Birkhoff Normal Forms and KAM Theory for Gumowski … · Gumowski-Mira Equation ... [12] for the...

Research ArticleBirkhoff Normal Forms and KAM Theory forGumowski-Mira Equation

M R S KulenoviT1 Z NurkanoviT2 and E Pilav3

1 Department of Mathematics University of Rhode Island Kingston RI 02881-0816 USA2Department of Mathematics University of Tuzla 75000 Tuzla Bosnia and Herzegovina3Department of Mathematics University of Sarajevo 71000 Sarajevo Bosnia and Herzegovina

Correspondence should be addressed to M R S Kulenovic mkulenovicmailuriedu

Received 29 August 2013 Accepted 19 October 2013 Published 16 January 2014

Academic Editors Z Guo Z Huang X Song and Y Xia

Copyright copy 2014 M R S Kulenovic et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation 119909119899+1

= (2119886119909119899)(1 +

1199092

119899)minus119909119899minus1

119899 = 0 1 where119909minus1 1199090isin (minusinfininfin) andwe obtain the Birkhoffnormal forms for this equation for different equilibrium

solutions

1 Introduction and Preliminaries

The Gumowski-Mira equation [1] is given by

119909119899+1

= 119910119899+ 119865 (119909

119899)

119910119899+1

= minus119909119899+ 119865 (119909

119899+1)

(119899 = 0 1 ) (1)

where 119865 is one of the functions

119886119906 + 119887

119906

1 + 119906

119887

119906

1 + 1199062 119887

1199062

1 + 1199062

120583119906 + (1 minus 120583) 119909119899

119899 = 2 3 120583 isin (minus1 1)

(2)

and the parameters 119886 and 119887 are positiveThese equations wereconsidered by Gumowski and Mira in a series of papers andthe book [1] System (1) implies119909119899+2

= 119910119899+1

+ 119865 (119909119899+1

) = minus119909119899+ 2119865 (119909

119899+1) 119899 = 0 1

(3)and so 119909

119899 satisfies the difference equation119909119899+2

= 2119865 (119909119899+1

) minus 119909119899 119899 = 0 1 (4)

In this paper we will consider (4) with 119865(119906) = 119886119906(1 + 1199062

)where 119886 gt 0 and the initial conditions are real numbers thatis we consider

119911119899+1

=

2119886119911119899

1 + 1199112

119899

minus 119911119899minus1

119899 = 0 1 (5)

where 119886 gt 0 and the initial conditions 119911minus1

and 1199110are real

numbersSeveral authors have studied the Gumowski-Mira equa-

tion (5) and have obtained some results on the stability ofthe equilibrium points the bifurcation of the global behaviorof solutions periodic and chaotic solutions and so forthsee [1ndash11] By using the methods of algebraic and projectivegeometry in [2 3 5 6] the authors obtained very precisedescription of complicated global behavior which includesfinding the feasible periods of all periodic solutions provingthe existence of chaotic solutions through conjugation ofmaps and so forthThesemethods were first used by Zeeman[12] for the study of so-called Lynessrsquo equation All thesemethods are based on the careful algebraic study of aninvariant that (5) possesses Our method here is purelyanalytic and is based on the asymptotic expansions Thismethod can be used to obtain similar results about periodicand chaotic solutions

Our method is based on the application of KAM theory(Kolmogorov-Arnold-Moser) which brings the consideredequation to certain normal form which in addition toinvestigation of stability of an equilibrium can be usedto find different periodic solutions chaotic solutions andso forth This technique was used successfully in [13ndash16]in the case of difference equations while there exists vast

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 819290 8 pageshttpdxdoiorg1011552014819290

2 The Scientific World Journal

literature in the case of differential equations see [17 18]Computer simulations of the trajectories of (5) indicate theexistence of an infinite nested family of invariant closedcurves surrounding an elliptic fixed point sequences ofperiodic islands in the regions between the invariant curvesand stochastic regions surrounding the periodic islands andbetween invariant closed curves Furthermore the entirestructure seems to exhibit self-similar structure

The main feature of (5) is that the corresponding mapis the area-preserving map of the plane having a nonde-generate elliptic fixed point We show that the complicatedorbit structure near the elliptic fixed point is an immediateconsequence of classical results from the KAM theory Awayfrom the elliptic fixed point the KAM theory does notapply and one has to study the geometric structures throughsome other analytical algebraic or geometric methods suchas those in [2 3 5 6 11 12] Most important geometricstructures of interest for (5) are periodic points which aretypically of hyperbolic or elliptic type the invariantmanifoldsassociated with hyperbolic periodic points KAM invariantcurves (around the elliptic fixed point or around ellipticperiodic orbits) and cantori which are remnant sets ofCantor type of destroyed invariant circles The buildingblocks for these structures are the periodic orbits as the othergeometric objects can be obtained as limits of periodic orbits

First we present the basic results that will be used in thesequel See [10 13 17ndash19]

Theorem 1 (Birkhoff normal form) Let F R2 rarr R2 be anarea-preserving 119862119899 map (119899 times continuously differentiable)with a fixed point at the origin whose complex-conjugateeigenvalues 120582 and 120582 are on the unit disk (elliptic fixed point)

Suppose there exists an integer 119897 such that

4 le 119897 le 119899 + 1 (6)

and suppose that the eigenvalues satisfy

120582119896

= 1 119891119900119903 119896 = 3 4 119897 (7)

Let 119903 = [1198972] be the integer part of 1198972Then there exists a smooth function 119892(119911 119911) that vanishes

with its derivatives up to order 119903 minus 1 at 119911 = 0 and there existsa real polynomial

120572 (119908) = 1205721119908 + 120572

21199082

+ sdot sdot sdot + 120572119903119908119903 (8)

such that the map F can be reduced to the normal form bysuitable change of complex coordinates

119911 997888rarr F (119911 119911) = 120582119911119890119894120572(119911119911)

+ 119892 (119911 119911) (9)

In other words the corresponding system of difference equations

x119899+1

= F (x119899) (10)

can be reduced to the form

[

119903119899+1

119904119899+1

] = [

cos120596 minus sin120596sin120596 cos120596 ][

119903119899

119904119899

] + [

119874119897

119874119897

] (11)

where

120596 =

119872

sum

119896=0

120574119896(1199032

119899+ 1199042

119899)

119896

119872 = [

119897

2

] minus 1 (12)

Here 119874119897denotes a convergent power series in 119903

119899and 119904119899with

terms of order greater than or equal to 119897 which vanishes at theorigin and [119909] denotes the least integer greater than or equal to119909

The numbers 1205741 120574

119896are called twist coefficients Using

Theorem 1 we can state the main stability result for an ellipticfixed point known as the KAM theorem (or Kolmogorov-Arnold-Moser theorem) see [10 18 19]

Theorem 2 (KAM theorem) Let F R2 rarr R2 be anarea-preserving map with an elliptic fixed point at the originsatisfying the conditions ofTheorem 1 If the polynomial 120572(|119911|2)is not identically zero then the origin is a stable equilibriumpoint In other words if for some 119896 isin 1 119872 one has 120574

119896= 0

in (12) then the origin is a stable equilibrium point

Remark 3 Consider an invariant annulus 119860120576= 119911 120576 lt

|119911| lt 2120576 in a neighbourhood of the elliptic fixed point for 120576a sufficiently small positive number Note that the linear partof normal form approximation leaves all circles invariantThemotion restricted to each of these circles is a rotation by someangle Also note that if at least one of the twist coefficients 120574

119896is

nonzero the angle of rotation will vary from circle to circleA radial line through the fixed point will undergo twistingunder the mapping The KAM theorem says that under theaddition of the remainder termmost of these invariant circleswill survive as invariant closed curves under the full map[17 18] Precisely the following result holds see [13 18]

Theorem 4 Assuming that 120572(119911119911) is not identically zero and120576 is sufficiently small then the map 119865 has a set of invariantclosed curves of positive Lebesgue measure close to the originalinvariant circles Moreover the relative measure of the setof surviving invariant curves approaches full measure as 120576approaches 0 The surviving invariant closed curves are filledwith dense irrational orbits

The following is a consequence of Moserrsquos twist maptheorem [18 19]

Theorem 5 Let F R2 rarr R2 be an area-preservingdiffeomorphism and (119909 119910) a nondegenerate elliptic fixed pointThere exist periodic points with arbitrarily large period in everyneighbourhood of (119909 119910)

Indeed the theorem implies that arbitrarily close to thefixed point there are always infinitely many gaps betweenconsecutive invariant curves that contain periodic pointsWithin these gaps one finds in general orbits of hyperbolicand elliptic periodic points These facts can hardly be seenfrom computer simulations since some periodic orbits canexist on very small scales

The linearized part of (11) represents a rotation for angle120596and so if 120596 is rational multiple of 120587 every solution is periodic

The Scientific World Journal 3

with the sameperiodwhile if120596 is irrationalmultiple of120587 therewill exist chaotic solutions In this paper we will not go intodetailed study of these behaviors

2 Equilibrium Solutions and LinearizedStability Analysis

Equation (5) has at most three equilibrium points 119910 = 0

for all values of parameter 119886 and 119910 = plusmnradic119886 minus 1 when 119886 gt

1 The linearized equation which corresponds to (5) at anyequilibrium point 119910 is

119911119899+1

minus 2119886

1 minus 1199102

(1 + 1199102

)

2119911119899+ 119911119899minus1

= 0 (13)

and its characteristic equation is

1205822

minus 2119886

1 minus 1199102

(1 + 1199102

)

2120582 + 1 = 0 (14)

which shows that the corresponding map is area preservingThe characteristic equation at 119910 = 0 is

1205822

minus 2119886120582 + 1 = 0 (15)

with characteristic roots 120582plusmn= 119886 plusmn radic119886

2minus 1 When 119886 gt 1

characteristic roots are positive and 120582+gt 1 120582

minusisin (0 1)which

shows that the zero equilibrium in the case 119886 gt 1 is a saddlepoint In the case minus1 lt 119886 lt 1 characteristic roots are complexconjugate numbers lying on the unit circle which means thatthe zero equilibrium is nonhyperbolic of the elliptic type see[10 19] If 119886 = 1 then the characteristic roots are both equal to1 and the zero equilibrium is nonhyperbolic of the parabolictype Similarly if 119886 = minus1 then the characteristic roots areboth equal to minus1 and the zero equilibrium is nonhyperbolicof the parabolic type In this paper we consider the case when119886 = plusmn 1

If 119886 lt minus1 then 120582plusmnlt 0 and we have |120582

+| = |119886 + radic119886

2minus 1| lt

1 |120582minus| = |119886 minus radic119886

2minus 1| gt 1 which shows that the zero

equilibrium is a saddle pointThe characteristic equation at 119910 = plusmnradic119886 minus 1 is

1205822

minus

2 (2 minus 119886)

119886

120582 + 1 = 0 (16)

with characteristic roots120582plusmn= (2minus119886plusmn2radic1 minus 119886)119886When 119886 gt 1

characteristic roots are complex conjugate numbers lying onthe unit circle which means that the non-zero equilibriumsolutions are nonhyperbolic of the elliptic type see [10 19]

3 KAM Theory Applied to (5) at ZeroEquilibrium for 119886 isin (minus1 1) minus(12)0

First we use the substitution

119911119899= 119910119899

119911119899+1

= 119909119899

(17)

to transform (5) to the system

119909119899+1

=

2119886119909119899

1 + 1199092

119899

minus 119910119899

119910119899+1

= 119909119899

119899 = 0 1 (18)

The corresponding linearized system at 1198640(0 0) is

119883119899+1

= 2119886119883119899minus 119884119899

119884119899+1

= 119883119899

119899 = 0 1 (19)

The characteristic equation of (19) is (15) with characteristicroots 120582

plusmn= 119886plusmnradic119886

2minus 1 We consider the case where minus1 lt 119886 lt

1 in which case 1198640is nonhyperbolic equilibrium of elliptic

typeA straightforward calculation gives the following expres-

sions for second third and fourth power of the characteristicroot

1205822

= (119886 + 119894radic1 minus 1198862)

2

= 21198862

minus 1 + 2119886119894radic1 minus 1198862

1205823

= 119886 (41198862

minus 3) + (41198862

minus 1) 119894radic1 minus 1198862

1205824

= 1 + 119886 (81198863

minus 8119886 + 4 (21198862

minus 1) 119894radic1 minus 1198862)

(20)

Clearly 1205823 = 1 and 1205824 = 1 for 119886 isin (minus1 1)minus(12) 0Thus theassumptions of Theorem 1 are satisfied for 119897 = 4 and we willfind the Birkhoff normal form of (18) by using the sequenceof transformations described in Section 1

31 First Transformation Notice that the matrix of thelinearized system (19) at the origin is given as

1198690= [

2119886 minus1

1 0] (21)

A straightforward calculation shows that thematrix of thecorresponding eigenvectors which correspond to 120582 and 120582 of1198690is

119875 = [

1 1

120582 120582

] (22)

In order to obtain the Birkhoff normal formof system (18)wewill expand the right-hand sides of the equations of system(18) at the equilibrium point (0 0) up to the order 119897 minus 1 = 3We obtain

119909119899+1

= 2119886119909119899minus 119910119899minus 2119886119909

3

119899+ 1198744

119910119899+1

= 119909119899

(23)

Now the change of variables

[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (24)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(25)


transforms system (23) into

119906119899+1

+ V119899+1

= 2119886 (119906119899+ V119899) minus (120582119906

119899+ 120582V119899)

minus 2119886(119906119899+ V119899)3

+ 1198744

120582119906119899+1

+ 120582V119899+1

= 119906119899+ V119899

(26)

which after some simplifications becomes

119906119899+1

= 120582119906119899+ 120590(119906

119899+ V119899)3

+ 1198744

V119899+1

= 120582V119899+ 120590(119906

119899+ V119899)3

+ 1198744

(27)

where

120590 = minus

2120582119886

120582 minus 120582

= 119886(minus1 +

119886

radic1 minus 1198862

119894)

1205822

minus 2119886120582 + 1 = 0 120582120582 = 1 120582 + 120582 = 2119886

(28)

32 SecondTransformation Theobjective of second transfor-mation is to obtain the nonlinear terms up to order 119897 minus 1 innormal form The change of variables

119906119899= 120585119899+ 1206012(120585119899 120578119899) + 1206013(120585119899 120578119899)

V119899= 120578119899+ 1205952(120585119899 120578119899) + 1205953(120585119899 120578119899)

(119899 = 0 1 )

(29)

where

120601119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119896minus119895

120578119895

120595119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119895

120578119896minus119895

(30)

for 119896 = 2 and 119896 = 3 yields

119906119899= 120585119899+ (119886201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

V119899= 120578119899+ (119886201205782

119899+ 11988621120585119899120578119899+ 119886221205852

119899)

+ (119886301205783

119899+ 119886311205851198991205782

119899+ 119886321205852

119899120578119899+ 119886331205853

119899)

1199062

119899= 1205852

119899+ 2119886201205853

119899+ 2119886211205852

119899120578119899+ 2119886221205851198991205782

119899+ 1198744

V2119899= 1205782

119899+ 2119886201205783

119899+ 2119886211205851198991205782

119899+ 2119886221205852

119899120578119899+ 1198744

1199063

119899= 1205853

119899+ 1198744

V3119899= 1205783

119899+ 1198744

1199062

119899V119899= 1205852

119899120578119899+ 1198744

119906119899V2119899= 1205851198991205782

119899+ 1198744

119906119899V119899= 119886221205853

119899+ (11988620+ 11988621) 1205852

119899120578119899

+ (11988621+ 11988620) 1205851198991205782

119899+ 120585119899120578119899+ 119886221205783

119899+ 1198744

(31)

Solving these equations for 120585119899+1

and 120578119899+1

one obtain

120585119899+1

= (120582120585119899+ 12057221205852

119899120578119899) + 1198744

120578119899+1

= (120582120578119899+ 12057221205851198991205782

119899) + 1198744

119899 = 0 1

(32)

By using (32) and 120582 = (2119886 minus 120582) in first equality of (31) andrescaling by replacing 119899 with 119899 + 1 we have

119906119899+1

= 120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899

+ (1205722+ 11988631120582) 1205852

119899120578119899+ 119886321205821205851198991205782

119899

+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

(33)

By using (33) in the left-hand side and (31) in the right-handside of (27) we obtains

120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899+ (1205722+ 11988631120582) 1205852

119899120578119899

+ 119886321205821205851198991205782

119899+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

= 120582120585119899+ 120582 (119886

201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ 120582 (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

