fj(2)1198 4651 y Fk(U)1264 4666 y
Fj(3)p 1102 4695 203 4 v 1137 4787 a Fk(@)5 b(t)1228
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y Fj(3)p 1444 4695 106 4 v 1472 4787 a Fi(4)1587 4719
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y Fe(\000)p Ff(it)2885 4622 y Fg(\021)2951 4719 y Fk(U)3017
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y Fk(\017)826 5146 y Fj(2)1051 5187 y Fi(:)1274 5119
y Fk(@)1330 5083 y Fj(2)1370 5119 y Fk(U)1436 5134 y
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y(8)p eop
%%Page: 9 9
9 8 bop 146 52 a Fo(The)25 b(solutions)e(of)i(\(41\))g(and)f(\(42\))h
(are)h(respecti)n(v)o(ely:)1247 393 y Fk(U)1313 408 y
Fj(0)1353 393 y Fi(\()p Fk(t;)17 b(\033)t Fi(\))28 b(=)f
Fk(A)1771 408 y Fj(0)1811 393 y Fi(\()p Fk(\033)t Fi(\))p
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1533 a Fd(2)1583 1548 y Fi(+)22 b Fk(A)1754 1507 y Fe(\003)1754
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1533 a Fd(2)2097 1548 y Fi(+)2205 1481 y Fk(a)2256 1444
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y Fk(A)2379 1563 y Fj(0)2418 1548 y Fk(e)2473 1480 y
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2440 y(From)25 b(\(42-46\))1229 2815 y Fh(\000)p Fk(i)1379
2747 y(d)p 1349 2791 110 4 v 1349 2883 a(d\033)1469 2815
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2830 y Fj(1)1852 2815 y Fi(\))28 b(=)g(\(4)p Fk(\016)2152
2830 y Fj(2)2213 2815 y Fi(+)2321 2747 y(3)p Fk(a)2421
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3059 y(from)d(\(49\))1235 3437 y Fk(B)1314 3372 y Fc(00)1309
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Fk(a)2163 3333 y Fj(2)p 2063 3413 140 4 v 2108 3505 a
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y Fo(is)f(in)h(form)1279 4053 y Fk(B)1353 4068 y Fj(0)1392
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Fj(4)1784 4053 y Fi(cos)17 b Fk(\026\033)26 b Fi(+)c
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y Fj(2)845 4295 y Fi(+)953 4256 y Fj(3)p Ff(a)1025 4237
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(parameter)-5 b(.)65 b(If)36 b(we)h(look)e(at)h(the)0
4429 y(stability)28 b(boundaries)h(,)h(stability)e(occurs)i(when)f
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4429 y Fk(>)3506 4389 y Fj(3)p 3489 4405 V 3489 4463
a(16)3569 4429 y Fo(.)45 b(Thus,)30 b(the)0 4549 y(higher)24
b(order)i(stability)d(boundary)h(is)g(gi)n(v)o(en)f(by:)1057
4909 y Fi(\001)28 b(=)g(1)22 b Fh(\006)1450 4841 y Fi(1)p
1450 4886 49 4 v 1450 4977 a(2)1509 4909 y Fk(\017)g
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a(16)1788 4909 y Fk(\017)1827 4868 y Fj(2)1889 4909 y
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1130 5500 y Fk(A)1203 5515 y Fj(1)1242 5500 y Fi(\()p
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b Fk(\026\033)t Fi(\))f Fk(:)1130 b Fo(\(54\))1983 5753
y(9)p eop
%%Page: 10 10
10 9 bop 146 52 a Fo(Using)24 b(the)h(initial)e(conditions)g(as)i
Fk(U)1427 67 y Fj(0)1467 52 y Fi(\(0\))i(=)h(0)p Fk(;)1843
27 y Fi(_)1816 52 y Fk(U)1882 67 y Fj(0)1921 52 y Fi(\(0\))f(=)h(1)p
Fk(;)116 b(U)2435 67 y Fj(1)2475 52 y Fi(\(0)p Fk(;)17
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%%Page: 11 11
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y(1)p 1626 1977 V 1626 2068 a(4)1684 2000 y Fk("cost)22
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a(16)2139 2000 y Fk(")2185 1959 y Fj(2)2224 2000 y Fk(sin)2361
1959 y Fj(2)2401 2000 y Fk(t)p Fi(\))p Fk(q)2521 1959
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g(the)g(hamiltonians)d(for)j(disturbed)f(VdP)g(oscillator)g(equation.)
