A three-dimensional system, with quadratic and cubic nonlinearities, undergoing a double-zero bifurcation.

q..

1 q1 q1 b1 q1

2 b2 q1 q1 b3 q1

2 q1 c1q2 q1 3 0

q2 kq2 c2q2 q1 3 0

k 0

Time scales and definitionsand definitions scales Time

OffGeneral::spell1 Notation`

Time scales

scales Time

SymbolizeT0; SymbolizeT1; SymbolizeT2; SymbolizeT3; SymbolizeT4;timeScales T0, T1, T2, T3, T4;dt1expr_ : Sum i

2 Dexpr, timeScalesi 1, i, 0, maxOrder;dt2expr_ : dt1dt1expr Expand . i_;imaxOrder 0;

conjugateRule A A, A A, , , Complex0, n_ Complex0, n;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;

2 my project.nb

Equations of motionsEquations motions of

Equations of motion

Equations motion of

EOM Subscriptq, 1''t Subscriptq, 1't Subscriptq, 1t b1 Subscriptq, 1t2 b2 Subscriptq, 1t Subscriptq, 1't b3 Subscriptq, 1t2 Subscriptq, 1't c1 Subscriptq, 2t Subscriptq, 1t3 0,

Subscriptq, 2't k Subscriptq, 2t c2 Subscriptq, 2t Subscriptq, 1t3 0;

EOM TableForm

q1t b1 q1t2 c1 q1t q2t3 q1t b2 q1t q1t b3 q1t2 q1t q1t

k q2t c2 q1t q2t3 q2t 0

Ordering of the dampings

dampings of Ordering the

smorzrule , ;Definition of the expansion of qi

Definition expansion of2 the qi

solRule qi_ Sum j2 qi,j11, 2, 3, 4, 5, j, 0, 5 &;

multiScales qi_t qi timeScales, Derivativen_q_t dtnq timeScales, t T0;Max order of the procedure

Max of order procedure the

maxOrder 4;

my project.nb 3

Expansion and scaling of the equationand equation Expansion of scaling the

q1T0, T1, T2, T3, T4, T5 . solRule

q1,1T0, T1, T2, T3, T4 q1,2T0, T1, T2, T3, T4 q1,3T0, T1, T2, T3, T4 32 q1,4T0, T1, T2, T3, T4 2 q1,5T0, T1, T2, T3, T4 52 q1,6T0, T1, T2, T3, T4

q1 't . multiScales

2 q10,0,0,0,1T0, T1, T2, T3, T4 32 q10,0,0,1,0T0, T1, T2, T3, T4 q10,0,1,0,0T0, T1, T2, T3, T4 q10,1,0,0,0T0, T1, T2, T3, T4 q11,0,0,0,0T0, T1, T2, T3, T4

Scaling of the variables

of Scaling the variables

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 q1t,q2t q2t, q1t q1t, q2t q2tModification of the equations of motion : substitution of the rules.Representation.

EOMa EOM . scaling . multiScales . smorzrule . solRule TrigToExp ExpandAll .n_;n3 0; EOMa . displayRule2 D0q1,1 52 D0q1,2 3 D0q1,3 D02q1,1 32 D02q1,2 2 D02q1,3 52 D02q1,4 3 D02q1,5 52 D1q1,1 3 D1q1,2 2 32 D0 D1q1,1 2 2 D0 D1q1,2 2 52 D0 D1q1,3 2 3 D0 D1q1,4 2 D1

2q1,1 52 D12q1,2 3 D12q1,3 3 D2q1,1 2 2 D0 D2q1,1 2 52 D0 D2q1,2 2 3 D0 D2q1,3 2 52 D1 D2q1,1 2 3 D1 D2q1,2 3 D22q1,1 2 52 D0 D3q1,1 2 3 D0 D3q1,2 2 3 D1 D3q1,1 2 3 D0 D4q1,1 2 q1,1 2 D0q1,1 b2 q1,1 52 D0q1,2 b2 q1,1 3 D0q1,3 b2 q1,1 52 D1q1,1 b2 q1,1 3 D1q1,2 b2 q1,1 3 D2q1,1 b2 q1,1 2 b1 q1,1

2 3 D0q1,1 b3 q1,12 3 c1 q1,1

3 52 q1,2 52 D0q1,1 b2 q1,2 3 D0q1,2 b2 q1,2 3 D1q1,1 b2 q1,2 2 52 b1 q1,1 q1,2 3 b1 q1,22

3 q1,3 3 D0q1,1 b2 q1,3 2 3 b1 q1,1 q1,3 3 3 c1 q1,12 q2,1 3 3 c1 q1,1 q2,1

2 3 c1 q2,13 0,

D0q2,1 32 D0q2,2 2 D0q2,3 52 D0q2,4 3 D0q2,5 32 D1q2,1 2 D1q2,2 52 D1q2,3 3 D1q2,4 2 D2q2,1 52 D2q2,2 3 D2q2,3 52 D3q2,1 3 D3q2,2 3 D4q2,1 3 c2 q1,13 k q2,1

3 3 c2 q1,12 q2,1 3 3 c2 q1,1 q2,1

2 3 c2 q2,13 k 32 q2,2 k 2 q2,3 k 52 q2,4 k 3 q2,5 0

4 my project.nb

EOMb ExpandEOMa1, 1 0, ExpandEOMa2, 1 0; EOMb . displayRule32 D0q1,1 2 D0q1,2 52 D0q1,3 D02q1,1 D0

2q1,2 32 D02q1,3 2 D02q1,4 52 D02q1,5 2 D1q1,1 52 D1q1,2 2 D0 D1q1,1 2 32 D0 D1q1,2 2 2 D0 D1q1,3 2 52 D0 D1q1,4 32 D12q1,1 2 D12q1,2 52 D12q1,3 52 D2q1,1 2 32 D0 D2q1,1 2 2 D0 D2q1,2 2 52 D0 D2q1,3 2 2 D1 D2q1,1 2 52 D1 D2q1,2 52 D22q1,1 2 2 D0 D3q1,1 2 52 D0 D3q1,2 2 52 D1 D3q1,1 2 52 D0 D4q1,1 32 q1,1 32 D0q1,1 b2 q1,1 2 D0q1,2 b2 q1,1 52 D0q1,3 b2 q1,1 2 D1q1,1 b2 q1,1 52 D1q1,2 b2 q1,1 52 D2q1,1 b2 q1,1 32 b1 q1,12 52 D0q1,1 b3 q1,12 52 c1 q1,13 2 q1,2

2 D0q1,1 b2 q1,2 52 D0q1,2 b2 q1,2 52 D1q1,1 b2 q1,2 2 2 b1 q1,1 q1,2 52 b1 q1,22 52 q1,3 52 D0q1,1 b2 q1,3 2 52 b1 q1,1 q1,3 3 52 c1 q1,12 q2,1 3 52 c1 q1,1 q2,12 52 c1 q2,13 0,

D0q2,1 D0q2,2 32 D0q2,3 2 D0q2,4 52 D0q2,5 D1q2,1 32 D1q2,2 2 D1q2,3 52 D1q2,4 32 D2q2,1 2 D2q2,2 52 D2q2,3 2 D3q2,1 52 D3q2,2 52 D4q2,1 52 c2 q1,13 k q2,1

3 52 c2 q1,12 q2,1 3 52 c2 q1,1 q2,12 52 c2 q2,13 k q2,2 k 32 q2,3 k 2 q2,4 k 52 q2,5 0Separation of the coefficients of the powers of

coefficients of3 powers Separation the2

eqEps RestThreadCoefficientListSubtract , 1

2 0 & EOMb Transpose;

Definition of the equations at orders of and representation

and at Definition equations of2 orders representation the

eqOrderi_ : 1 & eqEps1 . q_k_,1

qk,i 1 & eqEps1 . q_k_,1 qk,i 1 & eqEpsi Thread

my project.nb 5

Pertubation equationsequations Pertubation

eqOrder1 . displayRuleeqOrder2 . displayRuleeqOrder3 . displayRuleeqOrder4 . displayRuleeqOrder5 . displayRuleD02q1,1 0, D0q2,1 k q2,1 0D02q1,2 2 D0 D1q1,1, D0q2,2 k q2,2 D1q2,1D02q1,3 D0q1,1 2 D0 D1q1,2 D1

2q1,1 2 D0 D2q1,1 q1,1 D0q1,1 b2 q1,1 b1 q1,12 ,

D0q2,3 k q2,3 D1q2,2 D2q2,1D02q1,4 D0q1,2 D1q1,1 2 D0 D1q1,3 D12q1,2 2 D0 D2q1,2 2 D1 D2q1,1 2 D0 D3q1,1 D0q1,2 b2 q1,1

D1q1,1 b2 q1,1 q1,2 D0q1,1 b2 q1,2 2 b1 q1,1 q1,2, D0q2,4 k q2,4 D1q2,3 D2q2,2 D3q2,1D02q1,5 D0q1,3 D1q1,2 2 D0 D1q1,4 D12q1,3 D2q1,1

2 D0 D2q1,3 2 D1 D2q1,2 D22q1,1 2 D0 D3q1,2 2 D1 D3q1,1 2 D0 D4q1,1 D0q1,3 b2 q1,1

D1q1,2 b2 q1,1 D2q1,1 b2 q1,1 D0q1,1 b3 q1,12 c1 q1,1

3 D0q1,2 b2 q1,2 D1q1,1 b2 q1,2

b1 q1,22 q1,3 D0q1,1 b2 q1,3 2 b1 q1,1 q1,3 3 c1 q1,1

2 q2,1 3 c1 q1,1 q2,12 c1 q2,1

3 ,

D0q2,5 k q2,5 D1q2,4 D2q2,3 D3q2,2 D4q2,1 c2 q1,13 3 c2 q1,1

2 q2,1 3 c2 q1,1 q2,12 c2 q2,1

3

6 my project.nb

First Order ProblemFirst Order Problem

linearSys 1 & eqOrder1;linearSys . displayRule TableForm

D02q1,1

D0q2,1 k q2,1

Formal solution of the First Order Problem generating solution

generating Order Problem solution First Formal of solution the

sol1 q1,1 FunctionT0, T1, T2, T3, T4, A1T1, T2, T3, T4 ,q2,1 FunctionT0, T1, T2, T3, T4, 0q1,1 FunctionT0, T1, T2, T3, T4, A1T1, T2, T3, T4, q2,1 FunctionT0, T1, T2, T3, T4, 0

my project.nb 7

Second Order ProblemOrder Problem Second

Substitution of the solution on the Second Order Problem and representation

and Order Problem representation of on Second solution Substitution the2

eqOrder2 . displayRuleD02q1,2 2 D0 D1q1,1, D0q2,2 k q2,2 D1q2,1order2Eq eqOrder2 . sol1 ExpandAll;order2Eq . displayRuleD02q1,2 0, D0q2,2 k q2,2 0we eliminate secular terms then we obtain

eliminate obtain secular terms then we2

sol2 q1,2 FunctionT0, T1, T2, T3, T4, 0 , q2,2 FunctionT0, T1, T2, T3, T4, 0q1,2 FunctionT0, T1, T2, T3, T4, 0, q2,2 FunctionT0, T1, T2, T3, T4, 0

8 my project.nb

Third Order ProblemOrder Problem Third

Substitution in the Third Order Equations

Equations Order in Substitution the Third

order3Eq eqOrder3 . sol1 . sol2 ExpandAll;order3Eq . displayRuleD02q1,3 D1

2A1 A1 A12 b1, D0q2,3 k q2,3 0

ST31 order3Eq, 2 & 1;ST31 . displayRuleD12A1 A1 A12 b1SCond3 ST31 0;SCond3 . displayRuleD12A1 A1 A12 b1 0

SCond3 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42 A12,0,0,0T1, T2, T3, T4 0 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42 A12,0,0,0T1, T2, T3, T4 0 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42 A12,0,0,0T1, T2, T3, T4 0 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42 A12,0,0,0T1, T2, T3, T4 0 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42 A12,0,0,0T1, T2, T3, T4 0

SCond3Rule1

SolveSCond3, A12,0,0,0T1, T2, T3, T41 ExpandAll Simplify Expand;

SCond3Rule1 . displayRuleD12A1 A1 A12 b1

sol3 q1,3 FunctionT0, T1, T2, T3, T4, 0 , q2,3 FunctionT0, T1, T2, T3, T4, 0q1,3 FunctionT0, T1, T2, T3, T4, 0, q2,3 FunctionT0, T1, T2, T3, T4, 0

my project.nb 9

Fourth Order ProblemFourth Order Problem

Substitution in the Fourth Order Equations

Equations Order Fourth in Substitution the

order4Eq eqOrder4 . sol1 . sol2 . sol3 ExpandAll;order4Eq . displayRuleD02q1,4 D1A1 2 D1 D2A1 D1A1 A1 b2, D0q2,4 k q2,4 0ST41 order4Eq, 2 & 1;ST41 . displayRule D1A1 2 D1 D2A1 D1A1 A1 b2SCond4 ST41 0;SCond4 . displayRule D1A1 2 D1 D2A1 D1A1 A1 b2 0