+ 120590(120585119899+ 120578119899)3

+ 1198744

(34)

The last relation holds if the corresponding coefficients areequal which leads to the following set of equalities

1205852

119899 (1205822

minus 120582) 11988620= 0

1205853

119899 (1205823

minus 120582 minus 120590) 11988630= 0

120585119899120578119899 (1 minus 120582) 119886

21= 0

1205852

119899120578119899 1205722minus 3120590 = 0

(35)

Now

11988620= 11988621= 0

1205722= 3120590 = 3119886(minus1 + 119894

119886radic1 minus 1198862

1 minus 1198862

)

(36)

This implies

Re (1205722) = minus3119886 (37)

33ThirdTransformation Theobjective of the third transfor-mation consists in expressing the terms in (32) as real valuesThis is achieved by using the transformation

120585119899= 119903119899+ 119894119904119899

120578119899= 119903119899minus 119894119904119899

(38)


minus3 minus2 minus1 1 2 3

3

2

1

minus1

minus2

minus3

(a)

15

10

05

00

minus05

minus10

minus15

minus10 minus05 00 05 10

(b)

Figure 1 (a) A Phase portrait of three orbits for 119886 = 05 (b) A bifurcation diagram of typical solution of (5) for 119886 between minus1 and 1 The plotsare produced by Dynamica 3 [10]

Comparing the system obtained with (11) and using (12) for119897 = 4 we determine the twist coefficients 120574

0and 1205741

We have

cos 1205740= Re (120582) 120574

1=

minus1

sin 1205740

sdot Re (1205722) (39)

In order to apply the KAM theorem we have to show that1205741

= 0 Indeed

minus1 lt cos 1205740= 119886 lt 1

sin 1205740= plusmnradic1 minus 119886

2

1205741=

minus1

plusmnradic1 minus 1198862

(minus3119886) = plusmn

3119886


(40)

Since 119886 isin (minus1 1) minus(12) 0 this implies 1205741

= 0 (Figure 1)Thus we have proved the following result

Theorem 6 The zero equilibrium solution of (5) is stable for119886 isin (minus1 1) minus(12) 0

4 KAM Theory Applied to (5) at NonzeroEquilibrium Solutions for 119886 isin (1+infin) 2 4

In this section we will apply KAM theory to establish thestability of the non-zero equilibrium solutions for (5) Firstwe rescale (5) and then we use the substitution

119911119899= 119909119899+ radic119886 minus 1

119910119899= 119909119899minus1

119899 = 0 1 (41)

to obtain the system

119909119899+1

=

2119886 (119909119899+ radic119886 minus 1)

1 + (119909119899+ radic119886 minus 1)

2minus 119910119899minus 2radic119886 minus 1

119910119899+1

= 119909119899

(42)

with the corresponding equilibrium point at the origin

The linearized system of system (42) at (0 0) is

119883119899+1

=

2 (2 minus 119886)

119886

119883119899minus 119884119899

119884119899+1

= 119883119899

(119899 = 0 1 ) (43)

whose characteristic equation is (16) and the characteristicroots are 120582

plusmn= (2 minus 119886 plusmn 2radic1 minus 119886)119886 As we mentioned

earlier when 119886 gt 1 the characteristic roots are com-plex conjugate numbers lying on the unit circle whichmeans that the non-zero equilibrium is nonhyperbolic ofthe elliptic type and so KAM theory is a natural tool to beapplied

A straightforward calculation gives the following expres-sions for second third and fourth power of the characteristicroot

120582 =

2 minus 119886 + 2119894radic119886 minus 1

119886

1205822

=

1

1198862(8 minus 8119886 + 119886

2

+ 4119894 (2 minus 119886)radic119886 minus 1)

1205823

=

1

1198863((2 minus 119886) (16 minus 16119886 + 119886

2

)

+2 (16 minus 16119886 + 31198862

) 119894radic119886 minus 1)

1205823

= 1 lArrrArr 119886 = 4

1205824

=

1

1198864((8 minus 8119886 + 119886

2

)

2

minus 16(2 minus 119886)2

(119886 minus 1)

+8119894 (8 minus 8119886 + 1198862

) (2 minus 119886)radic119886 minus 1)

1205824

= 1 lArrrArr 119886 = 2

(44)

It can be shown that 1205823 = 1 and 1205824 = 1 for 119886 isin (1 +infin) 2 4

and so 119897 = 4


41 First Transformation Notice that the matrix of thelinearized system (43) is given as

1198690=[

[

2 (2 minus 119886)

119886

minus1

1 0

]

]

(45)

A straightforward calculation shows that the matrix of thecorresponding eigenvectors which correspond to 120582 and 120582 of1198690is

119875 = [

1 1

120582 120582

] (46)

In order to obtain the Birkhoff normal form of system (42)we will expand the right hand sides of the equations of system(42) at the equilibrium point (0 0) up to the order 119897 minus 1 = 3We obtain

119909119899+1

=

minus2 (119886 minus 2)

119886

119909119899minus 119910119899

+

2 (119886 minus 4)radic119886 minus 1

1198862

times (1199092

119899minus

(1198862

minus 8119886 + 8)


1199093

119899) + 119874

4

119910119899+1

= 119909119899

(47)

Next we use the change of variables

[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (48)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(49)

and after tedious simplification we obtain the transformedsystem up to the terms of order three in the form

119906119899+1

= 120582119906119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

V119899+1

= 120582V119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

(50)

where

120590 =

120582

120582 minus 120582

sdot


1198862

1205901= minus

(1198862

minus 8119886 + 8)


(51)

42 Second Transformation Similarly as in the case of thezero equilibrium we obtain

11988620=

120590

1205822minus 120582

=


1198862(120582 minus 1) (120582 minus 120582)

11988621=

minus2120590

120582 minus 1

=

(119886 minus 4) (radic119886 minus 1 + 119894 (119886 minus 1))

2119886 (119886 minus 1)

11988622=

120590

120582

2

minus 120582

2 (11988620+ 11988622) =


1198862

1

120582 minus 120582

(

2

120582 minus 1

minus

2

1205823minus 1

)

2 (11988620+ 11988622) = (


1198862

)(

minus119886119894

4radic119886 minus 1

)

times (

2119894119886radic119886 minus 1

(119886 minus 1) (119886 minus 4)

) =

1

radic119886 minus 1

1205722= 120590 (4Re (119886

21) + 2 (119886

20+ 11988622) + 3120590

1)

(52)

Now we simplify the right-hand sides of these expressionsbecause the coefficient 120572

2plays the crucial role in determin-

ing stability of the equilibrium We have

4Re (11988621) + 2 (119886

20+ 11988622) + 3120590

1

=


119886 (119886 minus 1)

+

1

radic119886 minus 1

minus

3 (1198862

minus 8119886 + 8)


=

4 (119886 + 2)

119886radic119886 minus 1 (119886 minus 4)

1205722= (

(2 minus 119886) 119894 minus 2radic119886 minus 1

minus4radic119886 minus 1

)(


1198862

)

times (

4 (119886 + 2)


)

=

2 (119886 + 2) (2radic119886 minus 1 + 119894 (119886 minus 2))

1198863radic119886 minus 1

(53)

The real part of last expression is

Re (1205722) =

4 (119886 + 2)

1198863

(54)



6

4

2

minus2

minus4

minus6

(a)

minus4 minus2 2 4

4

2

minus2

minus4

(b)

Figure 2 (a) A phase portrait of three orbits for 119886 = 15 (b) A phase portrait of three orbits for 119886 = 7 The plots are produced by Dynamica3 [10]

5

4

3

2

1

0540 545 550 555 560 565 570

Figure 3 A bifurcation diagram in the (119886 119909)-plane for 119909minus1

= 12

and 1199090= 10 with parameter 119886 in the range from 54 to 57 The plot

is produced by Dynamica 3 [10]

43 Third Transformation Using (39) we have

minus1 lt cos 1205740=

2 minus 119886

119886

lt 1

sin 1205740= plusmnradic1 minus (

2 minus 119886

119886

)

2

= plusmn

2

119886

radic119886 minus 1

1205741= plusmn

2 (119886 + 2)

1198862radic119886 minus 1

(55)

So we have that 1205741

= 0 for all 119886 isin (1 +infin) 2 4The computation for the other non-zero equilibrium

solution minusradic119886 minus 1 is similarThus we have proved the following result

