Time scales and definitions

Equations of motions and resonance conditions

Equations of motion

EOM q1't q1t c2 q1t q2t c3 q1t3 0, q2 't q2t bq1t2 0;EOM TableForm

q1t c3 q1t3 c2 q1t q2t q1t 0

bq1t2 q2t q2t 0

EOM Subscriptq, 1t c Subscriptq, 1t3 b1 Subscriptq, 1't3 b0 Subscriptq, 2't Subscriptq, 1't2 Subscriptq, 1't Subscriptq, 1''t 0,

22 Subscriptq, 2t c Subscriptq, 2t3 b2 Subscriptq, 2't3 b0 Subscriptq, 2't Subscriptq, 1't2

Subscriptq, 2't Subscriptq, 2''t 0; EOM TableForm

12 q1t c q1t3 q1t b1 q1t3 b0 q1t q2t2 q1t 0

22 q2t c q2t3 q2t b2 q2t3 b0 q1t q2t2 q2t 0

Ordering of the dampings

smorzrule , ;Definition of 2 (introduction of the detuning)

ombrule 2 2 2 2Definition of the expansion of qi

solRule qi_ Sumj1 qi,j11, 2, 3, j, 3 &;Some rules (don' t modify)

multiScales qi_t qi timeScales,

Derivativen_q_t dtnq timeScales, t T0;This is the result of "multiScales" and "solRule":

Homework2.nb 1

q1t . multiScales . solRule

q1,0T0, T1, T2 q1,1T0, T1, T2 2 q1,2T0, T1, T2Max order of the procedure

maxOrder 2;

Scaling of the variables

scaling q1t q1t, q2t q2t, q1 't q1't,q2't q2 't, q1''t q1 ''t, q2''t q2 ''tq1t q1t, q2t q2t, q1t q1t,q2t q2t, q1t q1t, q2t q2t

Modification of the equations of motion: substitution of the rules.

EOMa EOM . scaling . multiScales . smorzrule . ombrule . solRule TrigToExp ExpandAll . n_;n3 0;

Separation of the coefficients of the powers of

eqEps RestThreadCoefficientListSubtract , 0 & EOMa Transpose;

Definition of the equations at orders of and representation

eqOrderi_ : 1 & eqEps1 . q_k_,0 qk,i1 1 & eqEps1 . q_k_,0 qk,i1 1 & eqEpsi Thread

eqOrder1 . displayRuleeqOrder2 . displayRule

eqOrder3 . displayRuleD02q1,0 12 q1,0 0, D02q2,0 2

2 q2,0 0D02q1,1 12 q1,1 D0q1,0 2 D0 D1q1,0 D0q1,02 b0 2 D0q1,0 D0q2,0 b0 D0q2,0

2 b0,

D02q2,1 2

2 q2,1 D0q2,0 2 D0 D1q2,0 D0q1,02 b0 2 D0q1,0 D0q2,0 b0 D0q2,0

2 b0 2 2 q2,0D02q1,2 12 q1,2 D0q1,1 D1q1,0 2 D0 D1q1,1 D12q1,0 2 D0 D2q1,0 2 D0q1,0 D0q1,1 b0

2 D0q1,1 D0q2,0 b0 2 D0q1,0 D0q2,1 b0 2 D0q2,0 D0q2,1 b0 2 D0q1,0 D1q1,0 b0

2 D0q2,0 D1q1,0 b0 2 D0q1,0 D1q2,0 b0 2 D0q2,0 D1q2,0 b0 D0q1,03 b1 c q1,0

3 ,

D02q2,2 2

2 q2,2 D0q2,1 D1q2,0 2 D0 D1q2,1 D12q2,0 2 D0 D2q2,0 2 D0q1,0 D0q1,1 b0

2 D0q1,1 D0q2,0 b0 2 D0q1,0 D0q2,1 b0 2 D0q2,0 D0q2,1 b0 2 D0q1,0 D1q1,0 b0 2 D0q2,0 D1q1,0 b0

2 D0q1,0 D1q2,0 b0 2 D0q2,0 D1q2,0 b0 D0q2,03 b2

2 q2,0 c q2,03 2 2 q2,1

Resonance condition

ResonanceCond 1 1

22;

Involved frequencies

Homework2.nb 2

omgList 1, 2;Utility (don't modify)

omgRule SolveResonanceCond, , Flatten1 & omgList Reverse

2 2 1, 1 2

2

First-Order Problem

Second-Order Problem

Third-Order Problem

Substitution in the Third-Order Equations

order3Eq eqOrder3 . sol1 . sol2 ExpandAll;

Simplifications

order3Eqpr1, 2 Simplifyorder3Eq1, 2;order3Eqpr2, 2 Simplifyorder3Eq2, 2;

Terms of type ei1 T0 in the first equation of the Third-Order Problem and representation

ST311 Coefficientorder3Eqpr, 2 . expRule1, ExpI 1 T0 & 1;ST311 . displayRule 1

60 160 D1A1 1 60 D12A1 1 120 D2A1 12 120 D1A1 A2 b0 1 2 180 c A12 1 A1

120 D1A2 b0 12 A1 80 A1

2 b02 1

3 A1 180 A12 b1 1

4 A1 448 A1 A2 b02 1

2 2 A2 90 A1 A2 b02 1 2

2 A2Terms of type ei2 T0 in the first equation of the Third-Order Problem and representation

ST312 Coefficientorder3Eqpr, 2 . expRule1, ExpI 2 T0 & 1;ST312 . displayRule0

Terms of type ei1 T0 in the second equation of the Third-Order Problem and representation

Homework2.nb 3

ST321 Coefficientorder3Eqpr, 2 . expRule11, ExpI 1 T0 & 2;ST321 . displayRule 1

30 1 280 D1A1 A2 b0 12 2 60 D1A1 A2 b0 1 22 140 D1A2 b0 12 2 A1 40 A2 b0 12 2 A1

160 A2 b0 12 2 A1 40 A1

2 b02 1

3 2 A1 224 A1 A2 b02 1

2 22 A2 45 A1 A2 b0

2 1 23 A2

Terms of type ei2 T0 in the second equation of the Third-Order Problem and representation

ST322 Coefficientorder3Eqpr, 2 . expRule12, ExpI 2 T0 & 2;ST322 . displayRule 1

30 1 230 D1A2 1 2 30 D12A2 1 2 30 2 A2 1 2

60 D1A1 A1 b0 12 2 60 D2A2 1 2

2 64 A1 A2 b02 1

3 2 A1 45 A1 A2 b02 1

2 22 A1

90 c A22 1 2 A2 40 A2

2 b02 1 2

3 A2 16 A22 b0

2 24 A2 90 A2

2 b2 1 24 A2

Scalar product with the left eigenvectors: Second-Order AME; representation

SCond2 1, 0.ST311, ST321 0, 0, 1.ST312, ST322 0;SCond2 . displayRule 1

60 160 D1A1 1 60 D12A1 1 120 D2A1 12 120 D1A1 A2 b0 1 2 180 c A12 1 A1 120 D1A2b0 1

2 A1 80 A12 b0

2 13 A1 180 A1

2 b1 14 A1 448 A1 A2 b0

2 12 2 A2 90 A1 A2 b0

2 1 22 A2 0, 1

30 1 230 D1A2 1 2 30 D12A2 1 2 30 2 A2 1 2 60 D1A1 A1 b0 12 2

60 D2A2 1 22 64 A1 A2 b0

2 13 2 A1 45 A1 A2 b0

2 12 2

2 A1 90 c A22 1 2 A2

40 A22 b0

2 1 23 A2 16 A2

2 b02 2

4 A2 90 A22 b2 1 2

4 A2 0Algebraic manipulation to obtain D2 A1 and D2 A2

Homework2.nb 4

SCond2Rule1

SolveSCond2, A10,1T1, T2, A20,1T1, T21 . A12,0T1, T2 T1 SCond1Rule1

1, 2 . A22,0T1, T2 T1 SCond1Rule12, 2 . SCond1Rule1 .SCond1Rule1 . conjugateRule ExpandAll Simplify Expand;SCond2Rule1 . displayRuleD2A1

