Exercise: Multiple Bifurcations of Sample Dynamical Systems
- 1. Exercise.nb 1
Double Rayleigh-Duffing oscillator in nearly 1:2 resonance. MSM.
..
q1 q1 2 q1 b0
q2 q1 2 b1 q3 c q3 0
1 1 1
..
q q 2 q b q 2 b q3 c q3 0
q
2 2 2 2 0 2 1 2 2 2
2 2 1
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;
- 2. Exercise.nb 2
displayRule q_i_,j_ a__ __ RowTimes MapIndexedD1
211 &, a, qi,j ,
A_i_ a__ __ RowTimes MapIndexedD1
21 &, a, Ai ,
q_i_,j_ __ qi,j , A_i_ __ Ai ;
Equations of motions and resonance conditions
Equations of motion
EOM 1 2 Subscriptq, 1t c Subscriptq, 1t3
b1 Subscriptq, 1 't3 b0 Subscriptq, 2 't Subscriptq, 1 't2
Subscriptq, 1 't Subscriptq, 1 ''t 0,
2
2 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
2 q1 t c q1 t3 q1 t b1 q1 t3 b0 q1 t q2 t2 q1 t 0
2 q2 t c q2 t3 q2 t b2 q2 t3 b0 q1 t q2 t2 q2 t 0
1
2
Ordering of the dampings
smorzrule , ;
Definition of 2 (introduction of the detuning)
ombrule 2 2
2 2
Definition of the expansion of qi
solRule qi_ Sumj1 qi,j 1, 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
- 3. Exercise.nb 3
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 ''t
q1 t q1 t, q2 t q2 t, q1 t q1 t,
q2 t q2 t, q1 t q1 t, q2 t q2 t
Modification of the equations of motion: substitution of the rules. Representation.
EOMa EOM . scaling . multiScales . smorzrule . ombrule . solRule TrigToExp
ExpandAll . n_;n3 0; EOMa . displayRule
2 D0 q1,1 3 D0 q1,2 D2 q1,1 2 D2 q1,2 3 D2 q1,3 3 D1 q1,1
2 2 D0 D1 q1,1 2 3 D0 D1 q1,2 3 D2 q1,1 2 3 D0 D2 q1,1 2 D0 q1,1 2 b0
0 0 0
2 3 D0 q1,1 D0 q1,2 b0 2 2 D0 q1,1 D0 q2,1 b0 2 3 D0 q1,2 D0 q2,1 b0
1
2 D0 q2,1 2 b0 2 3 D0 q1,1 D0 q2,2 b0 2 3 D0 q2,1 D0 q2,2 b0
2 3 D0 q1,1 D1 q1,1 b0 2 3 D0 q2,1 D1 q1,1 b0 2 3 D0 q1,1 D1 q2,1 b0
2 3 D0 q2,1 D1 q2,1 b0 3 D0 q1,1 3 b1 2 q1,1 c 3 q3 2 2 q1,2 3 2 q1,3 0,
2 D0 q2,1 3 D0 q2,2 D2 q2,1 2 D2 q2,2 3 D2 q2,3 3 D1 q2,1
1 1,1 1 1
2 2 D0 D1 q2,1 2 3 D0 D1 q2,2 3 D2 q2,1 2 3 D0 D2 q2,1 2 D0 q1,1 2 b0
0 0 0
2 3 D0 q1,1 D0 q1,2 b0 2 2 D0 q1,1 D0 q2,1 b0 2 3 D0 q1,2 D0 q2,1 b0 2 D0 q2,1 2 b0
1
2 3 D0 q1,1 D0 q2,2 b0 2 3 D0 q2,1 D0 q2,2 b0 2 3 D0 q1,1 D1 q1,1 b0
2 3 D0 q2,1 D1 q1,1 b0 2 3 D0 q1,1 D1 q2,1 b0 2 3 D0 q2,1 D1 q2,1 b0 3 D0 q2,1 3 b2
3 2 q2,1 2 2 2 q2,1 2 q2,1 c 3 q3 2 3 2 q2,2 2 2 q2,2 3 2 q2,3 0
2 2,1 2 2
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
- 4. Exercise.nb 4
eqOrder1 . displayRule
eqOrder2 . displayRule
eqOrder3 . displayRule
D2 q1,1 2 q1,1 0, D2 q2,1 2 q2,1 0
0 1 0 2
D2 q1,2 2 q1,2 D0 q1,1 2 D0 D1 q1,1 D0 q1,1 2 b0 2 D0 q1,1 D0 q2,1 b0 D0 q2,1 2 b0 ,
0 1
D2 q2,2 2 q2,2
D0 q2,1 2 D0 D1 q2,1 D0 q1,1 2 b0 2 D0 q1,1 D0 q2,1 b0 D0 q2,1 2 b0 2 2 q2,1
0 2
D2 q1,3 2 q1,3
D0 q1,2 D1 q1,1 2 D0 D1 q1,2 D2 q1,1 2 D0 D2 q1,1 2 D0 q1,1 D0 q1,2 b0
0 1
2 D0 q1,2 D0 q2,1 b0 2 D0 q1,1 D0 q2,2 b0 2 D0 q2,1 D0 q2,2 b0 2 D0 q1,1 D1 q1,1 b0
1
2 D0 q2,1 D1 q1,1 b0 2 D0 q1,1 D1 q2,1 b0 2 D0 q2,1 D1 q2,1 b0 D0 q1,1 3 b1 c q3 ,
D2 q2,3 2 q2,3 D0 q2,2 D1 q2,1 2 D0 D1 q2,2 D2 q2,1 2 D0 D2 q2,1
1,1
2 D0 q1,1 D0 q1,2 b0 2 D0 q1,2 D0 q2,1 b0 2 D0 q1,1 D0 q2,2 b0
0 2 1
2 D0 q2,1 D0 q2,2 b0 2 D0 q1,1 D1 q1,1 b0 2 D0 q2,1 D1 q1,1 b0
2 D0 q1,1 D1 q2,1 b0 2 D0 q2,1 D1 q2,1 b0 D0 q2,1 3 b2 2 q2,1 c q3 2 2 q2,2
2,1
Resonance condition
ResonanceCond 1 2 ;
1
2
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
D2 q1,1 2 q1,1
0 1
D2 q2,1 2 q2,1
0 2
- 5. Exercise.nb 5
Formal solution of the First-Order Problem
sol1 q1,1 FunctionT0 , T1 , T2 , A1 T1 , T2 ExpI 1 T0 A 1 T1 , T2 Exp I 1 T0 ,
q2,1 FunctionT0 , T1 , T2 , A2 T1 , T2 ExpI 2 T0 A 2 T1 , T2 Exp I 2 T0
q1,1 FunctionT0 , T1 , T2 , A1 T1 , T2 1 T0 A1 T1 , T2 1 T0 ,
q2,1 FunctionT0 , T1 , T2 , A2 T1 , T2 2 T0 A2 T1 , T2 2 T0
Second-Order Problem
Substitution of the solution on the Second-Order Problem and representation
order2Eq eqOrder2 . sol1 ExpandAll;
order2Eq . displayRule
D2 q1,2 2 q1,2 2 T0 1 D1 A1 1 2 T0 1 D1 A1 1
0 1
T0 1 A1 1 2 T0 1 A2 b0 2 2 T0 1 T0 2 A1 A2 b0 1 2 2 T0 2 A2 b0 2
1 1 2 2
2
T0 1 1 A1 2 A1 b0 2 A1 2 T0 1 T0 2 A2 b0 1 2 A1 2 T0 1 b0 2 A1
1 1
2
2 T0 1 T0 2 A1 b0 1 2 A2 2 A2 b0 2 A2 2 T0 1 T0 2 b0 1 2 A1 A2 2 T0 2 b0 2 A2 ,
D2 q2,2 2 q2,2 2 T0 1 A2 b0 2 2 T0 2 D1 A2 2 2 T0 2 D1 A2 2 T0 2 A2 2
2 2
0 2 1 1
2 T0 2 A2 2 2 T0 1 T0 2 A1 A2 b0 1 2 2 T0 2 A2 b0 2 2 A1 b0 2 A1
2 2 1
2
2 T0 1 T0 2 A2 b0 1 2 A1 2 T0 1 b0 2 A1 T0 2 2 A2 2 T0 2 2 A2
2 T0 1 T0 2 A1 b0 1 2 A2 2 A2 b0 2 A2 2 T0 1 T0 2 b0 1 2 A1 A2 2 T0 2 b0 2 A2
1
2
2 2
Utility (don't modify)
expRule1i_ : Expa_ : ExpExpanda . omgRulei . T0 T1
Terms of type ei 1 T0 in the first equation of the Second-Order Problem and representation
ST11 Coefficientorder2Eq , 2 . expRule11, ExpI 1 T0 & 1;
ST11 . displayRule
2 D1 A1 1 A1 1 2 A2 b0 1 2 A1
Terms of type ei 2 T0 in the first equation of the Second-Order Problem and representation
ST12 Coefficientorder2Eq , 2 . expRule12, ExpI 2 T0 & 1;
ST12 . displayRule
A2 b0 2
1 1
Terms of type ei 1 T0 in the second equation of the Second-Order Problem and representation
- 6. Exercise.nb 6
ST21 Coefficientorder2Eq , 2 . expRule11, ExpI 1 T0 & 2;
ST21 . displayRule
2 A2 b0 1 2 A1
Terms of type ei 2 T0 in the second equation of the Second-Order Problem and representation
ST22 Coefficientorder2Eq , 2 . expRule12, ExpI 2 T0 & 2;
ST22 . displayRule
A2 b0 2 2 D1 A2 2 A2 2 2 A2 2
1 1
Scalar product with the left eigenvectors: First-Order AME; representation
SCond1 1, 0.ST11, ST21 0, 0, 1.ST12, ST22 0;
SCond1 . displayRule
2 D1 A1 1 A1 1 2 A2 b0 1 2 A1 0, A2 b0 2 2 D1 A2 2 A2 2 2 A2 2 0
1 1
Algebraic manioulation to obtain D1 A1 and D1 A2
SCond1Rule1 SolveSCond1, A1 T1 , T2 , A2 T1 , T2 1 ExpandAll;
1,0 1,0
SCond1Rule1 . displayRule TableForm
A1
D1 A1 A2 b0 2 A1
2
A2 A1 b0 2
2
D1 A2 A2 1
2 2 2
Substitution of the First - Order AME to the Second Order Equations
order2Eqm
order2Eq . SCond1Rule1 . SCond1Rule1 . conjugateRule . expRule1 &
1, 2 ExpandAll; order2Eqm . displayRule
D2 q1,2 2 q1,2 2 T0 1 A2 b0 2 2 3 T0 1 A1 A2 b0 1 2 4 T0 1 A2 b0 2
0 1 1 1 2 2
2 2
2 A1 b0 2 A1 2 T0 1 b0 2 A1 2 A2 b0 2 A2 2 3 T0 1 b0 1 2 A1 A2 4 T0 1 b0 2 A2 ,
1 1 2 2
3 1
T0 2 T0 2
D2 q2,2 2 q2,2 2 2
0 2 A1 A2 b0 1 2 2 T0 2 A2 b0 2 2 A1 b0 2 A1 2 2
2 2 1 A2 b0 1 2 A1
b0 1 2 A1 A2 2 T0 2 b0 2 A2
1 3
T0 2 T0 2 2
2 2 A1 b0 1 2 A2 2 A2 b0 2 A2 2
2
2
2
Assumpion on the coefficients
$Assumptions 1 , 2 , b0 , c, , , b1 , b2 Reals
1 2 b0 c b1 b2 Reals
Construction of the particular solution of the first equation of the Second-Order Problem and representation
- 7. Exercise.nb 7
solOrder2Eqm1
SimplifyTableq1,2 T0 , T1 , T2 . TrigToExpFlattenDSolveorder2Eqm1, 1
order2Eqm1, 2, i, q1,2 T0 , T1 , T2 , T0 . C1 0, C2 0,
i, 1, Lengthorder2Eqm1, 2; solOrder2Eqm1 . displayRule
1 3 T0 1 A1 A2 b0 2 4 T0 1 A2 b0 2
2 2
2 T0 1 A2 b0 ,
1 , , 2 A1 b0 A1 ,
3 4 1 15 2
1
2
1 2 2 A2 b0 2 A2
2 3 T0 1 b0 2 A1 A2 4 T0 1 b0 2 A2
2
2 T0 1
b0 A1 , , ,
3 2
1
4 1 15 2
1
solq21 ExpandTotalsolOrder2Eqm1;
solq21 . displayRule
1 3 T0 1 A1 A2 b0 2 4 T0 1 A2 b0 2
2 2
2 T0 1 A2 b0
1 2 A1 b0 A1
3 4 1 15 2
1
2
1 2 2 A2 b0 2 A2
2 3 T0 1 b0 2 A1 A2 4 T0 1 b0 2 A2
2
2 T0 1 b0 A1
3 2
1
4 1 15 2
1
Construction of the particular solution of the second equation of the Second-Order Problem
solOrder2Eqm2 Tableq2,2 T0 , T1 , T2 .
SimplifyTrigToExpFlattenDSolveorder2Eqm2, 1 order2Eqm2, 2, i,
q2,2 T0 , T1 , T2 , T0 . C1 0, C2 0,
i, 1, Lengthorder2Eqm2, 2; solOrder2Eqm2 . displayRule
3 1
T0 2 T0 2
8 2 A1 A2 b0 1 1 2 A1 b0 2 A1
1 8 2 A2 b0 1 A1
2 T0 2
, A2
2 b0 , , ,
5 2 3 2
2
3 2
2 T0 2 b0 A2
1 3
T0 2 T0 2
8 2 A1 b0 1 A2 8 2 b0 1 A1 A2 1 2
, 2 A2 b0 A2 , ,
3 2 5 2 3
solq22 ExpandTotalsolOrder2Eqm2;
solq22 . displayRule
3 1
T0 2 T0 2
1 8 2 A1 A2 b0 1 2 A1 b0 2 A1
1 8 2 A2 b0 1 A1
2 T0 2
A2
2 b0
3 5 2 2
2
3 2
1 3
T0 2 T0 2
8 2 A1 b0 1 A2 8 2 b0 1 A1 A2 1 2
2 A2 b0 A2 2 T0 2 b0 A2
3 2 5 2 3
Formal representation of the solution
- 8. Exercise.nb 8
sol2 q1,2 FunctionT0 , T1 , T2 , Evaluatesolq21 ,
q2,2 FunctionT0 , T1 , T2 , Evaluatesolq22;
sol2 . displayRule
q1,2 FunctionT0 , T1 , T2 ,
1 3 T0 1 b0 2 A1 A2 4 T0 1 b0 2 A2
2 2
2 T0 1 b0 A2
1
3 4 1 15 2
1
,
2
1 2 2 b0 2 A2 A2
2 3 T0 1 b0 2 A1 A2 4 T0 1 b0 2 A2
2
2 b0 A1 A1 2 T0 1 b0 A1
3 2
1
4 1 15 2
1
q2,2 FunctionT0 , T1 , T2 ,
3
T0 2
8 2 b0 1 A1 A2 1
2 T0 2 b0 A2
2
5 2 3
1 1
T0 2 T0 2
2 b0 2 A1 A1
1 8 2 b0 1 A2 A1 8 2 b0 1 A1 A2
2
2
3 2 3 2
2 T0 2 b0 A2
3
T0 2
8 2 b0 1 A1 A2 1 2
2 b0 A2 A2
5 2 3
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 ei 1 T0 in the first equation of the Third-Order Problem and representation
ST311 Coefficientorder3Eqpr , 2 . expRule1 , ExpI 1 T0 & 1;
ST311 . displayRule
1
60 1
60 D1 A1 1 60 D2 A1 1 120 D2 A1 2 120 D1 A1 A2 b0 1 2 180 c A2 1 A1 120
D1 A2 b0 2 A1 80 A2 b2 3 A1 180 A2 b1 4 A1 448 A1 A2 b2 2 2 A2 90 A1 A2 b2 1 2 A2
1 1 1
1 1 0 1 1 1 0 1 0 2
Terms of type ei 2 T0 in the first equation of the Third-Order Problem and representation
ST312 Coefficientorder3Eqpr , 2 . expRule1 , ExpI 2 T0 & 1;
ST312 . displayRule
0
- 9. Exercise.nb 9
Terms of type ei 1 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 2
80 D1 A1 A2 b0 2 2 60 D1 A1 A2 b0 1 2 140 D1 A2 b0 2 2 A1 40 A2 b0 2 2 A1
160 A2 b0 2 2 A1 40 A2 b2 3 2 A1 224 A1 A2 b2 2 2 A2 45 A1 A2 b2 1 3 A2
1 2 1 1
1 1 0 1 0 1 2 0 2
Terms of type ei 2 T0 in the second equation of the Third-Order Problem and representation
ST322 Coefficientorder3Eqpr , 2 . expRule12, ExpI 2 T0 & 2;
ST322 . displayRule
30 D1 A2 1 2 30 D2 A2 1 2 30 2 A2 1 2
1
1
30 1 2
60 D1 A1 A1 b0 2 2 60 D2 A2 1 2 64 A1 A2 b2 3 2 A1 45 A1 A2 b2 2 2 A1
90 c A2 1 2 A2 40 A2 b2 1 3 A2 16 A2 b2 4 A2 90 A2 b2 1 4 A2
1 2 0 1 0 1 2
2 2 0 2 2 0 2 2 2
Scalar product with the left eigenvectors: Second-Order AME; representation
SCond2 1, 0.ST311, ST321 0, 0, 1.ST312, ST322 0;
SCond2 . displayRule
60 D1 A1 1 60 D2 A1 1 120 D2 A1 2
1
1 1
60 1
120 D1 A1 A2 b0 1 2 180 c A2 1 A1 120 D1 A2 b0 2 A1 80 A2 b2 3 A1
180 A2 b1 4 A1 448 A1 A2 b2 2 2 A2 90 A1 A2 b2 1 2 A2 0,
1 1 1 0 1
1 1 0 1 0 2
30 D1 A2 1 2 30 D2 A2 1 2 30 2 A2 1 2 60 D1 A1 A1 b0 2 2
1
1 1
30 1 2
60 D2 A2 1 2 64 A1 A2 b2 3 2 A1 45 A1 A2 b2 2 2 A1
90 c A2 1 2 A2 40 A2 b2 1 3 A2 16 A2 b2 4 A2 90 A2 b2 1 4 A2 0
2 0 1 0 1 2
2 2 0 2 2 0 2 2 2
Algebraic manipulation to obtain D2 A1 and D2 A2
- 10. Exercise.nb 10
SCond2Rule1
SolveSCond2, A1 T1 , T2 , A2 T1 , T2 1 . A1 2,0 T1 , T2 T1 SCond1Rule1
0,1 0,1
1, 2 . A2 2,0 T1 , T2 T1 SCond1Rule12, 2 . SCond1Rule1 .
SCond1Rule1 . conjugateRule ExpandAll Simplify Expand;
SCond2Rule1 . displayRule
D2 A1
2 A1 1 3 c A2 A1
1 5 3 A2 b2 2 A1
1 0 1
A2 b0 A1 A2 b0 A1 A2 b2 1 A1
1 0 A2 b1 2 A1
1 1
8 1 2 2 1 12 2 2 2
A2 b0 2 A1 A2 b0 2 A1 A2 b0 2 A1 56 3 A1 A2 b2 2 A2
0 2
A1 A2 b2 2 A2
0 ,
2 1 4 1 2 1 15 4 1
A2 b0 2
1 1 A2 b0 2
1 1 A2 b0 2
1 1 2 A2 A2 b0 1
1 7
D2 A2 A1 A2 b2 1 A1
0
4 2
2 8 2
2 4 2
2
8 2 2 2 4
17 A1 A2 b2 2 A1
0 1 3 c A2 A2
2 2 3 4 A2 b2 2 A2
2 0 2
A2 b2 2 A2
2 0 A2 b2 2 A2
2 2
30 2 2 2 3 2 15 1
Reconstitution of the AME and of the solution
dA1
dt
ame1
ame1 A1 1,0 T1 , T2 A1 0,1 T1 , T2 .
JoinSCond1Rule1, SCond2Rule1 . omgRule1; ame1 . displayRule
A1 2 A1 3 c A2 A1
1
A2 b0 A1
2 8 1 2 1
2 3 67
A2 b2 1 A1
1 0 A2 b1 2 A1 A2 b0 2 A1
1 1 A1 A2 b2 1 A2
0
3 2 15
dA1
dt
ame2
ame2 A2 1,0 T1 , T2 A2 0,1 T1 , T2 .