SCond4 A11,0,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4 2 A11,1,0,0T1, T2, T3, T4 0 A11,0,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4 2 A11,1,0,0T1, T2, T3, T4 0 A11,0,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4 2 A11,1,0,0T1, T2, T3, T4 0 A11,0,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4 2 A11,1,0,0T1, T2, T3, T4 0 A11,0,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4 2 A11,1,0,0T1, T2, T3, T4 0

SCond4Rule1

SolveSCond4, A11,1,0,0T1, T2, T3, T41 ExpandAll Simplify Expand;

SCond4Rule1 . displayRuleD1 D2A1 D1A1

21

2D1A1 A1 b2

SCond3Rule1 . displayRuleD12A1 A1 A12 b1

SCond3Rule1A12,0,0,0T1, T2, T3, T4 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42A12,0,0,0T1, T2, T3, T4 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42A12,0,0,0T1, T2, T3, T4 A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42

10 my project.nb

SCond4Rule1A11,1,0,0T1, T2, T3, T4

1

2 A11,0,0,0T1, T2, T3, T4 1

2b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4

A11,1,0,0T1, T2, T3, T4

1

2 A11,0,0,0T1, T2, T3, T4 1

2b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4A11,1,0,0T1, T2, T3, T4

1

2 A11,0,0,0T1, T2, T3, T4 1

2b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4

A11,1,0,0T1, T2, T3, T4

1

2 A11,0,0,0T1, T2, T3, T4 1

2b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4A11,1,0,0T1, T2, T3, T4

1

2 A11,0,0,0T1, T2, T3, T4 1

2b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T42 A11,1,0,0T1, T2, T3, T4 A12,0,0,0T1, T2, T3, T4 . SCond3Rule1 . SCond4Rule1

A1T1, T2, T3, T4 b1 A1T1, T2, T3, T42 2

1

2 A11,0,0,0T1, T2, T3, T4 1

2b2 A1T1, T2, T3, T4 A11,0,0,0T1, T2, T3, T4

TimeRule1 A1T1, T2, T3, T4 A1t, A11,0,0,0T1, T2, T3, T4 A1 't A1T1, T2, T3, T4 A1t, A11,0,0,0T1, T2, T3, T4 A1t2 A11,1,0,0T1, T2, T3, T4 A12,0,0,0T1, T2, T3, T4 . SCond3Rule1 . SCond4Rule1 .TimeRule1 A1 ''t

A1t b1 A1t2 2 1

2 A1t 1

2b2 A1t A1t A1t

sol4 q1,4 FunctionT0, T1, T2, T3, T4, 0 , q2,4 FunctionT0, T1, T2, T3, T4, 0q1,4 FunctionT0, T1, T2, T3, T4, 0, q2,4 FunctionT0, T1, T2, T3, T4, 0

my project.nb 11

Fifth Order ProblemFifth Order Problem

order5Eq eqOrder5 . sol1 . sol2 . sol3 . sol4 ExpandAll;order5Eq . displayRuleD02q1,5 D2A1 D2

2A1 2 D1 D3A1 D2A1 A1 b2 A13 c1, D0q2,5 k q2,5 A1

3 c2ST51 order5Eq, 2 & 1;ST51 . displayRule D2A1 D22A1 2 D1 D3A1 D2A1 A1 b2 A13 c1SCond5 ST51 0;SCond5 . displayRule D2A1 D22A1 2 D1 D3A1 D2A1 A1 b2 A13 c1 0

SCond5c1 A1T1, T2, T3, T43 A10,1,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A10,1,0,0T1, T2, T3, T4 A10,2,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4 0c1 A1T1, T2, T3, T43 A10,1,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4A10,1,0,0T1, T2, T3, T4 A10,2,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4 0c1 A1T1, T2, T3, T43 A10,1,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A10,1,0,0T1, T2, T3, T4 A10,2,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4 0

SCond5Rule1

SolveSCond5, A10,2,0,0T1, T2, T3, T4 1 ExpandAll Simplify Expand;

SCond5Rule1 . displayRuleD22A1 D2A1 2 D1 D3A1 D2A1 A1 b2 A13 c1

SCond5Rule1A10,2,0,0T1, T2, T3, T4 c1 A1T1, T2, T3, T43 A10,1,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A10,1,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4A10,2,0,0T1, T2, T3, T4 c1 A1T1, T2, T3, T43 A10,1,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A10,1,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4A10,2,0,0T1, T2, T3, T4 c1 A1T1, T2, T3, T43 A10,1,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A10,1,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4