53 b(The)33 b(Hamilton-Jacobi)0 2353 y(equation)24 b(corresponding)g
(to)g(the)h(case)g Fk(")j Fi(=)f(0)e Fo(is)1323 2648
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y(Then)25 b(\(86\))g(separates)g(into)40 b Fi(_)-43 b
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4825 y(2)1648 4743 y Fh(p)p 1731 4743 V 82 x Fi(2)p Fk(\013)p
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2035 4870 357 4 v 2188 4961 a(2)2401 4893 y(\))1411 b
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%%Page: 14 14
14 13 bop 146 53 a Fo(and)53 b Fi(_)-47 b Fk(\013)43
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507 y Fk(\014)h Fi(+)22 b Fk("S)1584 522 y Fj(1)1623
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507 y Fk(S)2250 522 y Fj(2)2289 507 y Fi(\()p Fk(\013)2390
466 y Fe(\003)2429 507 y Fk(;)17 b(\014)6 b(;)17 b(t)p
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2189 y Fj(1)p 1430 2218 V 1462 2310 a Fk(@)g(t)1618 2241
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157 4 v 1206 2765 a Fk(@)g(\014)1353 2696 y(cos)1488
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v 1729 2765 a(2)1942 2696 y(\))p Fk(cost)22 b Fh(\000)2282
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eop
%%Page: 15 15
15 14 bop 146 52 a Fo(Therefore)26 b Fk(K)32 b Fo(can)25
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1432 y Fe(\003)2495 1468 y Fo(and)f Fk(\014)2724 1432
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Fk(!)d Fh(\000)e Fi(1\))p Fk(t)f Fi(+)g Fk(!)t(\014)2358
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1915 y(and)1720 2242 y Fk(\013)1783 2201 y Fe(\003)1850
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2876 a Ff(@)t(t)1792 2842 y Fo(and:)854 3168 y Fk(K)35
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%%Page: 16 16
16 15 bop 0 52 a Fl(5)143 b(Conclusions)0 276 y Fo(In)27
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23 b(equation)g(with)g(periodic)0 517 y(coef)n(\002cients.)44
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%%Page: 17 17
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2563 y Fi(23)p Fk(:)146 2684 y Fo(13.)29 b Fk(P)s(:E)6
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3045 y Fk(M)10 b(ath:Gen:)p Fi(36\(2003\))p Fk(;)17 b
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b(N)10 b(onl)r(inear)s(M)g(athematics;)17 b(M)10 b(c)p
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3286 y Fo(16.)69 b Fk(P)s(:Gl)r(enndining)t(;)17 b(S)6
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b(C)7 b(l)r(assical)r(M)j(echanics;)17 b(Addison)25 b
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4008 y Fo(20.)31 b Fk(D)s(:P)s(:K)7 b(r)s(usk)s(al)881
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y Fo(21.)29 b Fk(M)5 b(:K)i(r)s(usk)s(al)r(;)17 b(inD)s(y)t(namical)r
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b(otesinP)k(hy)t(sics)p Fi(38)p Fk(;)j(ed:by)t(J)n(:M)10
b(oser)s Fi(\()p Fk(S)c(pr)s(ing)t(er)s Fh(\000)0 4249
y Fk(V)22 b(er)s(l)r(ag)t(;)17 b(B)5 b(er)s(l)r(in;)17
b Fi(1975\))p Fk(:)1958 5753 y Fo(17)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF