exercise: multiple bifurcations of sample dynamical systems
DESCRIPTION
Multiple Bifurcations of Sample Dynamical SystemsTRANSCRIPT
Double Rayleigh-Duffing oscillator in nearly 1:2 resonance. MSM.
q..
1 q 1 12 q1 b0 q 2 q 1
2 b1 q 13 c q1
3 0
q..
2 q 2 22 q2 b0 q 2 q 1
2 b2 q 23 c q2
3 0
2 21
Time scales and definitions
Utility (don't modify)
OffGeneral::spell1 Notation`
Time scales
SymbolizeT0; SymbolizeT1; SymbolizeT2;timeScales T0, T1, T2;
Differentiation (don't modify)
dt1expr_ : Sumi Dexpr, timeScalesi 1, i, 0, maxOrder;dt2expr_ : dt1dt1expr Expand . i_;imaxOrder 0;
Some rules (don' t modify)
conjugateRule A A, A A, , , Complex0, n_ Complex0, n;
Exercise.nb 1
displayRule q_i_,j_a____ RowTimes MapIndexedD2111 &, a, qi,j,A_i_
a____ RowTimes MapIndexedD211 &, a, Ai,q_i_,j___ qi,j, A_i___ Ai;
Equations of motions and resonance conditions
Equations of motion
EOM 12 Subscriptq, 1t c Subscriptq, 1t3 b1 Subscriptq, 1't3 b0 Subscriptq, 2't Subscriptq, 1't2 Subscriptq, 1't Subscriptq, 1''t 0,
22Subscriptq, 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,j1, 2, 3, j, 3 &;Some rules (don' t modify)
multiScales qi_t qi timeScales,
Derivativen_q_t dtnq timeScales, t T0;Max order of the procedure
maxOrder 2;
Scaling of the variables
Exercise.nb 2
scaling Subscriptq, 1t Subscriptq, 1t,Subscriptq, 2t Subscriptq, 2t,Subscriptq, 1't Subscriptq, 1't,Subscriptq, 2't Subscriptq, 2't, Subscriptq, 1''t
Subscriptq, 1''t, Subscriptq, 2''t Subscriptq, 2''tq1t q1t, q2t q2t, q1t q1
t,q2
t q2t, q1
t q1t, q2
t q2t
Modification of the equations of motion: substitution of the rules. Representation.
EOMa EOM . scaling . multiScales . smorzrule . ombrule . solRule TrigToExp ExpandAll . n_;n3 0; EOMa . displayRule2 D0q1,1 3 D0q1,2 D02q1,1 2 D02q1,2 3 D02q1,3 3 D1q1,1
2 2 D0 D1q1,1 2 3 D0 D1q1,2 3 D12q1,1 2 3 D0 D2q1,1 2 D0q1,12 b0 2 3 D0q1,1 D0q1,2 b0 2 2 D0q1,1 D0q2,1 b0 2 3 D0q1,2 D0q2,1 b0 2 D0q2,12 b0 2 3 D0q1,1 D0q2,2 b0 2 3 D0q2,1 D0q2,2 b0 2 3 D0q1,1 D1q1,1 b0 2 3 D0q2,1 D1q1,1 b0 2 3 D0q1,1 D1q2,1 b0 2 3 D0q2,1 D1q2,1 b0 3 D0q1,13 b1 12 q1,1 c 3 q1,13 2 1
2 q1,2 3 1
2 q1,3 0,
2 D0q2,1 3 D0q2,2 D02q2,1 2 D02q2,2 3 D02q2,3 3 D1q2,1 2 2 D0 D1q2,1 2 3 D0 D1q2,2 3 D12q2,1 2 3 D0 D2q2,1 2 D0q1,12 b0 2 3 D0q1,1 D0q1,2 b0 2 2 D0q1,1 D0q2,1 b0 2 3 D0q1,2 D0q2,1 b0 2 D0q2,12 b0 2 3 D0q1,1 D0q2,2 b0 2 3 D0q2,1 D0q2,2 b0 2 3 D0q1,1 D1q1,1 b0 2 3 D0q2,1 D1q1,1 b0 2 3 D0q1,1 D1q2,1 b0 2 3 D0q2,1 D1q2,1 b0 3 D0q2,13 b2 3 2 q2,1 2
2 2 q2,1 22 q2,1 c
3 q2,13 2 3 2 q2,2
2 22 q2,2
3 22 q2,3 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_,1 qk,i 1 & eqEps1 . q_k_,1 qk,i 1 & eqEpsi Thread
Exercise.nb 3
eqOrder1 . displayRule
eqOrder2 . displayRuleeqOrder3 . displayRuleD02q1,1 12 q1,1 0, D0
2q2,1 22 q2,1 0D02q1,2 12 q1,2 D0q1,1 2 D0 D1q1,1 D0q1,12 b0 2 D0q1,1 D0q2,1 b0 D0q2,12 b0,
D02q2,2 2
2 q2,2
D0q2,1 2 D0 D1q2,1 D0q1,12 b0 2 D0q1,1 D0q2,1 b0 D0q2,12 b0 2 2 q2,1D02q1,3 12 q1,3 D0q1,2 D1q1,1 2 D0 D1q1,2 D12q1,1 2 D0 D2q1,1 2 D0q1,1 D0q1,2 b0 2 D0q1,2 D0q2,1 b0 2 D0q1,1 D0q2,2 b0 2 D0q2,1 D0q2,2 b0 2 D0q1,1 D1q1,1 b0 2 D0q2,1 D1q1,1 b0 2 D0q1,1 D1q2,1 b0 2 D0q2,1 D1q2,1 b0 D0q1,13 b1 c q1,13 ,
D02q2,3 2
2 q2,3 D0q2,2 D1q2,1 2 D0 D1q2,2 D12q2,1 2 D0 D2q2,1 2 D0q1,1 D0q1,2 b0 2 D0q1,2 D0q2,1 b0 2 D0q1,1 D0q2,2 b0 2 D0q2,1 D0q2,2 b0 2 D0q1,1 D1q1,1 b0 2 D0q2,1 D1q1,1 b0 2 D0q1,1 D1q2,1 b0 2 D0q2,1 D1q2,1 b0 D0q2,13 b2 2 q2,1 c q2,13 2 2 q2,2
Resonance condition
ResonanceCond 1 1
22;
Involved frequencies
omgList 1, 2;Utility (don't modify)
omgRule SolveResonanceCond, , Flatten1 & omgList Reverse
2 2 1, 1 2
2
First-Order Problem
Equations (=0)
linearSys 1 & eqOrder1;linearSys . displayRule TableForm
D02q1,1 1
2 q1,1
D02q2,1 2
2 q2,1
Exercise.nb 4
Formal solution of the First-Order Problem
sol1 q1,1 FunctionT0, T1, T2, A1T1, T2 ExpI 1 T0 A1T1, T2 ExpI 1 T0,q2,1 FunctionT0, T1, T2, A2T1, T2 ExpI 2 T0 A2T1, T2 ExpI 2 T0q1,1 FunctionT0, T1, T2, A1T1, T2 1 T0 A1T1, T2 1 T0,
q2,1 FunctionT0, T1, T2, A2T1, T2 2 T0 A2T1, T2 2 T0Second-Order Problem
Substitution of the solution on the Second-Order Problem and representation
order2Eq eqOrder2 . sol1 ExpandAll;
order2Eq . displayRuleD02q1,2 12 q1,2 2 T0 1 D1A1 1 2 T0 1 D1A1 1 T0 1 A1 1
2 T0 1 A12 b0 1
2 2 T0 1 T0 2 A1 A2 b0 1 2 2 T0 2 A2
2 b0 22
T0 1 1 A1 2 A1 b0 12 A1 2
T0 1 T0 2 A2 b0 1 2 A1 2 T0 1 b0 1
2 A12
2 T0 1 T0 2 A1 b0 1 2 A2 2 A2 b0 22 A2 2
T0 1 T0 2 b0 1 2 A1 A2 2 T0 2 b0 2
2 A22,
D02q2,2 2
2 q2,2 2 T0 1 A12 b0 1
2 2 T0 2 D1A2 2 2 T0 2 D1A2 2 T0 2 A2 2 2 T0 2 A2 2 2
T0 1 T0 2 A1 A2 b0 1 2 2 T0 2 A2
2 b0 22 2 A1 b0 1
2 A1
2 T0 1 T0 2 A2 b0 1 2 A1 2 T0 1 b0 1
2 A12 T0 2 2 A2 2
T0 2 2 A2
2 T0 1 T0 2 A1 b0 1 2 A2 2 A2 b0 22 A2 2
T0 1 T0 2 b0 1 2 A1 A2 2 T0 2 b0 2
2 A22
Utility (don't modify)
expRule1i_ : Expa_ : ExpExpanda . omgRulei . T0 T1Terms of type ei1 T0 in the first equation of the Second-Order Problem and representation
ST11 Coefficientorder2Eq, 2 . expRule11, ExpI 1 T0 & 1;ST11 . displayRule2 D1A1 1 A1 1 2 A2 b0 1 2 A1
Terms of type ei2 T0 in the first equation of the Second-Order Problem and representation
ST12 Coefficientorder2Eq, 2 . expRule12, ExpI 2 T0 & 1;ST12 . displayRuleA12 b0 12
Terms of type ei1 T0 in the second equation of the Second-Order Problem and representation
Exercise.nb 5
ST21 Coefficientorder2Eq, 2 . expRule11, ExpI 1 T0 & 2;ST21 . displayRule2 A2 b0 1 2 A1
Terms of type ei2 T0 in the second equation of the Second-Order Problem and representation
ST22 Coefficientorder2Eq, 2 . expRule12, ExpI 2 T0 & 2;ST22 . displayRuleA12 b0 12 2 D1A2 2 A2 2 2 A2 2
Scalar product with the left eigenvectors: First-Order AME; representation
SCond1 1, 0.ST11, ST21 0, 0, 1.ST12, ST22 0;SCond1 . displayRule2 D1A1 1 A1 1 2 A2 b0 1 2 A1 0, A12 b0 12 2 D1A2 2 A2 2 2 A2 2 0
Algebraic manioulation to obtain D1 A1 and D1 A2
SCond1Rule1 SolveSCond1, A11,0T1, T2, A21,0T1, T21 ExpandAll;
SCond1Rule1 . displayRule TableForm
D1A1 A12
A2 b0 2 A1
D1A2 A22
A2 A1
2 b0 12
2 2
Substitution of the First - Order AME to the Second Order Equations
order2Eqm order2Eq . SCond1Rule1 . SCond1Rule1 . conjugateRule . expRule1 & 1, 2 ExpandAll; order2Eqm . displayRuleD02q1,2 12 q1,2 2 T0 1 A12 b0 1
2 2 3 T0 1 A1 A2 b0 1 2 4 T0 1 A2
2 b0 22
2 A1 b0 12 A1
2 T0 1 b0 12 A1
2 2 A2 b0 2
2 A2 2 3 T0 1 b0 1 2 A1 A2
4 T0 1 b0 22 A2
2,
D02q2,2 2
2 q2,2 2 3
2 T0 2 A1 A2 b0 1 2
2 T0 2 A22 b0 2
2 2 A1 b0 12 A1 2
1
2 T0 2 A2 b0 1 2 A1
2 1
2 T0 2 A1 b0 1 2 A2 2 A2 b0 2
2 A2 2 3
2 T0 2 b0 1 2 A1 A2
2 T0 2 b0 22 A2
2Assumpion on the coefficients
$Assumptions 1, 2, b0, c, , , b1, b2 Reals1 2 b0 c b1 b2 Reals
Construction of the particular solution of the first equation of the Second-Order Problem and representation
Exercise.nb 6
solOrder2Eqm1
SimplifyTableq1,2T0, T1, T2 . TrigToExpFlattenDSolveorder2Eqm1, 1
order2Eqm1, 2, i, q1,2T0, T1, T2, T0 . C1 0, C2 0,i, 1, Lengthorder2Eqm1, 2; solOrder2Eqm1 . displayRule
132 T0 1 A1
2 b0, 3 T0 1 A1 A2 b0 2
4 1,4 T0 1 A2
2 b0 22
15 12
, 2 A1 b0 A1,
1
32 T0 1 b0 A1
2,2 A2 b0 2
2 A2
12
, 3 T0 1 b0 2 A1 A2
4 1,4 T0 1 b0 2
2 A22
15 12
solq21 ExpandTotalsolOrder2Eqm1;solq21 . displayRule
1
32 T0 1 A1
2 b0 3 T0 1 A1 A2 b0 2
4 14 T0 1 A2
2 b0 22
15 12
2 A1 b0 A1
1
32 T0 1 b0 A1
22 A2 b0 2
2 A2
12
3 T0 1 b0 2 A1 A2
4 14 T0 1 b0 2
2 A22
15 12
Construction of the particular solution of the second equation of the Second-Order Problem
solOrder2Eqm2 Tableq2,2T0, T1, T2 .SimplifyTrigToExpFlattenDSolveorder2Eqm2, 1 order2Eqm2, 2, i,
q2,2T0, T1, T2, T0 . C1 0, C2 0,i, 1, Lengthorder2Eqm2, 2; solOrder2Eqm2 . displayRule
8 3
2 T0 2 A1 A2 b0 1
5 2,
1
32 T0 2 A2
2 b0, 2 A1 b0 1
2 A1
22
,8
1
2 T0 2 A2 b0 1 A1
3 2,
8 1
2 T0 2 A1 b0 1 A2
3 2, 2 A2 b0 A2,
8 3
2 T0 2 b0 1 A1 A2
5 2,
1
32 T0 2 b0 A2
2solq22 ExpandTotalsolOrder2Eqm2;solq22 . displayRule
1
32 T0 2 A2
2 b0 8
3
2 T0 2 A1 A2 b0 1
5 22 A1 b0 1
2 A1
22
8
1
2 T0 2 A2 b0 1 A1
3 2
2 A2 b0 A2 8
1
2 T0 2 A1 b0 1 A2
3 28
3
2 T0 2 b0 1 A1 A2
5 21
32 T0 2 b0 A2
2
Formal representation of the solution
Exercise.nb 7
sol2 q1,2 FunctionT0, T1, T2, Evaluatesolq21 ,q2,2 FunctionT0, T1, T2, Evaluatesolq22;
sol2 . displayRule
q1,2 FunctionT0, T1, T2, 1
32 T0 1 b0 A1
2 3 T0 1 b0 2 A1 A2
4 14 T0 1 b0 2
2 A22
15 12
2 b0 A1 A1 1
32 T0 1 b0 A1
22 b0 2
2 A2 A2
12
3 T0 1 b0 2 A1 A2
4 14 T0 1 b0 2
2 A22
15 12
,q2,2 FunctionT0, T1, T2, 8
3
2 T0 2 b0 1 A1 A2
5 21
32 T0 2 b0 A2
2
2 b0 12 A1 A1
22
8
1
2 T0 2 b0 1 A2 A1
3 28
1
2 T0 2 b0 1 A1 A2
3 2
2 b0 A2 A2 8
3
2 T0 2 b0 1 A1 A2
5 21
32 T0 2 b0 A2
2Third-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 A12 b02 13 A1 180 A12 b1 14 A1 448 A1 A2 b02 12 2 A2 90 A1 A2 b02 1 22 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
Exercise.nb 8
Terms of type ei1 T0 in the second equation of the Third-Order Problem and representation
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 1
2 2 A1 40 A12 b0
2 13 2 A1 224 A1 A2 b0
2 12 2
2 A2 45 A1 A2 b02 1 2
3 A2Terms 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 22 64 A1 A2 b02 13 2 A1 45 A1 A2 b02 12 22 A1 90 c A2
2 1 2 A2 40 A22 b0
2 1 23 A2 16 A2
2 b02 2
4 A2 90 A22 b2 1 2
4 A2Scalar 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 D1A2 b0 12 A1 80 A12 b02 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 b02 13 2 A1 45 A1 A2 b02 12 22 A1 90 c A2