A10,2,0,0T1, T2, T3, T4 c1 c2 A1T1, T2, T3, T43

k A10,1,0,0T1, T2, T3, T4

b2 A1T1, T2, T3, T4 A10,1,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4A10,2,0,0T1, T2, T3, T4

c1 c2 A1T1, T2, T3, T43k

A10,1,0,0T1, T2, T3, T4 b2 A1T1, T2, T3, T4 A10,1,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4

12 my project.nb

A10,2,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4 . SCond5Rule1 . TimeRule1

c1 A1t3 A10,1,0,0T1, T2, T3, T4 b2 A1t A10,1,0,0T1, T2, T3, T4sol5 q1,5 FunctionT0, T1, T2, T3, T4, 0 , q2,5 FunctionT0, T1, T2, T3, T4, A13 c2

k

q1,5 FunctionT0, T1, T2, T3, T4, 0, q2,5 FunctionT0, T1, T2, T3, T4, A13 c2

k

my project.nb 13

Bifurcation equations and fixed pointsand Bifurcation equations fixed points

TimeRule2 A10,1,0,0T1, T2, T3, T4 0A10,1,0,0T1, T2, T3, T4 0RBFCE A10,2,0,0T1, T2, T3, T4 2 A11,0,1,0T1, T2, T3, T4

2 A11,1,0,0T1, T2, T3, T4 A12,0,0,0T1, T2, T3, T4 . SCond3Rule1 .SCond4Rule1 . SCond5Rule1 . TimeRule1 . TimeRule2 A1 ''t

A1t b1 A1t2 c1 A1t3 2 1

2 A1t 1

2b2 A1t A1t A1t

14 my project.nb

Fixed pointsPerfect system

Perfect system

perfectsyst A1t 0, A1t 0 A1t 0, A1t 0fix1 RBFCE . perfectsyst

A1t b1 A1t2 c1 A1t3fixpoint1 fix1 0;fixpoint1 . displayRule

A1 A12 b1 A1

3 c1 0

fixpoint1 A1t b1 A1t2 c1 A1t3 0

scalingRule2 A12,0,0,0T1, T2, T3, T4 A1t A12,0,0,0T1, T2, T3, T4 A1tA12,0,0,0T1, T2, T3, T4 A1tSolvefixpoint1, A1tA12,0,0,0T1, T2, T3, T4 A1tA1t 0, A1t

b1 b12 4 c1

2 c1, A1t

b1 b12 4 c1

2 c1

my project.nb 15

Reconstitution of the equation of the motionStepx1 A1T1, T2, T3, T4A1T1, T2, T3, T4Stepy1

c2 A1T1, T2, T3, T43k

c2 A1T1, T2, T3, T43k

ScalingRule1 A1T1, T2, T3, T4 A1tA1T1, T2, T3, T4 A1tx t A1T1, T2, T3, T4 . ScalingRule1

A1tscalingRule2 D1A11,0,0,0T1, T2, T3, T4 A1t , D2A1T1, T2, T3, T4 0 , D1A1T1, T2, T3, T4 A1tD1A11,0,0,0T1, T2, T3, T4 A1t, D2A1T1, T2, T3, T4 0, D1A1T1, T2, T3, T4 A1ty t

c2 A1T1, T2, T3, T43k

. scalingRule2 . ScalingRule1

c2 A1t3k

16 my project.nb

Numerical integrationsNumerical values for the perfect system

for Numerical perfect system the values

c1 1, k 2, b3 1

2, b1 1, b2 1, c2 1, 0.01, 0.9

1, 2,1

2, 1, 1, 1, 0.01, 0.9

Time of integration

integration of Time

ti 500;

Numerical Intergations of the reconstitutesolution and study of the motion around the equilibrium points

and around equilibrium Intergations motion Numerical of2 points reconstitute solution study the3

solramep1 NDSolveRBFCE 0, A10 0.01, A10 0.01 ,A1t, A1t, A1t, t, 0, tiA1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t

my project.nb 17

GraphicsArrayPlotSubscriptA, 1t . solramep1, t, 0, ti,PlotStyle Thick, PlotRange Automatic, All, Frame True,FrameLabel "t", "\\\SubscriptBox\"q\", \"1\"\t",PlotA1t . solramep1, t, 0, ti,PlotStyle Thick, PlotRange Automatic, Automatic,Frame True,FrameLabel "t","\\\SubscriptBoxOverscriptBox\"q\", \".\", \"1\"\t"

l TableSubscriptA, 1t . solramep1,A1t . solramep1, t, 0, ti;ListPlotTableExtractlj, 1, 1, Extractlj, 2, 1, j,1, ti 1, Joined True

