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)
OffGeneral::spell1
 Notation`
Time scales
SymbolizeT0 ; SymbolizeT1 ; SymbolizeT2 ;
timeScales  T0 , T1 , T2 ;
Differentiation (don't modify)
dt1expr_ : Sumi Dexpr, timeScalesi  1, i, 0, maxOrder;
dt2expr_ : dt1dt1expr  Expand . i_;imaxOrder  0;
Some rules (don' t modify)
conjugateRule  A  A, A  A,   ,   , Complex0, n_  Complex0,  n;

2. Exercise.nb 2
displayRule  q_i_,j_ a__ __  RowTimes  MapIndexedD1
211 &, a, qi,j ,
A_i_ a__ __  RowTimes  MapIndexedD1
21 &, a, Ai ,
q_i_,j_ __  qi,j , A_i_ __  Ai ;
Equations of motions and resonance conditions
Equations of motion
EOM  1 2 Subscriptq, 1t  c Subscriptq, 1t3 
b1 Subscriptq, 1 't3  b0 Subscriptq, 2 't  Subscriptq, 1 't2 
 Subscriptq, 1 't  Subscriptq, 1 ''t  0,
2
2 Subscriptq, 2t  c Subscriptq, 2t3  b2 Subscriptq, 2 't3 
b0 Subscriptq, 2 't  Subscriptq, 1 't2 
 Subscriptq, 2 't  Subscriptq, 2 ''t  0; EOM  TableForm
2 q1 t  c q1 t3   q1  t  b1 q1  t3  b0  q1  t  q2  t2  q1  t  0
2 q2 t  c q2 t3   q2  t  b2 q2  t3  b0  q1  t  q2  t2  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_  Sumj1 qi,j 1, 2, 3, j, 3 &;
Some rules (don' t modify)
multiScales  qi_ t  qi  timeScales,
Derivativen_q_t  dtnq  timeScales, t  T0 ;
Max order of the procedure
maxOrder  2;
Scaling of the variables

3. Exercise.nb 3
scaling  Subscriptq, 1t   Subscriptq, 1t,
Subscriptq, 2t   Subscriptq, 2t,
Subscriptq, 1 't   Subscriptq, 1 't,
Subscriptq, 2 't   Subscriptq, 2 't, Subscriptq, 1 ''t 
 Subscriptq, 1 ''t, Subscriptq, 2 ''t   Subscriptq, 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_;n3  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  RestThreadCoefficientListSubtract   ,   0 &  EOMa  Transpose;
Definition of the equations at orders of  and representation
eqOrderi_ :  1 &  eqEps1 . q_k_,1  qk,i  
 1 &  eqEps1 . q_k_,1  qk,i    1 &  eqEpsi  Thread

