Multiple Bifurcations of Sample Dynamical Systems

1. Homework2.nb 1
Time scales and definitions
Equations of motions and resonance conditions
Equations of motion
EOM    q1 't   q1 t  c2 q1 t q2 t  c3 q1 t3    0,  q2 't  q2 t  bq1 t2  0;
EOM  TableForm
   q1 t  c3 q1 t3  c2 q1 t q2 t  q1  t  0
bq1 t2  q2 t  q2  t  0
EOM    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 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 ;
This is the result of "multiScales" and "solRule":

2. Homework2.nb 2
q1 t . multiScales . solRule
q1,0 T0 , T1 , T2    q1,1 T0 , T1 , T2   2 q1,2 T0 , T1 , T2 
Max order of the procedure
maxOrder  2;
Scaling of the variables
scaling  q1 t   q1 t, q2 t   q2 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, 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.
EOMa  EOM . scaling . multiScales . smorzrule . ombrule . solRule  TrigToExp 
ExpandAll . n_;n3  0;
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_,0  qk,i1  
 1 &  eqEps1 . q_k_,0  qk,i1    1 &  eqEpsi  Thread
eqOrder1 . displayRule
eqOrder2 . displayRule
eqOrder3 . displayRule
D2 q1,0  2 q1,0  0, D2 q2,0  2 q2,0  0
0 1 0 2
D2 q1,1  2 q1,1   D0 q1,0  2 D0 D1 q1,0  D0 q1,0 2 b0  2 D0 q1,0 D0 q2,0 b0  D0 q2,0 2 b0 ,
D2 q2,1  2 q2,1   D0 q2,0  2 D0 D1 q2,0  D0 q1,0 2 b0  2 D0 q1,0 D0 q2,0 b0  D0 q2,0 2 b0  2  2 q2,0 
0 1
0 2
D2 q1,2  2 q1,2   D0 q1,1   D1 q1,0  2 D0 D1 q1,1  D2 q1,0  2 D0 D2 q1,0  2 D0 q1,0 D0 q1,1 b0 
0 1 1
2 D0 q1,1 D0 q2,0 b0  2 D0 q1,0 D0 q2,1 b0  2 D0 q2,0 D0 q2,1 b0  2 D0 q1,0 D1 q1,0 b0 
2 D0 q2,0 D1 q1,0 b0  2 D0 q1,0 D1 q2,0 b0  2 D0 q2,0 D1 q2,0 b0  D0 q1,0 3 b1  c q3 ,
1,0
D2 q2,2  2 q2,2   D0 q2,1   D1 q2,0  2 D0 D1 q2,1  D2 q2,0  2 D0 D2 q2,0  2 D0 q1,0 D0 q1,1 b0 
0 2 1
2 D0 q1,1 D0 q2,0 b0  2 D0 q1,0 D0 q2,1 b0  2 D0 q2,0 D0 q2,1 b0  2 D0 q1,0 D1 q1,0 b0  2 D0 q2,0 D1 q1,0 b0 
2 D0 q1,0 D1 q2,0 b0  2 D0 q2,0 D1 q2,0 b0  D0 q2,0 3 b2  2 q2,0  c q3  2  2 q2,1 
2,0
Resonance condition
ResonanceCond  1  2 ;
1
2
Involved frequencies

3. Homework2.nb 3
omgList  1 , 2 ;
Utility (don't modify)
omgRule  SolveResonanceCond,  ,    Flatten1 &  omgList  Reverse
2  2 1 , 1  
2
2
First-Order Problem
Second-Order Problem
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
 60  D1 A1 1  60 D2 A1 1  120  D2 A1 2  120  D1 A1 A2 b0 1 2  180 c A2 1 A1 
1
1 1 1
60 1
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 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
Terms of type ei 1 T0 in the second equation of the Third-Order Problem and representation

4. Homework2.nb 4
ST321  Coefficientorder3Eqpr , 2 . expRule11, ExpI 1 T0  &  2;
ST321 . displayRule
  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 
1
1 2 1 1
30 1 2
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 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  120  D1 A1 A2 b0 1 2  180 c A2 1 A1  120  D1 A2
1
1 1 1
60 1
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 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

5. Homework2.nb 5
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
Polar form of the AME
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  200;
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
NDSolve::ndode : Input is not an ordinary differential equation. 
a1t, a2t, t, t, 0, ti
NDSolverame1  0, rame2  0, rame3  0, a10  0.1, a20  0.1, 0  0.1,

6. Homework2.nb 6
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
a1 t
1.5
t
a2 t
0.0 0.0
0.2 1.0
0.5 0.4 0.5
0.6 0.0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
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., 200., t,
a2t  InterpolatingFunction0., 200., t,
1t  InterpolatingFunction0., 200., t,
2t  InterpolatingFunction0., 200., 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 2.0 2.0
0.5 0.2
a1 t
1 t
2 t
1.5 1.5
a2 t
0.0 0.0
1.0 1.0
0.2
0.5 0.5
0.5 0.4
0.6 0.0 0.0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150
t t t t
Graphics of the reconstituted solution

7. Homework2.nb 7
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 50 100 150 200 0 50 100 150 200
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., 200., t,
q2 t  InterpolatingFunction0., 200., 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
q1 t
0.0
q2 t
0.0
0.2
0.5 0.4
0.6
0 50 100 150 200 0 50 100 150 200
t t