final result : Multiple Bifurcations of Sample Dynamical Systems

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

q..1 q

1 q1 b1 q

12 c1 q1 q2 b2 q1 q1 b3 q12 q

1 0

q2 kq2 c2 q12 0

k 0

Time scales and definitions

OffGeneral::spell1 Notation`

Time scales

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 4project_final_result.nb

Equations of motions

Equations of motion

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

c2 Subscriptq, 1t2 k Subscriptq, 2t Subscriptq, 2't 0; EOM TableForm

q1t b1 q1t2 c1 q1t q2t q1t b2 q1t q1t b3 q1t2 q1t q1t 0

c2 q1t2 k q2t q2t 0

Ordering of the dampings

smorzrule , ;ombrule 3 3 Definition of the expansion of 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

maxOrder 4;

4project_final_result.nb 3

Expansion and scaling of the equation

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

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 . ombrule . solRule TrigToExp ExpandAll .n_;n3 0; EOMa . displayRule3 2 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 D12q1,1 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 D2

2q1,1 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 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

2 c1 q1,1 q2,1 52 c1 q1,2 q2,1 3 c1 q1,3 q2,1 52 c1 q1,1 q2,2 3 c1 q1,2 q2,2 3 c1 q1,1 q2,3 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 2 c2 q1,12

2 52 c2 q1,1 q1,2 3 c2 q1,22 2 3 c2 q1,1 q1,3 k q2,1 k 32 q2,2 k 2 q2,3 k 52 q2,4 k 3 q2,5 0

4 4project_final_result.nb

EOMb ExpandEOMa1, 1 0, ExpandEOMa2, 1 0; EOMb . displayRule52 32 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 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 32 c1 q1,1 q2,1 2 c1 q1,2 q2,1 52 c1 q1,3 q2,1 2 c1 q1,1 q2,2 52 c1 q1,2 q2,2 52 c1 q1,1 q2,3 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 32 c2 q1,12 2 2 c2 q1,1 q1,2

52 c2 q1,22 2 52 c2 q1,1 q1,3 k q2,1 k q2,2 k 32 q2,3 k 2 q2,4 k 52 q2,5 0Separation of the coefficients of the powers of

eqEps RestThreadCoefficientListSubtract , 1

2 0 & EOMb 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

4project_final_result.nb 5

Pertubation equations

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 c1 q1,1 q2,1,

D0q2,3 k q2,3 D1q2,2 D2q2,1 c2 q1,12 D02q1,4 D0q1,2 D1q1,1 2 D0 D1q1,3 D1

2q1,2 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 c1 q1,2 q2,1 c1 q1,1 q2,2,

D0q2,4 k q2,4 D1q2,3 D2q2,2 D3q2,1 2 c2 q1,1 q1,2D02q1,5 D0q1,3 D1q1,2 2 D0 D1q1,4 D1

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

D0q2,5 k q2,5 D1q2,4 D2q2,3 D3q2,2 D4q2,1 c2 q1,22 2 c2 q1,1 q1,3

6 4project_final_result.nb

First Order ProblemEquations 0)

linearSys 1 & eqOrder1;linearSys . displayRule TableForm

D02q1,1

D0q2,1 k q2,1

Formal solution of the First Order Problem generating solution

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

4project_final_result.nb 7

Second Order Problem

Substitution of the solution on the Second Order Problem and representation

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

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 4project_final_result.nb

Third Order Problem

Substitution in the Third Order Equations

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

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

2 c2ST31 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

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, A1T1, T2, T3, T42 c2 kq1,3 FunctionT0, T1, T2, T3, T4, 0,

q2,3 FunctionT0, T1, T2, T3, T4, A1T1, T2, T3, T42 c2

k

4project_final_result.nb 9

Fourth Order Problem

Substitution in the Fourth Order Equations

order4Eq eqOrder4 . sol1 . sol2 . sol3 ExpandAll;order4Eq . displayRuleD02q1,4 D1A1 2 D1 D2A1 D1A1 A1 b2, D0q2,4 k q2,4

2 D1A1 A1 c2

k

ST41 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

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, T42SCond4Rule1A11,1,0,0T1, 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

10 4project_final_result.nb

2 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, 2 D1A1T1, T2, T3, T4 A1 T1, T2, T3, T4 c2

kq1,4 FunctionT0, T1, T2, T3, T4, 0,

q2,4 FunctionT0, T1, T2, T3, T4, 2 D1A1T1, T2, T3, T4 A1T1, T2, T3, T4 c2k

4project_final_result.nb 11

Fifth Order Problem

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

2A1 2 D1 D3A1 D2A1 A1 b2 A13 c1 c2

k,

D0q2,5 k q2,5 2 D2A1 A1 c2

k2 D1A1 c2 D1A1T1, T2, T3, T4

k2 A1 c2 D1A11,0,0,0T1, T2, T3, T4

k

ST51 order5Eq, 2 & 1;ST51 . displayRule D2A1 D22A1 2 D1 D3A1 D2A1 A1 b2 A1

3 c1 c2

k

SCond5 ST51 0;SCond5 . displayRule D2A1 D22A1 2 D1 D3A1 D2A1 A1 b2 A1

3 c1 c2

k 0

SCond5 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 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 c2

k

SCond5Rule1A10,2,0,0T1, 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 2 A11,0,1,0T1, T2, T3, T4 . SCond5Rule1 . TimeRule1

c1 c2 A1t3

k A10,1,0,0T1, T2, T3, T4 b2 A1t A10,1,0,0T1, T2, T3, T4

12 4project_final_result.nb

sol5 q1,5 FunctionT0, T1, T2, T3, T4, 0 , q2,5 FunctionT0, T1, T2, T3, T4,2 D2A1 A1 c2

k22 D1A1 c2 D1A1T1, T2, T3, T4

k22 A1 c2 D1A11,0,0,0T1, T2, T3, T4

k2q1,5 FunctionT0, T1, T2, T3, T4, 0, q2,5 FunctionT0, T1, T2, T3, T4,

2 D2A1 A1 c2

k22 D1A1 c2 D1A1T1, T2, T3, T4

k22 A1 c2 D1A11,0,0,0T1, T2, T3, T4

k2

4project_final_result.nb 13

Bifurcation equations and 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 c2 A1t3k

21

2 A1t 1

2b2 A1t A1t A1t

IF we neglected the contribution of the passive variable the bifurcation equation becomes

MBFCE 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

c1 c2 A1T1, T2, T3, T43k

. SCond3Rule1 .SCond4Rule1 . SCond5Rule1 . TimeRule1 . TimeRule2 A1 ''t

A1t b1 A1t2 2 1

2 A1t 1

2b2 A1t A1t A1t

14 4project_final_result.nb

Fixed points both for the

perfect and imperfect system

Perfect system

perfectsyst 0, A1t 0, A1t 0 0, A1t 0, A1t 0fix1 MBFCE . perfectsyst

A1t b1 A1t2fixpoint1 fix1 0;fixpoint1 . displayRule

A1 A12 b1 0

fixpoint1 A1t b1 A1t2 0

Imperfect system

imperfectsyst A1t 0, A1t 0 A1t 0, A1t 0fix2 MBFCE . imperfectsyst

A1t b1 A1t2fixpoint2 fix2 0;fixpoint2 . displayRule

A1 A12 b1 0

fixpoint2 A1t b1 A1t2 0

scalingRule2 A12,0,0,0T1, T2, T3, T4 A1t A12,0,0,0T1, T2, T3, T4 A1t

4project_final_result.nb 15

Reconstitution of the equation of the motion

Stepx1 A1T1, T2, T3, T4A1T1, T2, T3, T4Stepy1

2 D2A1T1, T2, T3, T4 A1T1, T2, T3, T4 c2k2

2 D1A1 T1, T2, T3, T4 c2 D1A1T1, T2, T3, T4

k2

2 A1 T1, T2, T3, T4 c2 D1A11,0,0,0T1, T2, T3, T4k2

2 D1A1T1, T2, T3, T4 A1T1, T2, T3, T4 c2k

A1T1, T2, T3, T42 c2

k

2 c2 D1A1T1, T2, T3, T42

k22 c2 D1A1T1, T2, T3, T4 A1T1, T2, T3, T4

k

2 c2 D2A1T1, T2, T3, T4 A1T1, T2, T3, T4k2

c2 A1T1, T2, T3, T42

k

2 c2 A1T1, T2, T3, T4 D1A11,0,0,0T1, T2, T3, T4k2

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

Set::write : Tag Times in t x is Protected.

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 A1t

16 4project_final_result.nb

y t 2 c2 D1A1T1, T2, T3, T42

k22 c2 D1A1T1, T2, T3, T4 A1T1, T2, T3, T4

k

2 c2 D2A1T1, T2, T3, T4 A1T1, T2, T3, T4k2

c2 A1T1, T2, T3, T42

k

2 c2 A1T1, T2, T3, T4 D1A11,0,0,0T1, T2, T3, T4k2

. scalingRule2 . ScalingRule1

Set::write : Tag Times in t y is Protected.

c2 A1t2

k2 c2 A1t A1t

k2 c2 A1t2

k22 c2 A1t A1t

k2

4project_final_result.nb 17

Numerical integrations

Numerical values for the perfect system

c1 1, k 2, b3 1

2, b1 1, b2 1, c2 1, 0.01, 0.09, 0

1, 2,1

2, 1, 1, 1, 0.01, 0.09, 0

Time of integration

ti 500;

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

solramep1 NDSolveMBFCE 0, A10 0.01, A10 0.01 ,A1t, A1t, A1t, t, 0, tiNDSolve::ndsz : At t 74.65045623139203`, step size is effectively zero; singularity or stiff system suspected. A1t InterpolatingFunction0., 74.6505, t,A1t InterpolatingFunction0., 74.6505, t,A1t InterpolatingFunction0., 74.6505, t