JoinSCond1Rule1, SCond2Rule1 . omgRule1; ame2 . displayRule
A2 3 1 1 2 A2
A2 A2 b0
1 A2 b0
1 A2 b0
1
2 16 32 16 16 1
A2 b0 2
1 1 61 3 c A2 A2
2 12
A1 A2 b2 1 A1
0 A2 b2 1 A2 6 A2 b2 2 A2
2 0 2 1
2 2 30 4 1 5
Better representation
- 11. Exercise.nb 11
Collectame1, A1 T1 , T2 , A1 T1 , T2 2 A 1 T1 , T2 ,
A1 T1 , T2 A2 T1 , T2 A 2 T1 , T2 , A2 T1 , T2 A 1 T1 , T2 . displayRule
b1 2 A1 A2 b0 b0 2 A1
2 3c 2 3 67
A1 A2
1 b2 1
0 1 A1 A2 b2 1 A2
0
2 8 1 2 1 3 2 15
Collectame2, A2 T1 , T2 , A1 T1 , T2 2 , A1 T1 , T2 2 A 1 T1 , T2 ,
A1 T1 , T2 A2 T1 , T2 A 2 T1 , T2 , A2 T1 , T2 2 A 2 T1 , T2 . displayRule
3 b0 b0 1 b0 2
1
A2
1 b0
16 32 16 2 2
2 61 3c 12
A2 A1 b2 1 A1 A2
0 2 b2 1 6 b2 2 A2
0 1
2 16 1 30 4 1 5
Polar form of the AME
Utility
traspRule
A1 T1 , T2 A1 t, A 1 T1 , T2 A 1 t, A2 T1 , T2 A2 t, A 2 T1 , T2 A 2 t
A1 T1 , T2 A1 t, A1 T1 , T2 A1 t, A2 T1 , T2 A2 t, A2 T1 , T2 A2 t
Rewriting of the AME
amem1 ame1 . traspRule A1 't
2 A1 t 3 c A1 t2 A1 t
A1 t b2 1 A1 t2 A1 t b1 2 A1 t2 A1 t
1 2 3
0 1
2 8 1 2 1 3 2
b0 A2 t A1 t b0 2 A2 t A1 t b2 1 A1 t A2 t A2 t A1 t
67
0
15
amem2 ame2 . traspRule A2 't
b0 2 A1 t2
b0 A1 t2 b0 A1 t2 b0 A1 t2
3 1 1 1
16 32 16 2 2
2 A2 t
A2 t A2 t b2 1 A1 t A2 t A1 t
1 61
0
2 16 1 30
3 c A2 t2 A2 t
b2 1 A2 t2 A2 t 6 b2 2 A2 t2 A2 t A2 t
12
0 1
4 1 5
Polar form of the complex amplitudes
- 12. Exercise.nb 12
polRule A1 t
1
a1t 1t ,
2
A2 t a2t 2t , A 1 t a1t 1t , A 2 t a2t 2t
1 1 1
2 2 2
A1 t 1t a1t, A2 t
1 1
2t a2t,
2 2
A1 t a1t, A2 t
1 1
1t 2t a2t
2 2
polRuleD JoinpolRule, DpolRule, t
A1 t 1t a1t, A2 t 2t a2t, A1 t
1 1 1
1t a1t,
2 2 2
A2 t 2t a2t, A1 t 1t a1 t 1t a1t 1 t,
1 1 1
2 2 2
A2 t 2t a2 t 2t a2t 2 t,
1 1
2 2
A1 t a1 t 1t a1t 1 t,
1 1
1t
2 2
A2 t a2 t 2t a2t 2 t
1 1
2t
2 2
Substitution of the complex amplitudes with the polar form
eq1 amem1 . polRuleD
1 1 1t 2 a1t 3 c 1t a1t3
1t a1t 1t 2t a1t a2t b0
4 4 16 1 16 1
1 67 3
1t a1t3 b2 1
0 1t a1t a2t2 b2 1
1t a1t3 b1 2
0 1
12 120 16
1t 2t a1t a2t b0 2 1t a1 t 1t a1t 1 t
1 1 1
4 2 2
eq2 amem2 . polRuleD
1 1 3
2t a2t 2t a2t 2 1t a1t2 b0
4 2 64
1 1 2t 2 a2t
2 1t a1t2 b0 2 1t a1t2 b0
128 64 32 1
3 c 2t a2t3 61 3
2t a1t2 a2t b2 1
0 2t a2t3 b2 1
0
32 1 240 10
2t a2 t 2t a2t 2 t
3 2 1t a1t2 b0 2
1 1 1
2t a2t3 b2 2
1
4 8 2 2 2
Autonomous equations and separations of the real and imaginary parts
- 13. Exercise.nb 13
eqa1 ComplexExpandExpandeq1 1t
1 1
a1t a1t a2t Cos2 1t 2t b0
a1 t
4 4
3 1
a1t3 b1 2
1 a1t a2t Sin2 1t 2t b0 2
16 4 2
1 2 a1t 3 c a1t3 1
a1t a2t Sin2 1t 2t b0 a1t3 b2 1
0
4 16 1 16 1 12
a1t 1 t
67 1 1
a1t a2t2 b2 1
0 a1t a2t Cos2 1t 2t b0 2