2 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
Exercise.nb 9
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
dA1
dt ame1
ame1 A11,0T1, T2 A10,1T1, T2 .JoinSCond1Rule1, SCond2Rule1 . omgRule1; ame1 . displayRule
A1
2 2 A1
8 1 A2 b0 A1
3 c A12 A1
2 1
2
3 A1
2 b02 1 A1
3
2A12 b1 1
2 A1 A2 b0 2 A1 67
15 A1 A2 b0
2 1 A2
dA1
dt ame2
ame2 A21,0T1, T2 A20,1T1, T2 .JoinSCond1Rule1, SCond2Rule1 . omgRule1; ame2 . displayRule
A2
2 A2
3
16 A1
2 b0 1
32 A1
2 b0 1
16 A1
2 b0 2 A2
16 1
A12 b0 1
2
2 261
30 A1 A2 b0
2 1 A1 3 c A2
2 A2
4 112
5 A2
2 b02 1 A2 6 A2
2 b2 12 A2
Better representation
Exercise.nb 10
Collectame1, A1T1, T2, A1T1, T22 A1T1, T2,A1T1, T2 A2T1, T2 A2T1, T2, A2T1, T2 A1T1, T2 . displayRule
A1
2 2
8 1 A1
23 c
2 12
3 b0
2 1 3
2b1 1
2 A1 A2 b0 b0 2 A1 67
15 A1 A2 b0
2 1 A2
Collectame2, A2T1, T2, A1T1, T22, A1T1, T22 A1T1, T2,A1T1, T2 A2T1, T2 A2T1, T2, A2T1, T22 A2T1, T2 . displayRule
A12
3 b0
16 b0
32
1
16 b0
b0 12
2 2
A2
2
2
16 161
30 A1 b0
2 1 A1 A22
3 c
4 112
5 b0
2 1 6 b2 12 A2
Polar form of the AME
Utility
traspRule A1T1, T2 A1t, A1T1, T2 A1t, A2T1, T2 A2t, A2T1, T2 A2tA1T1, T2 A1t, A1T1, T2 A1t, A2T1, T2 A2t, A2T1, T2 A2t
Rewriting of the AME
amem1 ame1 . traspRule A1't1
2 A1t 2 A1t
8 13 c A1t2 A1t
2 12
3 b0
2 1 A1t2 A1t 3
2b1 1
2 A1t2 A1t b0 A2t A1t b0 2 A2t A1t 67
15 b0
2 1 A1t A2t A2t A1tamem2 ame2 . traspRule A2't
3
16 b0 A1t2 1
32 b0 A1t2 1
16 b0 A1t2 b0 1
2 A1t22 2
1
2 A2t A2t 2 A2t
16 161
30 b0
2 1 A1t A2t A1t 3 c A2t2 A2t
4 112
5 b0
2 1 A2t2 A2t 6 b2 12 A2t2 A2t A2tPolar form of the complex amplitudes
Exercise.nb 11
polRule A1t 1
2a1t 1t,
A2t 1
2a2t 2t, A1t
1
2a1t 1t, A2t
1
2a2t 2t
A1t 1
2 1t a1t, A2t
1
2 2t a2t,
A1t 1
2 1t a1t, A2t
1
2 2t a2t
polRuleD JoinpolRule, DpolRule, tA1t 1
2 1t a1t, A2t
1
2 2t a2t, A1t
1
2 1t a1t,
A2t 1
2 2t a2t, A1
t 1
2 1t a1t 1
2 1t a1t 1t,
A2t
1
2 2t a2t 1
2 2t a2t 2t,
A1t
1
2 1t a1t 1
2 1t a1t 1t,
A2t
1
2 2t a2t 1
2 2t a2t 2t
Substitution of the complex amplitudes with the polar form
eq1 amem1 . polRuleD
1
4 1t a1t 1
4 1t 2t a1t a2t b0 1t 2 a1t
16 13 c 1t a1t3
16 1
1
12 1t a1t3 b02 1 67
120 1t a1t a2t2 b02 1 3
16 1t a1t3 b1 12
1
4 1t 2t a1t a2t b0 2 1
2 1t a1t 1
2 1t a1t 1t
eq2 amem2 . polRuleD
1
4 2t a2t 1
2 2t a2t 3