Theorem 7 The equilibrium solution 119910plusmn= plusmnradic119886 minus 1 of (5) is

stable for 119886 isin (1 +infin) 2 4

Remark 8 The cases 119886 = 2 and 119886 = 4 can be treated by themethod of local Lyapunov function as in [6 9 10 20] To doso one needs the corresponding invariant

119868 (119909119899 119909119899minus1

) = 1199092

1198991199092

119899minus1+ 1199092

119899+ 1199092

119899minus1minus 119886119909119899119909119899minus1

(56)

which assumes a minimum value at the isolated equilibriumpoint (119910 119910) and by Morsersquos lemma see [19 20] the levelsets 119868(119909 119910) = 119862 are diffeomorphic to the circles in theneighborhood of (119910 119910)Thismethod can be extended furtherto give some global results on the dynamics of (5) as it wasdone in [6 21] (Figures 2 and 3)

Remark 9 Theorems 6 and 7 show that (5) undergoes a bifur-cation as the parameter 119886 passes through 1 Precisely as theparameter 119886 passes through 1 the zero equilibrium changesits local character from a non-hyperbolic equilibrium pointof elliptic type when minus1 lt 119886 lt 1 to a saddle point when119886 isin (minusinfin minus1) cup (1 +infin) where at the critical value 119886 = plusmn1zero equilibrium is a non-hyperbolic equilibrium point ofparabolic type The positive equilibrium solutions plusmnradic119886 minus 1

are always the non-hyperbolic equilibrium points of elliptictype The global change of behavior is that zero equilibriumloses its stability as the parameter 119886 passes through plusmn1 andits stability is picked up by the positive equilibrium So thisbifurcation can be described as the exchange of stabilitybifurcation The remaining case is dynamics at 119886 = plusmn1 inwhich case the zero equilibrium which is unique is a non-hyperbolic equilibrium point of parabolic type Finally if 119886 =0 (5) reduces to

119911119899+1

= minus119911119899minus1

119899 = 0 1 (57)

which has the unique zero equilibrium and all solutions areperiodic of period two in which case we have some trivialstability of equilibrium


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Gumowski and C Mira Recurrences and Discrete DynamicSystems Lecture Notes in Mathematics Springer Berlin Ger-many 1980

[2] G Bastien and M Rogalski ldquoOn the algebraic differenceequations 119906

119899+2119906119899= 120595(119906

119899+1) in R+

lowast related to a family of elliptic

quartics in the planerdquo Advances in Difference Equations vol2005 no 3 pp 227ndash261 2005


119899+2+ 119906119899

= 120595(119906119899+1

) in R related to a family ofelliptic quartics in the planerdquo Journal of Mathematical Analysisand Applications vol 326 no 2 pp 822ndash844 2007

[4] F Beukers and R Cushman ldquoZeemanrsquos monotonicity conjec-turerdquo Journal of Differential Equations vol 143 no 1 pp 191ndash200 1998

[5] A Cima A Gasull and V Manosa ldquoDynamics of rational dis-crete dynamical systems via first integralsrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 16 pp 631ndash645 2006

[6] A Cima A Gasull and V Manosa ldquoNon-autonomous two-periodic Gumowski-Mira difference equationrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 14 pages 2012

[7] C A Clark E J Janowski and M R S Kulenovic ldquoStabilityof the Gumowski-Mira equation with period-two coefficientrdquoJournal of Mathematical Analysis and Applications vol 307 no1 pp 292ndash304 2005

[8] E J Janowski M R S Kulenovic and Z Nurkanovic ldquoStabilityof the kTH order lynessrsquo equation with a period-k coefficientrdquoInternational Journal of Bifurcation and Chaos vol 17 no 1 pp143ndash152 2007

[9] M R S Kulenovic ldquoInvariants and related Liapunov functionsfor difference equationsrdquo Applied Mathematics Letters vol 13no 7 pp 1ndash8 2000

[10] M R S Kulenovic and O Merino Discrete Dynamical Systemsand Difference Equations with Mathematica Chapman andHallCRC Boca Raton Fla USA 2002

[11] R S MacKay Renormalization in Area-Preserving Maps WorldScientific River Edge NJ USA 1993

[12] E C Zeeman Geometric Unfolding of A Difference EquationHertford College Oxford UK 1996

[13] M Gidea J D Meiss James I Ugarcovici and H WeissldquoApplications of KAM theory to population dynamicsrdquo Journalof Biological Dynamics vol 5 no 1 pp 44ndash63 2011

[14] V L Kocic G Ladas G Tzanetopoulos and EThomas ldquoOn thestability of Lynessrsquo equationrdquo Dynamics of Continuous Discreteand Impulsive Systems vol 1 pp 245ndash254 1995

[15] M R S Kulenovic and Z Nurkanovic ldquoStability of Lynessrsquoequationwith period-two coeffcient viaKAMtheoryrdquo Journal ofConcrete and Applicable Mathematics vol 6 pp 229ndash245 2008

[16] G Ladas G Tzanetopoulos and A Tovbis ldquoOnMayrsquos host par-asitoidmodelrdquo Journal of Difference Equations and Applicationsvol 2 pp 195ndash204 1996

[17] M Tabor Chaos and Integrability in Nonlinear Dynamics AnIntroduction Wiley-Interscience New York NY USA 1989

[18] C Siegel and J Moser Lectures on Celestial Mechanics SpringerNew York NY USA 1971

[19] J K Hale and H Kocak Dynamics and Bifurcation SpringerNew York NY USA 1991

[20] G Bastien and M Rogalski ldquoLevel sets lemmas and unicityof critical points of invariants tools for local stability andtopological properties of dynamical systemsrdquo Sarajevo Journalof Mathematics vol 21 pp 273ndash282 2012

[21] J DuistermaatDiscrete Integrable Systems QRTMaps and Ellip-tic Surfaces Springer Monographs in Mathematics SpringerNew York NY USA 2010


literature in the case of differential equations see [17 18]Computer simulations of the trajectories of (5) indicate theexistence of an infinite nested family of invariant closedcurves surrounding an elliptic fixed point sequences ofperiodic islands in the regions between the invariant curvesand stochastic regions surrounding the periodic islands andbetween invariant closed curves Furthermore the entirestructure seems to exhibit self-similar structure

The main feature of (5) is that the corresponding mapis the area-preserving map of the plane having a nonde-generate elliptic fixed point We show that the complicatedorbit structure near the elliptic fixed point is an immediateconsequence of classical results from the KAM theory Awayfrom the elliptic fixed point the KAM theory does notapply and one has to study the geometric structures throughsome other analytical algebraic or geometric methods suchas those in [2 3 5 6 11 12] Most important geometricstructures of interest for (5) are periodic points which aretypically of hyperbolic or elliptic type the invariantmanifoldsassociated with hyperbolic periodic points KAM invariantcurves (around the elliptic fixed point or around ellipticperiodic orbits) and cantori which are remnant sets ofCantor type of destroyed invariant circles The buildingblocks for these structures are the periodic orbits as the othergeometric objects can be obtained as limits of periodic orbits

First we present the basic results that will be used in thesequel See [10 13 17ndash19]

Theorem 1 (Birkhoff normal form) Let F R2 rarr R2 be anarea-preserving 119862119899 map (119899 times continuously differentiable)with a fixed point at the origin whose complex-conjugateeigenvalues 120582 and 120582 are on the unit disk (elliptic fixed point)

Suppose there exists an integer 119897 such that

4 le 119897 le 119899 + 1 (6)

and suppose that the eigenvalues satisfy

120582119896

= 1 119891119900119903 119896 = 3 4 119897 (7)

Let 119903 = [1198972] be the integer part of 1198972Then there exists a smooth function 119892(119911 119911) that vanishes

with its derivatives up to order 119903 minus 1 at 119911 = 0 and there existsa real polynomial

120572 (119908) = 1205721119908 + 120572

21199082

+ sdot sdot sdot + 120572119903119908119903 (8)

such that the map F can be reduced to the normal form bysuitable change of complex coordinates

119911 997888rarr F (119911 119911) = 120582119911119890119894120572(119911119911)

+ 119892 (119911 119911) (9)