120 4 2
eqa2 ComplexExpandExpandeq2 2t
1 3
a2t a1t2 Cos2 1t 2t b0
4 64
1 1
a1t2 Cos2 1t 2t b0 a1t2 Sin2 1t 2t b0
a2 t
128 64
3 a1t2 Sin2 1t 2t b0 2
1
a2t3 b2 2
1
4 8 2 2
1 1 3
a2t a1t2 Cos2 1t 2t b0 a1t2 Sin2 1t 2t b0
2 64 64
1 2 a2t 3 c a2t3 61
a1t2 Sin2 1t 2t b0 a1t2 a2t b2 1
0
128 32 1 32 1 240
a2t 2 t
3 a1t2 Cos2 1t 2t b0 2
1 1
a2t3 b2 1
0
10 8 2 2
Definition of
phiRule 2 1t 2t t
2 1t 2t t
phiRuleD 2 't 2 1 't 't
2 t 2 1 t t
Polar Amplitude Modulation Equations
amep1 CollectComplexExpand2 eqa1 . 0, a1t, a2t
3
a1t3 b1 2
1
8
a1 t
1 1
a1t a2t Cos2 1t 2t b0 Sin2 1t 2t b0 2
2 2 2
- 14. Exercise.nb 14
amep2 CollectComplexExpand2 eqa2 . 0, a1t, a2t
1 3
a2t a2t3 b2 2
1
2 2
3 1
a1t2 Cos2 1t 2t b0 Cos2 1t 2t b0
32 64
a2 t
1 Sin2 1t 2t b0 2
1
Sin2 1t 2t b0
32 4 2
amep3 CollectComplexExpand 2 eqa1 . 0, a1t, a2t
3c 1 2 67
a1t3 b2 1 a1t
0 a2t2 b2 1
0
8 1 6 8 1 60
Cos2 1t 2t b0 2 1 t
1 1
a2t Sin2 1t 2t b0
2 2
amep4 CollectComplexExpand 2 eqa2 . 0, a1t, a2t
3c 3
a2t3 b2 1 a1t2
0
16 1 5
1 3 1
Cos2 1t 2t b0 Sin2 1t 2t b0 Sin2 1t 2t b0
32 32 64
2 t
61 Cos2 1t 2t b0 2
1 2
a2t b2 1
0 a2t
120 4 2 16 1
Reduced Amplitude Modulation Equations
rame1 Collectamep1 . phiRule, a1t, a2t, t
a1 t
3 1 1
a1t3 b1 2 a1t
1 a2t Cost b0 Sint b0 2
8 2 2 2
rame2 Collectamep2 . phiRule, a1t, a2t, t
1 3
a2t a2t3 b2 2
1
2 2
a2 t
3 1 1 Sint b0 2
1
a1t2 Cost b0 Cost b0 Sint b0
32 64 32 4 2
- 15. Exercise.nb 15
rame3 Expand2 a2t amep3 a1t amep4 . phiRule . phiRuleD
1 3
a1t a2t a1t3 Cost b0 a1t3 Sint b0
32 32
1 2 a1t a2t 2 a1t a2t
a1t3 Sint b0 a1t a2t2 Sint b0
64 4 1 16 1
3 c a1t3 a2t 3 c a1t a2t3 7 49
a1t3 a2t b2 1
0 a1t a2t3 b2 1
0
4 1 16 1 40 30
a1t a2t2 Cost b0 2 a1t a2t t
a1t3 Cost b0 2
1
4 2
fixRule 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 3 1 1
a13 b1 2 a1 a2
1 Cos b0 Sin b0 2
2 8 2 2
fix2
CollectExpandSimplifyrame2 . a1 't 0, a2 't 0, 't 0, a1t, a2t,
a1t3 , a2t3 , a1t2 a2t, a1t a2t2 , b0 , Cost, Sint . fixRule
a2 3 3 2
1
a23 b2 2 a12 b0
1 Cos Sin
2 2 32 64 32 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
2 2 3c 49 3c 7
a1 a2 a1 a23 b2 1 a13 a2
0 b2 1
0
4 1 16 1 16 1 30 4 1 40
a1 a22 b0 Sin Cos 2
3 2
1
a13 b0 Sin Cos
32 64 32 4 2
Equilibrium paths in the case a1 0
ep1 SimplifySolvefix2 . a1 0 0, a2, 1 0
a2 0, a2 , a2
3 b2 1 3 b2 1
Reconstitution of the solution
- 16. Exercise.nb 16
qr1 t_ ComplexExpand
A1 T1 , T2 1 t A 1 T1 , T2 1 t solq21 . traspRule . polRule . T0 t
1 1
a1t Cost 1 1t a1t2 b0 a1t2 Cos2 t 1 2 1t b0
2 6
a1t a2t Cos3 t 1 1t 2t b0 2 a2t2 b0 2
2 a2t2 Cos4 t 1 2 2t b0 2
2
8 1 2 2
1 30 2
1
qr2 t_ ComplexExpand