2 A1

8 11

2 A2 b0 A1 A2 b0 A1

3 c A12 A1

2 1

5

12 A1

2 b02 1 A1

3

2A12 b1 1

2 A1 A1

2 b02 1

2 A1

2 2

A2 b0 2 A1

2 1 A2 b0 2 A1

4 1 A2 b0 2 A1

2 156

15 A1 A2 b0

2 2 A2 3 A1 A2 b0

2 22 A2

4 1,

D2A2 A1

2 b0 12

4 22

A1

2 b0 12

8 22

A1

2 b0 12

4 22

2 A2

8 2 A1

2 b0 1

2 27

4 A1 A2 b0

2 1 A1

17 A1 A2 b02 1

2 A1

30 23 c A2

2 A2

2 22

3 A2

2 b02 2 A2

3

2A22 b2 2

2 A2 4 A2

2 b02 2

2 A2

15 1

Reconstitution of the AME and of the solution

Polar form of the AME

Numerical integrations

Numerical values

1 1, 2 2, b0 1

2, b1 1, b2 1, c 1, 0.05, 0.05, 0, 2 2 1

1, 2,1

2, 1, 1, 1, 0.05, 0.05, 0, 2

Time of integration

ti 200;

Numerical Intergations of the RAME

solrame1 NDSolverame1 0, rame2 0, rame3 0,a10 0.1, a20 0.1, 0 0.1, a1t, a2t, t, t, 0, ti

NDSolve::ndode : Input is not an ordinary differential equation.

NDSolverame1 0, rame2 0, rame3 0, a10 0.1, a20 0.1, 0 0.1,a1t, a2t, t, t, 0, ti

Homework2.nb 5

GraphicsArrayPlota1t . solrame1, t, 0, ti, PlotStyle Thick,

PlotRange Automatic, 0.8, 0.8, Frame True, FrameLabel "t", "a1t",Plota2t . solrame1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "a2t",

Plott . solrame1, t, 0, ti, PlotStyle Thick, Frame True,

AxesOrigin 0, 0, FrameLabel "t", "t"

0 50 100 150 200

0.5

0.0

0.5

t

a1t

0 50 100 150 2000.60.40.20.00.20.4

ta2t

0 50 100 150 2000.0

0.5

1.0

1.5

2.0

t

t

Numerical Intergations of the AME

solramep1 NDSolveamep1 0, amep2 0, amep3 0, amep4 0, a10 0.1,a20 0.1, 10 0.1, 20 0.1, a1t, a2t, 1t, 2t, t, 0, tia1t InterpolatingFunction0., 200., t,

a2t InterpolatingFunction0., 200., t,1t InterpolatingFunction0., 200., t,2t InterpolatingFunction0., 200., t

GraphicsArrayPlota1t . solramep1, t, 0, ti, PlotStyle Thick,

PlotRange Automatic, 0.8, 0.8, Frame True, FrameLabel "t", "a1t",Plota2t . solramep1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "a2t",

Plot1t . solramep1, t, 0, ti, PlotStyle Thick,

Frame True, AxesOrigin 0, 0, FrameLabel "t", "1t",Plot2t . solramep1, t, 0, ti, PlotStyle Thick, Frame True,AxesOrigin 0, 0, FrameLabel "t", "2t"

0 50 100 150 200

0.5

0.0

0.5

t

a1t

0 50 100 150 2000.60.40.20.00.20.4

t

a2t

0 50 100 150 2000.00.51.01.52.0

t

1t

0 50 100 1500.00.51.01.52.0

t

2t

Graphics of the reconstituted solution

Homework2.nb 6

GraphicsArrayPlotqr1t . solramep1, t, 0, ti, PlotStyle Thick,

PlotRange Automatic, 0.8, 0.8, Frame True, FrameLabel "t", "q1t",Plotqr2t . solramep1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "q2t"

0 50 100 150 200

0.5

0.0

0.5

t

q1t

0 50 100 150 2000.6

0.4

0.2

0.0

0.2

0.4

t

q2t

Numerical Intergations of the original equations

solorig1 NDSolveJoinEOM, q10 0.1, q20 0.1, q1'0 0.1, q2 '0 0.1,q1t, q2t, t, 0, ti, MaxSteps 1000000q1t InterpolatingFunction0., 200., t,q2t InterpolatingFunction0., 200., t

GraphicsArrayPlotq1t . solorig1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 0.8, 0.8, Frame True, FrameLabel "t", "q1t",

Plotq2t . solorig1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "q2t"

0 50 100 150 200

0.5

0.0

0.5

t

q1t

0 50 100 150 2000.6

0.4

0.2

0.0

0.2

0.4

t

q2t

Homework2.nb 7