642 1t a1t2 b0
1
1282 1t a1t2 b0 1
64 2 1t a1t2 b0 2t 2 a2t
32 1
3 c 2t a2t332 1
61
240 2t a1t2 a2t b02 1 3
10 2t a2t3 b02 1
3
4 2t a2t3 b2 12 2 1t a1t2 b0 12
8 21
2 2t a2t 1
2 2t a2t 2t
Autonomous equations and separations of the real and imaginary parts
Exercise.nb 12
eqa1 ComplexExpandExpandeq1 1t1
4 a1t 1
4 a1t a2t Cos2 1t 2t b0
3
16a1t3 b1 12 1
4a1t a2t Sin2 1t 2t b0 2 a1t
2
1
4 a1t a2t Sin2 1t 2t b0 2 a1t
16 13 c a1t3
16 1
1
12a1t3 b02 1
67
120a1t a2t2 b02 1 1
4a1t a2t Cos2 1t 2t b0 2 1
2a1t 1t
eqa2 ComplexExpandExpandeq2 2t1
4 a2t 3
64 a1t2 Cos2 1t 2t b0
1
128 a1t2 Cos2 1t 2t b0 1
64 a1t2 Sin2 1t 2t b0
3
4a2t3 b2 12 a1t2 Sin2 1t 2t b0 12
8 2a2t
2
1
2 a2t 1
64 a1t2 Cos2 1t 2t b0 3
64 a1t2 Sin2 1t 2t b0
1
128 a1t2 Sin2 1t 2t b0 2 a2t
32 13 c a2t3
32 1
61
240a1t2 a2t b02 1
3
10a2t3 b02 1 a1t2 Cos2 1t 2t b0 12
8 21
2a2t 2t
Definition of
phiRule 2 1t 2t t2 1t 2t tphiRuleD 2't 2 1't 't2t 2 1t t
Polar Amplitude Modulation Equations
amep1 CollectComplexExpand2 eqa1 . 0, a1t, a2t3
8a1t3 b1 12
a1t
2 a2t
1
2 Cos2 1t 2t b0 1
2Sin2 1t 2t b0 2 a1t
Exercise.nb 13
amep2 CollectComplexExpand2 eqa2 . 0, a1t, a2t1
2 a2t 3
2a2t3 b2 12
a1t2 3
32 Cos2 1t 2t b0 1
64 Cos2 1t 2t b0
1
32 Sin2 1t 2t b0 Sin2 1t 2t b0 12
4 2 a2t
amep3 CollectComplexExpand 2 eqa1 . 0, a1t, a2ta1t3 3 c
8 11
6b02 1 a1t
2
8 167
60a2t2 b02 1
a2t 1
2 Sin2 1t 2t b0 1
2Cos2 1t 2t b0 2 1t
amep4 CollectComplexExpand 2 eqa2 . 0, a1t, a2ta2t3 3 c
16 13
5b02 1 a1t2
1
32 Cos2 1t 2t b0 3
32 Sin2 1t 2t b0 1
64 Sin2 1t 2t b0
61
120a2t b02 1 Cos2 1t 2t b0 12
4 2 a2t
2
16 1 2t
Reduced Amplitude Modulation Equations
rame1 Collectamep1 . phiRule, a1t, a2t, t3
8a1t3 b1 12 a1t
2 a2t
1
2 Cost b0 1
2Sint b0 2 a1t
rame2 Collectamep2 . phiRule, a1t, a2t, t1
2 a2t 3
2a2t3 b2 12
a1t2 3
32 Cost b0 1
64 Cost b0 1
32 Sint b0 Sint b0 12
4 2 a2t
Exercise.nb 14
rame3 Expand2 a2t amep3 a1t amep4 . phiRule . phiRuleD a1t a2t 1
32 a1t3 Cost b0 3
32 a1t3 Sint b0
1
64 a1t3 Sint b0 a1t a2t2 Sint b0 2 a1t a2t
4 12 a1t a2t
16 1
3 c a1t3 a2t4 1
3 c a1t a2t3
16 1
7
40a1t3 a2t b02 1 49
30a1t a2t3 b02 1
a1t3 Cost b0 124 2
a1t a2t2 Cost b0 2 a1t a2t tfixRule a1t a1, a2t a2, t a1t a1, a2t a2, t
Equations to find Fixed Points
fix1 CollectExpandSimplifyrame1 . a1't 0, a2't 0, 't 0,a1t, a2t, a1t a2t, a1t3, a2t3, a1t2 a2t, a1t a2t2 . fixRule