In other words the corresponding system of difference equations

x119899+1

= F (x119899) (10)

can be reduced to the form

[

119903119899+1

119904119899+1

] = [

cos120596 minus sin120596sin120596 cos120596 ][

119903119899

119904119899

] + [

119874119897

119874119897

] (11)

where

120596 =

119872

sum

119896=0

120574119896(1199032

119899+ 1199042

119899)

119896

119872 = [

119897

2

] minus 1 (12)

Here 119874119897denotes a convergent power series in 119903

119899and 119904119899with

terms of order greater than or equal to 119897 which vanishes at theorigin and [119909] denotes the least integer greater than or equal to119909

The numbers 1205741 120574

119896are called twist coefficients Using

Theorem 1 we can state the main stability result for an ellipticfixed point known as the KAM theorem (or Kolmogorov-Arnold-Moser theorem) see [10 18 19]

Theorem 2 (KAM theorem) Let F R2 rarr R2 be anarea-preserving map with an elliptic fixed point at the originsatisfying the conditions ofTheorem 1 If the polynomial 120572(|119911|2)is not identically zero then the origin is a stable equilibriumpoint In other words if for some 119896 isin 1 119872 one has 120574

119896= 0

in (12) then the origin is a stable equilibrium point

Remark 3 Consider an invariant annulus 119860120576= 119911 120576 lt

|119911| lt 2120576 in a neighbourhood of the elliptic fixed point for 120576a sufficiently small positive number Note that the linear partof normal form approximation leaves all circles invariantThemotion restricted to each of these circles is a rotation by someangle Also note that if at least one of the twist coefficients 120574

119896is

nonzero the angle of rotation will vary from circle to circleA radial line through the fixed point will undergo twistingunder the mapping The KAM theorem says that under theaddition of the remainder termmost of these invariant circleswill survive as invariant closed curves under the full map[17 18] Precisely the following result holds see [13 18]

Theorem 4 Assuming that 120572(119911119911) is not identically zero and120576 is sufficiently small then the map 119865 has a set of invariantclosed curves of positive Lebesgue measure close to the originalinvariant circles Moreover the relative measure of the setof surviving invariant curves approaches full measure as 120576approaches 0 The surviving invariant closed curves are filledwith dense irrational orbits

The following is a consequence of Moserrsquos twist maptheorem [18 19]

Theorem 5 Let F R2 rarr R2 be an area-preservingdiffeomorphism and (119909 119910) a nondegenerate elliptic fixed pointThere exist periodic points with arbitrarily large period in everyneighbourhood of (119909 119910)

Indeed the theorem implies that arbitrarily close to thefixed point there are always infinitely many gaps betweenconsecutive invariant curves that contain periodic pointsWithin these gaps one finds in general orbits of hyperbolicand elliptic periodic points These facts can hardly be seenfrom computer simulations since some periodic orbits canexist on very small scales

The linearized part of (11) represents a rotation for angle120596and so if 120596 is rational multiple of 120587 every solution is periodic







119911119899+1

minus 2119886

1 minus 1199102

(1 + 1199102

)

2119911119899+ 119911119899minus1

= 0 (13)


1205822

minus 2119886

1 minus 1199102

(1 + 1199102

)

2120582 + 1 = 0 (14)


1205822

minus 2119886120582 + 1 = 0 (15)







+| = |119886 + radic119886

2minus 1| lt




1205822

minus

2 (2 minus 119886)

119886

120582 + 1 = 0 (16)





119911119899= 119910119899

119911119899+1

= 119909119899

(17)


119909119899+1

=

2119886119909119899

1 + 1199092

119899

minus 119910119899

119910119899+1

= 119909119899

119899 = 0 1 (18)


119883119899+1

= 2119886119883119899minus 119884119899

119884119899+1

= 119883119899

119899 = 0 1 (19)







1205822

= (119886 + 119894radic1 minus 1198862)

2

= 21198862


1205823

= 119886 (41198862

minus 3) + (41198862


1205824

= 1 + 119886 (81198863

minus 8119886 + 4 (21198862


(20)



1198690= [

2119886 minus1

1 0] (21)


119875 = [

1 1

120582 120582

] (22)


119909119899+1

= 2119886119909119899minus 119910119899minus 2119886119909

3

119899+ 1198744

119910119899+1

= 119909119899

(23)


[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (24)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(25)



119906119899+1

+ V119899+1

= 2119886 (119906119899+ V119899) minus (120582119906

119899+ 120582V119899)

minus 2119886(119906119899+ V119899)3

+ 1198744

120582119906119899+1

+ 120582V119899+1

= 119906119899+ V119899

(26)


119906119899+1

= 120582119906119899+ 120590(119906

119899+ V119899)3

+ 1198744

V119899+1

= 120582V119899+ 120590(119906

119899+ V119899)3

+ 1198744

(27)

where

120590 = minus

2120582119886

120582 minus 120582

= 119886(minus1 +

119886


119894)

1205822

minus 2119886120582 + 1 = 0 120582120582 = 1 120582 + 120582 = 2119886

(28)


119906119899= 120585119899+ 1206012(120585119899 120578119899) + 1206013(120585119899 120578119899)

V119899= 120578119899+ 1205952(120585119899 120578119899) + 1205953(120585119899 120578119899)

(119899 = 0 1 )

(29)

where

120601119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119896minus119895

120578119895

120595119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119895

120578119896minus119895

(30)

for 119896 = 2 and 119896 = 3 yields

119906119899= 120585119899+ (119886201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

V119899= 120578119899+ (119886201205782

119899+ 11988621120585119899120578119899+ 119886221205852

119899)

+ (119886301205783

119899+ 119886311205851198991205782

119899+ 119886321205852

119899120578119899+ 119886331205853

119899)

1199062

119899= 1205852

119899+ 2119886201205853

119899+ 2119886211205852

119899120578119899+ 2119886221205851198991205782

119899+ 1198744

V2119899= 1205782

119899+ 2119886201205783

119899+ 2119886211205851198991205782

119899+ 2119886221205852

119899120578119899+ 1198744

1199063

119899= 1205853

119899+ 1198744

V3119899= 1205783

119899+ 1198744

1199062

119899V119899= 1205852

119899120578119899+ 1198744

119906119899V2119899= 1205851198991205782

119899+ 1198744

119906119899V119899= 119886221205853

119899+ (11988620+ 11988621) 1205852

119899120578119899

+ (11988621+ 11988620) 1205851198991205782

119899+ 120585119899120578119899+ 119886221205783

119899+ 1198744

(31)


and 120578119899+1

one obtain

120585119899+1

= (120582120585119899+ 12057221205852

119899120578119899) + 1198744

120578119899+1

= (120582120578119899+ 12057221205851198991205782

119899) + 1198744

119899 = 0 1

(32)


119906119899+1

= 120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899

+ (1205722+ 11988631120582) 1205852

119899120578119899+ 119886321205821205851198991205782

119899

+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

(33)


120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899+ (1205722+ 11988631120582) 1205852

119899120578119899

+ 119886321205821205851198991205782

119899+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

= 120582120585119899+ 120582 (119886

201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ 120582 (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

+ 120590(120585119899+ 120578119899)3

+ 1198744

(34)


1205852

119899 (1205822

minus 120582) 11988620= 0

1205853

119899 (1205823

minus 120582 minus 120590) 11988630= 0

120585119899120578119899 (1 minus 120582) 119886

21= 0

1205852

119899120578119899 1205722minus 3120590 = 0

(35)

Now

11988620= 11988621= 0

1205722= 3120590 = 3119886(minus1 + 119894

119886radic1 minus 1198862

1 minus 1198862

)

(36)

This implies

Re (1205722) = minus3119886 (37)


120585119899= 119903119899+ 119894119904119899

120578119899= 119903119899minus 119894119904119899

(38)



3

2

1

minus1

minus2

minus3

(a)

15

10

05

00

minus05

minus10

minus15


(b)



0and 1205741

We have

cos 1205740= Re (120582) 120574

1=

minus1

sin 1205740

sdot Re (1205722) (39)


= 0 Indeed

minus1 lt cos 1205740= 119886 lt 1


2

1205741=

minus1



3119886


(40)






119911119899= 119909119899+ radic119886 minus 1

119910119899= 119909119899minus1

119899 = 0 1 (41)


119909119899+1

=

2119886 (119909119899+ radic119886 minus 1)

1 + (119909119899+ radic119886 minus 1)


119910119899+1

= 119909119899

(42)



119883119899+1

=

2 (2 minus 119886)