A2 T1 , T2 2 t A 2 T1 , T2 2 t solq22 . traspRule . polRule . T0 t
1 1 a1t2 b0 2
1
a2t Cost 2 2t a2t2 b0 a2t2 Cos2 t 2 2 2t b0
2 6 2 2
2
t 2 3 t 2
4 a1t a2t Cos 1t 2t b0 1 4 a1t a2t Cos 1t 2t b0 1
2 2
3 2 5 2
Numerical integrations
Numerical values
1 1, 2 2, b0
1
, b1 1, b2 1, c 1, 0.05, 0.05, 0, 2 2 1
2
1, 2,
1
, 1, 1, 1, 0.05, 0.05, 0, 2
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, ti
a1t InterpolatingFunction0., 500., t,
a2t InterpolatingFunction0., 500., t,
t InterpolatingFunction0., 500., t
- 17. Exercise.nb 17
GraphicsArrayPlota1t . solrame1, t, 0, ti, PlotStyle Thick,
PlotRange Automatic, 0.8, 0.8, Frame True, FrameLabel "t", "a1 t",
Plota2t . solrame1, t, 0, ti, PlotStyle Thick,
PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "a2 t",
Plott . solrame1, t, 0, ti, PlotStyle Thick, Frame True,
AxesOrigin 0, 0, FrameLabel "t", "t"
0.4
0.5 2.0
0.2
1.5
a1 t
t
a2 t
0.0 0.0
0.2 1.0
0.5 0.4 0.5
0.6 0.0
0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500
t 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, ti
a1t 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", "a1 t",
Plota2t . solramep1, t, 0, ti, PlotStyle Thick,
PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "a2 t",
Plot1t . solramep1, t, 0, ti, PlotStyle Thick,
Frame True, AxesOrigin 0, 0, FrameLabel "t", "1 t",
Plot2t . solramep1, t, 0, ti, PlotStyle Thick, Frame True,
AxesOrigin 0, 0, FrameLabel "t", "2 t"
0.4 4 6
0.5 0.2 5
a1 t
1 t
2 t
3
a2 t
0.0 4
0.0 2 3
0.2 2
0.5 0.4 1 1
0.6 0 0
0 100200300400500 0 100 300 500
200 400 0 100200300400500 0 100200300400500
t t t t
Graphics of the reconstituted solution
- 18. Exercise.nb 18
GraphicsArrayPlotqr1 t . solramep1, t, 0, ti, PlotStyle Thick,
PlotRange Automatic, 0.8, 0.8, Frame True, FrameLabel "t", "q1 t",
Plotqr2 t . solramep1, t, 0, ti, PlotStyle Thick,
PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "q2 t"
0.4
0.5
0.2
q1 t
0.0
q2 t
0.0
0.2
0.5 0.4
0.6
0 100 200 300 400 500 0 100 200 300 400 500
t t
Numerical Intergations of the original equations
solorig1 NDSolveJoinEOM, q1 0 0.1, q2 0 0.1, q1 '0 0.1, q2 '0 0.1,
q1 t, q2 t, t, 0, ti, MaxSteps 1 000 000
q1 t InterpolatingFunction0., 500., t,
q2 t InterpolatingFunction0., 500., t
GraphicsArrayPlotq1 t . solorig1, t, 0, ti, PlotStyle Thick,
PlotRange Automatic, 0.8, 0.8, Frame True, FrameLabel "t", "q1 t",
Plotq2 t . solorig1, t, 0, ti, PlotStyle Thick,
PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "t", "q2 t"
0.4
0.5
0.2
0.0
q1 t
q2 t
0.0
0.2
0.5 0.4
0.6
0 100 200 300 400 500 0 100 200 300 400 500
t t