0 100 200 300 400 500

0.4

0.2

0.0

0.2

t

q 1t

0 100 200 300 400 500

0.150.100.05

0.000.050.100.15

t

q 1t

0.15 0.10 0.05 0.05 0.10 0.15

0.20

0.15

0.10

0.05

0.05

0.10

0.15

Graphics of the reconstituted solution

Graphics of reconstituted solution the

18 my project.nb

solramep1 NDSolveRBFCE 0, A10 0.01, A10 0.01 ,A1t, A1t, A1t, t, 0, tiGraphicsArrayPlotA1t . solramep1, t, 0, ti, PlotStyle Thick,

PlotRange Automatic, 3.8, 3.8, Frame True, FrameLabel "t", "xt",Plot c2 A1t3

k. solramep1, t, 0, ti, PlotStyle Thick,

PlotRange Automatic, 2.6, 2.4, Frame True, FrameLabel "t", "yt"A1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t

0 100 200 300 400 500

321

0123

t

xt

0 100 200 300 400 500

21

012

t

yt

Numerical Intergations of the original equations

equations Intergations Numerical of original the

solorig1 NDSolveJoinEOM, q10 0.01, q20 0.01, q1 '0 0.01,q1t, q2t, t, 0, ti, MaxSteps 1 000 000q1t 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

q 1t

0 100 200 300 400 5000.60.40.2

0.00.20.4

t

q 2t

my project.nb 19

Another ExampleNumerical values for the perfect syst emc1 0.5, k 1, b3

1

2, b1 0.5, b2 0.5, c2 0.5, 0.01, 0.9

Time of integrationti 500;Numerical Intergations of the reconstitutesolution and study of the motion around the equilibrium points

solramep1 NDSolveRBFCE 0, A10 0.01, A10 0.01 ,A1t, A1t, A1t, t, 0, tiGraphicsArrayPlotSubscriptA, 1t . solramep1, t, 0, ti,PlotStyle Thick, PlotRange Automatic, All, Frame True,FrameLabel "t", "\\\SubscriptBox\"q\", \"1\"\t",PlotA1t . solramep1, t, 0, ti,PlotStyle Thick, PlotRange Automatic, Automatic,Frame True,FrameLabel "t","\\\SubscriptBoxOverscriptBox\"q\", \".\", \"1\"\t"

l TableSubscriptA, 1t . solramep1,A1t . solramep1, t, 0, ti;ListPlotTableExtractlj, 1, 1, Extractlj, 2, 1, j,1, ti 1, Joined True

Another Example

em for Numerical perfect syst the values

0.5, 1,1

2, 0.5, 0.5, 0.5, 0.01, 0.9

integration of Time

and around equilibrium Intergations motion Numerical of2 points reconstitute solution study the3A1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t

0 100 200 300 400 5000.2

0.1

0.0

0.1

0.2

t

q 1t

0 100 200 300 400 500

0.150.100.05

0.000.050.100.15

t

q 1t

20 my project.nb

0.20 0.15 0.10 0.05 0.05 0.10 0.15

0.15

0.10

0.05

0.05

0.10

0.15

Graphics of the reconstituted solution

Graphics of reconstituted solution the

solramep1 NDSolveRBFCE 0, A10 0.01, A10 0.01 ,A1t, A1t, A1t, t, 0, tiGraphicsArrayPlotA1t . solramep1, t, 0, ti, PlotStyle Thick,

PlotRange Automatic, 3.8, 3.8, Frame True, FrameLabel "t", "xt",Plot c2 A1t3

k. solramep1, t, 0, ti, PlotStyle Thick,

PlotRange Automatic, 2.6, 2.4, Frame True, FrameLabel "t", "yt"A1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t,A1t InterpolatingFunction0., 500., t

0 100 200 300 400 500

321

0123

t

xt

0 100 200 300 400 500

21

012

t

yt

my project.nb 21

Numerical Intergations of the original equationssolorig1 NDSolveJoinEOM, q10 0.01, q20 0.01, q1 '0 0.01,q1t, q2t, t, 0, ti, MaxSteps 1 000 000GraphicsArrayPlotq1t . 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"

equations Intergations Numerical of original theq1t InterpolatingFunction0., 500., t,q2t InterpolatingFunction0., 500., t

0 100 200 300 400 500

0.5

0.0

0.5

t

q 1t

0 100 200 300 400 5000.60.40.2

0.00.20.4

t

q 2t

22 my project.nb