4. Exercise.nb 4
eqOrder1 . displayRule
eqOrder2 . displayRule
eqOrder3 . 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  SolveResonanceCond,  ,    Flatten1 &  omgList  Reverse
2  2 1 , 1  
2
2
First-Order Problem
Equations (=0)
linearSys   1 &  eqOrder1;
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  FunctionT0 , T1 , T2 , A1 T1 , T2  ExpI 1 T0   A 1 T1 , T2  Exp I 1 T0 ,
q2,1  FunctionT0 , T1 , T2 , A2 T1 , T2  ExpI 2 T0   A 2 T1 , T2  Exp I 2 T0 
q1,1  FunctionT0 , T1 , T2 , A1 T1 , T2   1 T0  A1 T1 , T2   1 T0 ,
q2,1  FunctionT0 , 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  eqOrder2 . 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)
expRule1i_ : Expa_ : ExpExpanda . omgRulei .  T0  T1 
Terms of type ei 1 T0 in the first equation of the Second-Order Problem and representation
ST11  Coefficientorder2Eq , 2 . expRule11, ExpI 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  Coefficientorder2Eq , 2 . expRule12, ExpI 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  Coefficientorder2Eq , 2 . expRule11, ExpI 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  Coefficientorder2Eq , 2 . expRule12, ExpI 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  SolveSCond1, 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
solOrder2Eqm1 
SimplifyTableq1,2 T0 , T1 , T2  . TrigToExpFlattenDSolveorder2Eqm1, 1 
order2Eqm1, 2, i, q1,2 T0 , T1 , T2 , T0  . C1  0, C2  0,
i, 1, Lengthorder2Eqm1, 2; solOrder2Eqm1 . 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  ExpandTotalsolOrder2Eqm1;
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
solOrder2Eqm2  Tableq2,2 T0 , T1 , T2  .
SimplifyTrigToExpFlattenDSolveorder2Eqm2, 1  order2Eqm2, 2, i,
q2,2 T0 , T1 , T2 , T0  . C1  0, C2  0,
i, 1, Lengthorder2Eqm2, 2; solOrder2Eqm2 . 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  ExpandTotalsolOrder2Eqm2;
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  FunctionT0 , T1 , T2 , Evaluatesolq21 ,
q2,2  FunctionT0 , T1 , T2 , Evaluatesolq22;
sol2 . displayRule
q1,2  FunctionT0 , 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  FunctionT0 , 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  eqOrder3 . sol1 . sol2  ExpandAll;
Simplifications
order3Eqpr1, 2  Simplifyorder3Eq1, 2;
order3Eqpr2, 2  Simplifyorder3Eq2, 2;
Terms of type ei 1 T0 in the first equation of the Third-Order Problem and representation
ST311  Coefficientorder3Eqpr , 2 . expRule1 , ExpI 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  Coefficientorder3Eqpr , 2 . expRule1 , ExpI 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  Coefficientorder3Eqpr , 2 . expRule11, ExpI 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  Coefficientorder3Eqpr , 2 . expRule12, ExpI 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 
SolveSCond2, 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 SCond1Rule12, 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
 ame1
ame1  A1 1,0 T1 , T2   A1 0,1 T1 , T2  .
JoinSCond1Rule1, SCond2Rule1 . omgRule1; ame1 . 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
 ame2
ame2  A2 1,0 T1 , T2   A2 0,1 T1 , T2  .
JoinSCond1Rule1, SCond2Rule1 . omgRule1; ame2 . 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
Collectame1, 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 3c 2 3 67
A1   A2
1   b2 1 
0 1  A1 A2 b2 1 A2
0
2 8 1 2 1 3 2 15
Collectame2, 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 3c 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
amem1  ame1 . traspRule  A1 't
 2 A1 t 3  c A1 t2 A1 t
 A1 t   b2 1 A1 t2 A1 t  b1 2 A1 t2 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
amem2  ame2 . traspRule  A2 't
 b0 2 A1 t2
 b0 A1 t2   b0 A1 t2    b0 A1 t2 
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 t2 A2 t
 b2 1 A2 t2 A2 t  6 b2 2 A2 t2 A2 t  A2  t
12
 0 1
4 1 5
Polar form of the complex amplitudes

12. Exercise.nb 12
polRule  A1 t 
1
a1t  1t ,
2
A2 t  a2t  2t , A 1 t  a1t  1t , A 2 t  a2t  2t 
1 1 1
2 2 2
A1 t   1t a1t, A2 t 
1 1
 2t a2t,
2 2
A1 t  a1t, A2 t 
1 1
 1t  2t a2t
2 2
polRuleD  JoinpolRule, DpolRule, t
A1 t   1t a1t, A2 t   2t a2t, A1 t 
1 1 1
 1t a1t,
2 2 2
A2 t   2t a2t, A1  t   1t a1 t    1t a1t 1 t,
1 1 1
2 2 2
A2  t   2t a2 t    2t a2t 2 t,
1 1
2 2
A1 t  a1 t    1t a1t 1 t,
 1 1
 1t
2 2
A2 t  a2 t    2t a2t 2 t
 1 1
 2t
2 2
Substitution of the complex amplitudes with the polar form
eq1  amem1 . polRuleD
1 1   1t 2 a1t 3  c  1t a1t3
 1t  a1t   1t 2t  a1t a2t b0   
4 4 16 1 16 1
1 67 3
  1t a1t3 b2 1 
0   1t a1t a2t2 b2 1 
 1t a1t3 b1 2 
0 1
12 120 16
  1t 2t a1t a2t b0 2   1t a1 t    1t a1t 1 t
1 1 1
4 2 2
eq2  amem2 . polRuleD
1 1 3
 2t  a2t    2t  a2t  2  1t  a1t2 b0 
4 2 64
1 1   2t 2 a2t
2  1t  a1t2 b0   2  1t  a1t2 b0  
128 64 32 1
3  c  2t a2t3 61 3
   2t a1t2 a2t b2 1 
0   2t a2t3 b2 1 
0
32 1 240 10
 2t a2 t    2t a2t 2 t
3  2  1t a1t2 b0 2
1 1 1
 2t a2t3 b2 2 
1 
4 8 2 2 2
Autonomous equations and separations of the real and imaginary parts

13. Exercise.nb 13
eqa1  ComplexExpandExpandeq1  1t 
1 1
 a1t   a1t a2t Cos2 1t  2t b0 
a1 t
4 4
3 1
a1t3 b1 2 
1 a1t a2t Sin2 1t  2t b0 2  
16 4 2
1 2 a1t 3 c a1t3 1
  a1t a2t Sin2 1t  2t b0    a1t3 b2 1 
0
4 16 1 16 1 12
a1t 1 t
67 1 1
a1t a2t2 b2 1 
0 a1t a2t Cos2 1t  2t b0 2 
120 4 2
eqa2  ComplexExpandExpandeq2  2t 
1 3
 a2t   a1t2 Cos2 1t  2t b0 
4 64
1 1
 a1t2 Cos2 1t  2t b0   a1t2 Sin2 1t  2t b0 
a2 t
128 64
3 a1t2 Sin2 1t  2t b0 2
1
a2t3 b2 2 
1  
4 8 2 2
1 1 3
  a2t   a1t2 Cos2 1t  2t b0   a1t2 Sin2 1t  2t b0 
2 64 64
1 2 a2t 3 c a2t3 61
 a1t2 Sin2 1t  2t b0    a1t2 a2t b2 1 
0
128 32 1 32 1 240
a2t 2 t
3 a1t2 Cos2 1t  2t b0 2
1 1
a2t3 b2 1 
0 
10 8 2 2
Definition of 
phiRule  2 1t  2t  t
2 1t  2t  t
phiRuleD  2 't  2 1 't   't
2 t  2 1 t   t
Polar Amplitude Modulation Equations
amep1  CollectComplexExpand2 eqa1 .   0, a1t, a2t
3
 a1t3 b1 2 
1
8
 a1 t
 1 1
a1t  a2t   Cos2 1t  2t b0  Sin2 1t  2t b0 2
2 2 2

14. Exercise.nb 14
amep2  CollectComplexExpand2 eqa2 .   0, a1t, a2t
1 3
 a2t  a2t3 b2 2 
1
2 2
3 1
a1t2   Cos2 1t  2t b0   Cos2 1t  2t b0 
32 64
 a2 t
1 Sin2 1t  2t b0 2
1
 Sin2 1t  2t b0 
32 4 2
amep3  CollectComplexExpand  2 eqa1 .   0, a1t, a2t
3c 1 2 67
a1t3  b2 1  a1t 
0  a2t2 b2 1 
0
8 1 6 8 1 60
Cos2 1t  2t b0 2  1 t
1 1
a2t  Sin2 1t  2t b0 
2 2
amep4  CollectComplexExpand  2 eqa2 .   0, a1t, a2t
3c 3
a2t3  b2 1  a1t2
0
16 1 5
1 3 1
 Cos2 1t  2t b0   Sin2 1t  2t b0   Sin2 1t  2t b0 
32 32 64
 2 t
61 Cos2 1t  2t b0 2
1 2
a2t b2 1 
0  a2t  
120 4 2 16 1
Reduced Amplitude Modulation Equations
rame1  Collectamep1 . phiRule, a1t, a2t, t
 a1 t
3  1 1
 a1t3 b1 2  a1t
1  a2t   Cost b0  Sint b0 2
8 2 2 2
rame2  Collectamep2 . phiRule, a1t, a2t, t
1 3
 a2t  a2t3 b2 2 
1
2 2
 a2 t
3 1 1 Sint b0 2
1
a1t2   Cost b0   Cost b0   Sint b0 
32 64 32 4 2