119886

119883119899minus 119884119899

119884119899+1

= 119883119899

(119899 = 0 1 ) (43)





120582 =


119886

1205822

=

1

1198862(8 minus 8119886 + 119886

2


1205823

=

1

1198863((2 minus 119886) (16 minus 16119886 + 119886

2

)

+2 (16 minus 16119886 + 31198862

) 119894radic119886 minus 1)

1205823

= 1 lArrrArr 119886 = 4

1205824

=

1

1198864((8 minus 8119886 + 119886

2

)

2


(119886 minus 1)

+8119894 (8 minus 8119886 + 1198862


1205824

= 1 lArrrArr 119886 = 2

(44)


and so 119897 = 4



1198690=[

[

2 (2 minus 119886)

119886

minus1

1 0

]

]

(45)


119875 = [

1 1

120582 120582

] (46)


119909119899+1

=


119886

119909119899minus 119910119899

+


1198862

times (1199092

119899minus

(1198862

minus 8119886 + 8)


1199093

119899) + 119874

4

119910119899+1

= 119909119899

(47)


[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (48)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(49)


119906119899+1

= 120582119906119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

V119899+1

= 120582V119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

(50)

where

120590 =

120582

120582 minus 120582

sdot


1198862

1205901= minus

(1198862

minus 8119886 + 8)


(51)


11988620=

120590

1205822minus 120582

=


1198862(120582 minus 1) (120582 minus 120582)

11988621=

minus2120590

120582 minus 1

=


2119886 (119886 minus 1)

11988622=

120590

120582

2

minus 120582

2 (11988620+ 11988622) =


1198862

1

120582 minus 120582

(

2

120582 minus 1

minus

2

1205823minus 1

)

2 (11988620+ 11988622) = (


1198862

)(

minus119886119894


)

times (

2119894119886radic119886 minus 1

(119886 minus 1) (119886 minus 4)

) =

1

radic119886 minus 1

1205722= 120590 (4Re (119886

21) + 2 (119886

20+ 11988622) + 3120590

1)

(52)




4Re (11988621) + 2 (119886

20+ 11988622) + 3120590

1

=


119886 (119886 minus 1)

+

1

radic119886 minus 1

minus

3 (1198862

minus 8119886 + 8)


=

4 (119886 + 2)


1205722= (



)(


1198862

)

times (

4 (119886 + 2)


)

=


1198863radic119886 minus 1

(53)


Re (1205722) =

4 (119886 + 2)

1198863

(54)



6

4

2

minus2

minus4

minus6

(a)

minus4 minus2 2 4

4

2

minus2

minus4

(b)


5

4

3

2

1

0540 545 550 555 560 565 570


= 12





2 minus 119886

119886

lt 1


2 minus 119886

119886

)

2

= plusmn

2

119886

radic119886 minus 1

1205741= plusmn

2 (119886 + 2)

1198862radic119886 minus 1

(55)







119868 (119909119899 119909119899minus1

) = 1199092

1198991199092

119899minus1+ 1199092

119899+ 1199092


(56)




119911119899+1


119899 = 0 1 (57)





References



119899+2119906119899= 120595(119906

119899+1) in R+




119899+2+ 119906119899

= 120595(119906119899+1


























119911119899+1

minus 2119886

1 minus 1199102

(1 + 1199102

)

2119911119899+ 119911119899minus1

= 0 (13)


1205822

minus 2119886

1 minus 1199102

(1 + 1199102

)

2120582 + 1 = 0 (14)


1205822

minus 2119886120582 + 1 = 0 (15)







+| = |119886 + radic119886

2minus 1| lt




1205822

minus

2 (2 minus 119886)

119886

120582 + 1 = 0 (16)





119911119899= 119910119899

119911119899+1

= 119909119899

(17)


119909119899+1

=

2119886119909119899

1 + 1199092

119899

minus 119910119899

119910119899+1

= 119909119899

119899 = 0 1 (18)


119883119899+1

= 2119886119883119899minus 119884119899

119884119899+1

= 119883119899

119899 = 0 1 (19)







1205822

= (119886 + 119894radic1 minus 1198862)

2

= 21198862


1205823

= 119886 (41198862

minus 3) + (41198862


1205824

= 1 + 119886 (81198863

minus 8119886 + 4 (21198862


(20)



1198690= [

2119886 minus1

1 0] (21)


119875 = [

1 1

120582 120582

] (22)


119909119899+1

= 2119886119909119899minus 119910119899minus 2119886119909

3

119899+ 1198744

119910119899+1

= 119909119899

(23)


[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (24)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(25)



119906119899+1

+ V119899+1

= 2119886 (119906119899+ V119899) minus (120582119906

119899+ 120582V119899)

minus 2119886(119906119899+ V119899)3

+ 1198744

120582119906119899+1

+ 120582V119899+1

= 119906119899+ V119899

(26)


119906119899+1

= 120582119906119899+ 120590(119906

119899+ V119899)3

+ 1198744

V119899+1

= 120582V119899+ 120590(119906

119899+ V119899)3

+ 1198744

(27)

where

120590 = minus

2120582119886

120582 minus 120582

= 119886(minus1 +

119886


119894)

1205822

minus 2119886120582 + 1 = 0 120582120582 = 1 120582 + 120582 = 2119886

(28)


119906119899= 120585119899+ 1206012(120585119899 120578119899) + 1206013(120585119899 120578119899)

V119899= 120578119899+ 1205952(120585119899 120578119899) + 1205953(120585119899 120578119899)

(119899 = 0 1 )

(29)

where

120601119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119896minus119895

120578119895

120595119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119895

120578119896minus119895

(30)

for 119896 = 2 and 119896 = 3 yields

119906119899= 120585119899+ (119886201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

V119899= 120578119899+ (119886201205782

119899+ 11988621120585119899120578119899+ 119886221205852

119899)

+ (119886301205783

119899+ 119886311205851198991205782

119899+ 119886321205852

119899120578119899+ 119886331205853

119899)

1199062

119899= 1205852

119899+ 2119886201205853

119899+ 2119886211205852

119899120578119899+ 2119886221205851198991205782

119899+ 1198744

V2119899= 1205782

119899+ 2119886201205783

119899+ 2119886211205851198991205782

119899+ 2119886221205852

119899120578119899+ 1198744

1199063

119899= 1205853

119899+ 1198744

V3119899= 1205783

119899+ 1198744

1199062

119899V119899= 1205852

119899120578119899+ 1198744

119906119899V2119899= 1205851198991205782

119899+ 1198744

119906119899V119899= 119886221205853

119899+ (11988620+ 11988621) 1205852

119899120578119899

+ (11988621+ 11988620) 1205851198991205782

119899+ 120585119899120578119899+ 119886221205783

119899+ 1198744

(31)


and 120578119899+1

one obtain

120585119899+1

= (120582120585119899+ 12057221205852

119899120578119899) + 1198744

120578119899+1

= (120582120578119899+ 12057221205851198991205782

119899) + 1198744

119899 = 0 1

(32)


119906119899+1

= 120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899

+ (1205722+ 11988631120582) 1205852

119899120578119899+ 119886321205821205851198991205782

119899

+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

(33)


120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899+ (1205722+ 11988631120582) 1205852

119899120578119899

+ 119886321205821205851198991205782

119899+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

= 120582120585119899+ 120582 (119886

201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ 120582 (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

+ 120590(120585119899+ 120578119899)3

+ 1198744

(34)


1205852

119899 (1205822

minus 120582) 11988620= 0

1205853

119899 (1205823

minus 120582 minus 120590) 11988630= 0

120585119899120578119899 (1 minus 120582) 119886

21= 0

1205852

119899120578119899 1205722minus 3120590 = 0

(35)

Now

11988620= 11988621= 0

1205722= 3120590 = 3119886(minus1 + 119894

119886radic1 minus 1198862

1 minus 1198862

)

(36)

This implies

Re (1205722) = minus3119886 (37)


120585119899= 119903119899+ 119894119904119899

120578119899= 119903119899minus 119894119904119899

(38)



3

2

1

minus1

minus2

minus3

(a)

15

10

05

00

minus05

minus10

minus15


(b)



0and 1205741

We have

cos 1205740= Re (120582) 120574

1=

minus1

sin 1205740

sdot Re (1205722) (39)