a1
23
8a13 b1 1
2 a1 a2 1
2 Cos b0 1
2Sin b0 2
fix2
CollectExpandSimplifyrame2 . a1't 0, a2't 0, 't 0, a1t, a2t,a1t3, a2t3, a1t2 a2t, a1t a2t2, b0, Cost, Sint . fixRule
a2
23
2a23 b2 1
2 a12 b0 3
32
64Cos Sin
32
12
4 2
fix3 CollectExpandSimplifyrame3 . a1't 0, a2't 0, 't 0,a1t, a2t, a1t3, a2t3, a1t2 a2t, a1t a2t2,a1t3 a2t, a1t a2t3, b0, Cost, Sint . fixRule
a1 a2 2
4 1
2
16 1 a1 a23
3 c
16 149
30b02 1 a13 a2
3 c
4 1
7
40b02 1
a13 b03
32
64Sin Cos
32
12
4 2 a1 a22 b0 Sin Cos 2
Equilibrium paths in the case a1 0
ep1 SimplifySolvefix2 . a1 0 0, a2, 1 0a2 0, a2
3 b2 1
, a2
3 b2 1
Reconstitution of the solution
Exercise.nb 15
qr1t_ ComplexExpandA1T1, T2 1 t A1T1, T2 1 t solq21 . traspRule . polRule . T0 ta1t Cost 1 1t 1
2a1t2 b0 1
6a1t2 Cos2 t 1 2 1t b0
a1t a2t Cos3 t 1 1t 2t b0 28 1
a2t2 b0 22
2 12
a2t2 Cos4 t 1 2 2t b0 22
30 12
qr2t_ ComplexExpandA2T1, T2 2 t A2T1, T2 2 t solq22 . traspRule . polRule . T0 ta2t Cost 2 2t 1
2a2t2 b0 1
6a2t2 Cos2 t 2 2 2t b0 a1t2 b0 12
2 22
4 a1t a2t Cos t 22
1t 2t b0 13 2
4 a1t a2t Cos 3 t 2
2 1t 2t b0 1
5 2
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 500;
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, tia1t InterpolatingFunction0., 500., t,
a2t InterpolatingFunction0., 500., t,t InterpolatingFunction0., 500., t
Exercise.nb 16
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"
0 100 200 300 400 500
0.5
0.0
0.5
t
a1t
0 100 200 300 400 5000.6
0.4
0.2
0.0
0.2
0.4
ta2t
0 100 200 300 400 5000.0
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., 500., t,a2t InterpolatingFunction0., 500., t,1t InterpolatingFunction0., 500., t,2t InterpolatingFunction0., 500., 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 100200300400500
0.5
0.0
0.5
t
a1t
0 1002003004005000.60.40.20.00.20.4
t
a2t
0 10020030040050001234
t
1t
0 1002003004005000123456
t
2t
Graphics of the reconstituted solution
Exercise.nb 17
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"
0 100 200 300 400 500
0.5
0.0
0.5
t
q1t
0 100 200 300 400 5000.6
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., 500., t,q2t InterpolatingFunction0., 500., 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"
0 100 200 300 400 500
0.5
0.0
0.5
t
q1t
0 100 200 300 400 5000.6
0.4
0.2
0.0
0.2
0.4
t
q2t
Exercise.nb 18