18 4project_final_result.nb

GraphicsArrayPlotA1t . solramep1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 3.8, 3.8, Frame True, FrameLabel "t", "xt",

PlotA1t . solramep1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 3.6, 3.4, Frame True, FrameLabel "t", "x't"

ParametricPlotA1t . solramep1, A1t . solramep1, t, 0, 25, PlotStyle Thick,PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "xt", "x't"

0 100 200 300 400 500

321

0123

t

xt

0 100 200 300 400 500

321

0123

t

x't

0.0 0.2 0.4 0.6 0.8 1.00.6

0.4

0.2

0.0

0.2

0.4

xt

x't

Graphics of the reconstituted solution

4project_final_result.nb 19

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 A1t2k

2 c2 A1t A1t

k2 c2 A1t2

k22 c2 A1t A1t

k2. solramep1,t, 0, ti, PlotStyle Thick, PlotRange Automatic, 2.6, 2.4,

Frame True, FrameLabel "t", "yt"NDSolve::ndsz : At t 73.98791597135596`, step size is effectively zero; singularity or stiff system suspected. A1t InterpolatingFunction0., 73.9879, t,A1t InterpolatingFunction0., 73.9879, t,A1t InterpolatingFunction0., 73.9879, 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

solorig1 NDSolveJoinEOM, q10 0.01, q20 0.01, q1 '0 0.01,q1t, q2t, t, 0, ti, MaxSteps 1 000 000NDSolve::nderr : Error test failure at t 95.2991109549007`; unable to continue. q1t InterpolatingFunction0., 95.2991, t,

q2t InterpolatingFunction0., 95.2991, tGraphicsArrayPlotq1t . 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

Numerical value for the imperfect system

20 4project_final_result.nb

c1 1, k 2, b3 1

2, b1 1, b2 1, c2 1, 0.01, 0.01, 0.01 0.001

1, 2,1

2, 1, 1, 1, 0.01, 0.01, 0.00001

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

solramep1 NDSolveMBFCE 0, A10 0.01, A10 0.01 ,A1t, A1t, A1t, t, 0, tiNDSolve::ndsz : At t 26.093351632129746`, step size is effectively zero; singularity or stiff system suspected. A1t InterpolatingFunction0., 26.0934, t,A1t InterpolatingFunction0., 26.0934, t,A1t InterpolatingFunction0., 26.0934, t

4project_final_result.nb 21

GraphicsArrayPlotA1t . solramep1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 3.8, 3.8, Frame True, FrameLabel "t", "xt",

PlotA1t . solramep1, t, 0, ti, PlotStyle Thick,PlotRange Automatic, 3.6, 3.4, Frame True, FrameLabel "t", "x't"

ParametricPlotA1t . solramep1, A1t . solramep1, t, 0, 25, PlotStyle Thick,PlotRange Automatic, 0.6, 0.4, Frame True, FrameLabel "xt", "x't"

0 100 200 300 400 500

321

0123

t

xt

0 100 200 300 400 500

321

0123

t

x't

0.0 0.2 0.4 0.6 0.8 1.00.6

0.4

0.2

0.0

0.2

0.4

xt

x't

Graphics of the reconstituted solution

solramep1 NDSolveRBFCE 0, A10 0.01, A10 0.01 ,A1t, A1t, A1t, t, 0, tiNDSolve::ndsz : At t 25.75990904121472`, step size is effectively zero; singularity or stiff system suspected. A1t InterpolatingFunction0., 25.7599, t,A1t InterpolatingFunction0., 25.7599, t,A1t InterpolatingFunction0., 25.7599, t

22 4project_final_result.nb

GraphicsArrayPlotA1t . solramep1, t, 0, ti, PlotStyle Thick,

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

c2 A1t2k

2 c2 A1t A1t

k2 c2 A1t2

k22 c2 A1t A1t

k2. solramep1,t, 0, ti, PlotStyle Thick, PlotRange Automatic, 2.6, 2.4,

Frame True, FrameLabel "t", "yt"

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

solorig1 NDSolveJoinEOM, q10 0.01, q20 0.01, q1 '0 0.01,q1t, q2t, t, 0, ti, MaxSteps 1 000 000NDSolve::nderr : Error test failure at t 43.327453744383035`; unable to continue. q1t InterpolatingFunction0., 43.3275, t,

q2t InterpolatingFunction0., 43.3275, tGraphicsArrayPlotq1t . 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

Proof that the system has an infinite value in finite time

c1 1, k 2, b3 4, b1 1, b2 1, c2 1, 0.01, 0.095, 0.01 0.0011, 2, 4, 1, 1, 1, 0.01, 0.095, 0.00001

4project_final_result.nb 23

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

24 4project_final_result.nb