= 0 Indeed

minus1 lt cos 1205740= 119886 lt 1


2

1205741=

minus1



3119886


(40)






119911119899= 119909119899+ radic119886 minus 1

119910119899= 119909119899minus1

119899 = 0 1 (41)


119909119899+1

=

2119886 (119909119899+ radic119886 minus 1)

1 + (119909119899+ radic119886 minus 1)


119910119899+1

= 119909119899

(42)



119883119899+1

=

2 (2 minus 119886)

119886

119883119899minus 119884119899

119884119899+1

= 119883119899

(119899 = 0 1 ) (43)





120582 =


119886

1205822

=

1

1198862(8 minus 8119886 + 119886

2


1205823

=

1

1198863((2 minus 119886) (16 minus 16119886 + 119886

2

)

+2 (16 minus 16119886 + 31198862

) 119894radic119886 minus 1)

1205823

= 1 lArrrArr 119886 = 4

1205824

=

1

1198864((8 minus 8119886 + 119886

2

)

2


(119886 minus 1)

+8119894 (8 minus 8119886 + 1198862


1205824

= 1 lArrrArr 119886 = 2

(44)


and so 119897 = 4



1198690=[

[

2 (2 minus 119886)

119886

minus1

1 0

]

]

(45)


119875 = [

1 1

120582 120582

] (46)


119909119899+1

=


119886

119909119899minus 119910119899

+


1198862

times (1199092

119899minus

(1198862

minus 8119886 + 8)


1199093

119899) + 119874

4

119910119899+1

= 119909119899

(47)


[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (48)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(49)


119906119899+1

= 120582119906119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

V119899+1

= 120582V119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

(50)

where

120590 =

120582

120582 minus 120582

sdot


1198862

1205901= minus

(1198862

minus 8119886 + 8)


(51)


11988620=

120590

1205822minus 120582

=


1198862(120582 minus 1) (120582 minus 120582)

11988621=

minus2120590

120582 minus 1

=


2119886 (119886 minus 1)

11988622=

120590

120582

2

minus 120582

2 (11988620+ 11988622) =


1198862

1

120582 minus 120582

(

2

120582 minus 1

minus

2

1205823minus 1

)

2 (11988620+ 11988622) = (


1198862

)(

minus119886119894


)

times (

2119894119886radic119886 minus 1

(119886 minus 1) (119886 minus 4)

) =

1

radic119886 minus 1

1205722= 120590 (4Re (119886

21) + 2 (119886

20+ 11988622) + 3120590

1)

(52)




4Re (11988621) + 2 (119886

20+ 11988622) + 3120590

1

=


119886 (119886 minus 1)

+

1

radic119886 minus 1

minus

3 (1198862

minus 8119886 + 8)


=

4 (119886 + 2)


1205722= (



)(


1198862

)

times (

4 (119886 + 2)


)

=


1198863radic119886 minus 1

(53)


Re (1205722) =

4 (119886 + 2)

1198863

(54)



6

4

2

minus2

minus4

minus6

(a)

minus4 minus2 2 4

4

2

minus2

minus4

(b)


5

4

3

2

1

0540 545 550 555 560 565 570


= 12





2 minus 119886

119886

lt 1


2 minus 119886

119886

)

2

= plusmn

2

119886

radic119886 minus 1

1205741= plusmn

2 (119886 + 2)

1198862radic119886 minus 1

(55)







119868 (119909119899 119909119899minus1

) = 1199092

1198991199092

119899minus1+ 1199092

119899+ 1199092


(56)




119911119899+1


119899 = 0 1 (57)





References



119899+2119906119899= 120595(119906

119899+1) in R+




119899+2+ 119906119899

= 120595(119906119899+1






















119906119899+1

+ V119899+1

= 2119886 (119906119899+ V119899) minus (120582119906

119899+ 120582V119899)

minus 2119886(119906119899+ V119899)3

+ 1198744

120582119906119899+1

+ 120582V119899+1

= 119906119899+ V119899

(26)


119906119899+1

= 120582119906119899+ 120590(119906

119899+ V119899)3

+ 1198744

V119899+1

= 120582V119899+ 120590(119906

119899+ V119899)3

+ 1198744

(27)

where

120590 = minus

2120582119886

120582 minus 120582

= 119886(minus1 +

119886


119894)

1205822

minus 2119886120582 + 1 = 0 120582120582 = 1 120582 + 120582 = 2119886

(28)


119906119899= 120585119899+ 1206012(120585119899 120578119899) + 1206013(120585119899 120578119899)

V119899= 120578119899+ 1205952(120585119899 120578119899) + 1205953(120585119899 120578119899)

(119899 = 0 1 )

(29)

where

120601119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119896minus119895

120578119895

120595119896(120585 120578) =

119896

sum

119895=0

119886119896119895120585119895

120578119896minus119895

(30)

for 119896 = 2 and 119896 = 3 yields

119906119899= 120585119899+ (119886201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

V119899= 120578119899+ (119886201205782

119899+ 11988621120585119899120578119899+ 119886221205852

119899)

+ (119886301205783

119899+ 119886311205851198991205782

119899+ 119886321205852

119899120578119899+ 119886331205853

119899)

1199062

119899= 1205852

119899+ 2119886201205853

119899+ 2119886211205852

119899120578119899+ 2119886221205851198991205782

119899+ 1198744

V2119899= 1205782

119899+ 2119886201205783

119899+ 2119886211205851198991205782

119899+ 2119886221205852

119899120578119899+ 1198744

1199063

119899= 1205853

119899+ 1198744

V3119899= 1205783

119899+ 1198744

1199062

119899V119899= 1205852

119899120578119899+ 1198744

119906119899V2119899= 1205851198991205782

119899+ 1198744

119906119899V119899= 119886221205853

119899+ (11988620+ 11988621) 1205852

119899120578119899

+ (11988621+ 11988620) 1205851198991205782

119899+ 120585119899120578119899+ 119886221205783

119899+ 1198744

(31)


and 120578119899+1

one obtain

120585119899+1

= (120582120585119899+ 12057221205852

119899120578119899) + 1198744

120578119899+1

= (120582120578119899+ 12057221205851198991205782

119899) + 1198744

119899 = 0 1

(32)


119906119899+1

= 120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899

+ (1205722+ 11988631120582) 1205852

119899120578119899+ 119886321205821205851198991205782

119899

+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

(33)


120582120585119899+ 119886201205822

1205852

119899+ 119886301205823

1205853

119899+ 11988621120585119899120578119899+ (1205722+ 11988631120582) 1205852

119899120578119899

+ 119886321205821205851198991205782

119899+ 1198862

22120582

2

1205782

119899+ 1198863

33120582

3

1205783

119899

= 120582120585119899+ 120582 (119886

201205852

119899+ 11988621120585119899120578119899+ 119886221205782

119899)

+ 120582 (119886301205853

119899+ 119886311205852

119899120578119899+ 119886321205851198991205782

119899+ 119886331205783

119899)

+ 120590(120585119899+ 120578119899)3

+ 1198744

(34)


1205852

119899 (1205822

minus 120582) 11988620= 0

1205853

119899 (1205823

minus 120582 minus 120590) 11988630= 0

120585119899120578119899 (1 minus 120582) 119886

21= 0

1205852

119899120578119899 1205722minus 3120590 = 0

(35)

Now

11988620= 11988621= 0

1205722= 3120590 = 3119886(minus1 + 119894

119886radic1 minus 1198862

1 minus 1198862

)

(36)

This implies

Re (1205722) = minus3119886 (37)


120585119899= 119903119899+ 119894119904119899

120578119899= 119903119899minus 119894119904119899

(38)



3

2

1

minus1

minus2

minus3

(a)

15

10

05

00

minus05

minus10

minus15


(b)



0and 1205741

We have

cos 1205740= Re (120582) 120574

1=

minus1

sin 1205740

sdot Re (1205722) (39)


= 0 Indeed

minus1 lt cos 1205740= 119886 lt 1


2

1205741=

minus1



3119886


(40)






119911119899= 119909119899+ radic119886 minus 1

119910119899= 119909119899minus1

119899 = 0 1 (41)


119909119899+1

=

2119886 (119909119899+ radic119886 minus 1)

1 + (119909119899+ radic119886 minus 1)


119910119899+1

= 119909119899

(42)



119883119899+1

=

2 (2 minus 119886)

