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# Exercise: Multiple Bifurcations of Sample Dynamical Systems

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Multiple Bifurcations of Sample Dynamical Systems

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### Exercise: Multiple Bifurcations of Sample Dynamical Systems

1. 1. Exercise.nb 1Double 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 1Time scales and definitions Utility (dont modify) OffGeneral::spell1  Notation`Time scales SymbolizeT0 ; SymbolizeT1 ; SymbolizeT2 ; timeScales  T0 , T1 , T2 ; Differentiation (dont 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. 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 conditionsEquations 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 2Ordering 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. 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 2Separation 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. 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  2,1Resonance condition ResonanceCond  1  2 ; 1 2Involved frequencies omgList  1 , 2 ; Utility (dont modify) omgRule  SolveResonanceCond,  ,    Flatten1 &  omgList  Reverse 2  2 1 , 1   2 2First-Order ProblemEquations (=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. 5. Exercise.nb 5Formal 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 ProblemSubstitution 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 (dont 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 1Terms of type ei 1 T0 in the second equation of the Second-Order Problem and representation
6. 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 1Scalar 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 1Algebraic 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 2Substitution 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 2Assumpion on the coefficients \$Assumptions  1 , 2 , b0 , c, , , b1 , b2   Reals 1 2 b0 c   b1 b2   RealsConstruction of the particular solution of the first equation of the Second-Order Problem and representation
7. 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 1Construction 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 3Formal representation of the solution
8. 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 3Third-Order ProblemSubstitution 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 2Terms 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. 9. Exercise.nb 9Terms 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 2Terms 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 2Scalar 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 2Algebraic manipulation to obtain D2 A1 and D2 A2
10. 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 1Reconstitution of the AME and of the solutiondA1 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 15dA1 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 5Better representation
11. 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 5Polar 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 5Polar form of the complex amplitudes
12. 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 2Substitution 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 2Autonomous equations and separations of the real and imaginary parts
13. 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 2Definition 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. 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 1Reduced 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. 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 2Equilibrium 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 1Reconstitution of the solution
16. 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 2Numerical integrationsNumerical 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 2Time 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. 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 tNumerical 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 tGraphics of the reconstituted solution
18. 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 tNumerical 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