15. Exercise.nb 15
rame3  Expand2 a2t amep3  a1t amep4 . phiRule . phiRuleD
1 3
  a1t a2t   a1t3 Cost b0   a1t3 Sint b0 
32 32
1 2 a1t a2t 2 a1t a2t
 a1t3 Sint b0   a1t a2t2 Sint b0   
64 4 1 16 1
3 c a1t3 a2t 3 c a1t a2t3 7 49
  a1t3 a2t b2 1 
0 a1t a2t3 b2 1 
0
4 1 16 1 40 30
 a1t a2t2 Cost b0 2  a1t a2t  t
a1t3 Cost b0 2
1
4 2
fixRule  a1t  a1, a2t  a2, t  
a1t  a1, a2t  a2, t  
Equations to find Fixed Points
fix1  CollectExpandSimplifyrame1 . a1 't  0, a2 't  0,  't  0,
a1t, a2t, a1t a2t, a1t3 , a2t3 , a1t2 a2t, a1t a2t2  . fixRule
a1  3 1 1
 a13 b1 2  a1 a2 
1  Cos b0  Sin b0 2
2 8 2 2
fix2 
CollectExpandSimplifyrame2 . a1 't  0, a2 't  0,  't  0, a1t, a2t,
a1t3 , a2t3 , a1t2 a2t, a1t a2t2 , b0 , Cost, Sint . fixRule
a2  3 3   2
1
 a23 b2 2  a12 b0
1   Cos  Sin  
2 2 32 64 32 4 2
fix3  CollectExpandSimplifyrame3 . a1 't  0, a2 't  0,  't  0,
a1t, a2t, a1t3 , a2t3 , a1t2 a2t, a1t a2t2 ,
a1t3 a2t, a1t a2t3 , b0 , Cost, Sint . 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
ep1  SimplifySolvefix2 . 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
a1t Cost 1  1t  a1t2 b0  a1t2 Cos2 t 1  2 1t b0 
2 6
a1t a2t Cos3 t 1  1t  2t b0 2 a2t2 b0 2
2 a2t2 Cos4 t 1  2 2t 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 a1t2 b0 2
1
a2t Cost 2  2t  a2t2 b0  a2t2 Cos2 t 2  2 2t b0  
2 6 2 2
2
t 2 3 t 2
4 a1t a2t Cos  1t  2t b0 1 4 a1t a2t Cos  1t  2t 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
solrame1  NDSolverame1  0, rame2  0, rame3  0,
a10  0.1, a20  0.1, 0  0.1, a1t, a2t, t, t, 0, ti
a1t  InterpolatingFunction0., 500., t,
a2t  InterpolatingFunction0., 500., t,
t  InterpolatingFunction0., 500., t

17. Exercise.nb 17
GraphicsArrayPlota1t . solrame1, t, 0, ti, PlotStyle  Thick,
PlotRange  Automatic,  0.8, 0.8, Frame  True, FrameLabel  "t", "a1 t",
Plota2t . solrame1, t, 0, ti, PlotStyle  Thick,
PlotRange  Automatic,  0.6, 0.4, Frame  True, FrameLabel  "t", "a2 t",
Plott . solrame1, 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
solramep1  NDSolveamep1  0, amep2  0, amep3  0, amep4  0, a10  0.1,
a20  0.1, 10  0.1, 20  0.1, a1t, a2t, 1t, 2t, t, 0, ti
a1t  InterpolatingFunction0., 500., t,
a2t  InterpolatingFunction0., 500., t,
1t  InterpolatingFunction0., 500., t,
2t  InterpolatingFunction0., 500., t
GraphicsArrayPlota1t . solramep1, t, 0, ti, PlotStyle  Thick,
PlotRange  Automatic,  0.8, 0.8, Frame  True, FrameLabel  "t", "a1 t",
Plota2t . solramep1, t, 0, ti, PlotStyle  Thick,
PlotRange  Automatic,  0.6, 0.4, Frame  True, FrameLabel  "t", "a2 t",
Plot1t . solramep1, t, 0, ti, PlotStyle  Thick,
Frame  True, AxesOrigin  0, 0, FrameLabel  "t", "1 t",
Plot2t . solramep1, 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
GraphicsArrayPlotqr1 t . solramep1, t, 0, ti, PlotStyle  Thick,
PlotRange  Automatic,  0.8, 0.8, Frame  True, FrameLabel  "t", "q1 t",
Plotqr2 t . solramep1, 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
solorig1  NDSolveJoinEOM, 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  InterpolatingFunction0., 500., t,
q2 t  InterpolatingFunction0., 500., t
GraphicsArrayPlotq1 t . solorig1, t, 0, ti, PlotStyle  Thick,
PlotRange  Automatic,  0.8, 0.8, Frame  True, FrameLabel  "t", "q1 t",
Plotq2 t . solorig1, 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