119886

119883119899minus 119884119899

119884119899+1

= 119883119899

(119899 = 0 1 ) (43)





120582 =


119886

1205822

=

1

1198862(8 minus 8119886 + 119886

2


1205823

=

1

1198863((2 minus 119886) (16 minus 16119886 + 119886

2

)

+2 (16 minus 16119886 + 31198862

) 119894radic119886 minus 1)

1205823

= 1 lArrrArr 119886 = 4

1205824

=

1

1198864((8 minus 8119886 + 119886

2

)

2


(119886 minus 1)

+8119894 (8 minus 8119886 + 1198862


1205824

= 1 lArrrArr 119886 = 2

(44)


and so 119897 = 4



1198690=[

[

2 (2 minus 119886)

119886

minus1

1 0

]

]

(45)


119875 = [

1 1

120582 120582

] (46)


119909119899+1

=


119886

119909119899minus 119910119899

+


1198862

times (1199092

119899minus

(1198862

minus 8119886 + 8)


1199093

119899) + 119874

4

119910119899+1

= 119909119899

(47)


[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (48)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(49)


119906119899+1

= 120582119906119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

V119899+1

= 120582V119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

(50)

where

120590 =

120582

120582 minus 120582

sdot


1198862

1205901= minus

(1198862

minus 8119886 + 8)


(51)


11988620=

120590

1205822minus 120582

=


1198862(120582 minus 1) (120582 minus 120582)

11988621=

minus2120590

120582 minus 1

=


2119886 (119886 minus 1)

11988622=

120590

120582

2

minus 120582

2 (11988620+ 11988622) =


1198862

1

120582 minus 120582

(

2

120582 minus 1

minus

2

1205823minus 1

)

2 (11988620+ 11988622) = (


1198862

)(

minus119886119894


)

times (

2119894119886radic119886 minus 1

(119886 minus 1) (119886 minus 4)

) =

1

radic119886 minus 1

1205722= 120590 (4Re (119886

21) + 2 (119886

20+ 11988622) + 3120590

1)

(52)




4Re (11988621) + 2 (119886

20+ 11988622) + 3120590

1

=


119886 (119886 minus 1)

+

1

radic119886 minus 1

minus

3 (1198862

minus 8119886 + 8)


=

4 (119886 + 2)


1205722= (



)(


1198862

)

times (

4 (119886 + 2)


)

=


1198863radic119886 minus 1

(53)


Re (1205722) =

4 (119886 + 2)

1198863

(54)



6

4

2

minus2

minus4

minus6

(a)

minus4 minus2 2 4

4

2

minus2

minus4

(b)


5

4

3

2

1

0540 545 550 555 560 565 570


= 12





2 minus 119886

119886

lt 1


2 minus 119886

119886

)

2

= plusmn

2

119886

radic119886 minus 1

1205741= plusmn

2 (119886 + 2)

1198862radic119886 minus 1

(55)







119868 (119909119899 119909119899minus1

) = 1199092

1198991199092

119899minus1+ 1199092

119899+ 1199092


(56)




119911119899+1


119899 = 0 1 (57)





References



119899+2119906119899= 120595(119906

119899+1) in R+




119899+2+ 119906119899

= 120595(119906119899+1






















3

2

1

minus1

minus2

minus3

(a)

15

10

05

00

minus05

minus10

minus15


(b)



0and 1205741

We have

cos 1205740= Re (120582) 120574

1=

minus1

sin 1205740

sdot Re (1205722) (39)


= 0 Indeed

minus1 lt cos 1205740= 119886 lt 1


2

1205741=

minus1



3119886


(40)






119911119899= 119909119899+ radic119886 minus 1

119910119899= 119909119899minus1

119899 = 0 1 (41)


119909119899+1

=

2119886 (119909119899+ radic119886 minus 1)

1 + (119909119899+ radic119886 minus 1)


119910119899+1

= 119909119899

(42)



119883119899+1

=

2 (2 minus 119886)

119886

119883119899minus 119884119899

119884119899+1

= 119883119899

(119899 = 0 1 ) (43)





120582 =


119886

1205822

=

1

1198862(8 minus 8119886 + 119886

2


1205823

=

1

1198863((2 minus 119886) (16 minus 16119886 + 119886

2

)

+2 (16 minus 16119886 + 31198862

) 119894radic119886 minus 1)

1205823

= 1 lArrrArr 119886 = 4

1205824

=

1

1198864((8 minus 8119886 + 119886

2

)

2


(119886 minus 1)

+8119894 (8 minus 8119886 + 1198862


1205824

= 1 lArrrArr 119886 = 2

(44)


and so 119897 = 4



1198690=[

[

2 (2 minus 119886)

119886

minus1

1 0

]

]

(45)


119875 = [

1 1

120582 120582

] (46)


119909119899+1

=


119886

119909119899minus 119910119899

+


1198862

times (1199092

119899minus

(1198862

minus 8119886 + 8)


1199093

119899) + 119874

4

119910119899+1

= 119909119899

(47)


[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (48)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(49)


119906119899+1

= 120582119906119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

V119899+1

= 120582V119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

(50)

where

120590 =

120582

120582 minus 120582

sdot


1198862

1205901= minus

(1198862

minus 8119886 + 8)


(51)


11988620=

120590

1205822minus 120582

=


1198862(120582 minus 1) (120582 minus 120582)

11988621=

minus2120590

120582 minus 1

=


2119886 (119886 minus 1)

11988622=

120590

120582

2

minus 120582

2 (11988620+ 11988622) =


1198862

1

120582 minus 120582

(

2

120582 minus 1

minus

2

1205823minus 1

)

2 (11988620+ 11988622) = (


1198862

)(

minus119886119894


)

times (

2119894119886radic119886 minus 1

(119886 minus 1) (119886 minus 4)

) =

1

radic119886 minus 1

1205722= 120590 (4Re (119886

21) + 2 (119886

20+ 11988622) + 3120590

1)

(52)




4Re (11988621) + 2 (119886

20+ 11988622) + 3120590

1

=


119886 (119886 minus 1)

+

1

radic119886 minus 1

minus

3 (1198862

minus 8119886 + 8)


=

4 (119886 + 2)


1205722= (



)(


1198862

)

times (

4 (119886 + 2)


)

=


1198863radic119886 minus 1

(53)


Re (1205722) =

4 (119886 + 2)

1198863

(54)



6

4

2

minus2

minus4

minus6

(a)

minus4 minus2 2 4

4

2

minus2

minus4

(b)


5

4

3

2

1

0540 545 550 555 560 565 570


= 12





2 minus 119886

119886

lt 1


2 minus 119886

119886

)

2

= plusmn

2

119886

radic119886 minus 1

1205741= plusmn

2 (119886 + 2)

1198862radic119886 minus 1

(55)







119868 (119909119899 119909119899minus1

) = 1199092

1198991199092

119899minus1+ 1199092

119899+ 1199092


(56)




119911119899+1


119899 = 0 1 (57)





References



119899+2119906119899= 120595(119906

119899+1) in R+




119899+2+ 119906119899

= 120595(119906119899+1






















1198690=[

[

2 (2 minus 119886)

119886

minus1

1 0

]

]

(45)


119875 = [

1 1

120582 120582

] (46)


119909119899+1

=


119886

119909119899minus 119910119899

+


1198862

times (1199092

119899minus

(1198862

minus 8119886 + 8)


1199093

119899) + 119874

4

119910119899+1

= 119909119899

(47)


[

119909119899

119910119899

] = 119875 sdot [

119906119899

V119899

] = [

119906119899+ V119899

120582119906119899+ 120582V119899

] (48)

or

119909119899= 119906119899+ V119899

119910119899= 120582119906119899+ 120582V119899

(49)


119906119899+1

= 120582119906119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

V119899+1

= 120582V119899+ 120590 ((119906

119899+ V119899)2

+ 1205901(119906119899+ V119899)3

)

(50)

where

120590 =

120582

120582 minus 120582

sdot


1198862

1205901= minus

(1198862

minus 8119886 + 8)


(51)


11988620=

120590

1205822minus 120582

=


1198862(120582 minus 1) (120582 minus 120582)

11988621=

minus2120590

120582 minus 1

=


2119886 (119886 minus 1)

11988622=

120590

120582

2

minus 120582

2 (11988620+ 11988622) =


1198862

1

120582 minus 120582

(

2

120582 minus 1

minus

2

1205823minus 1

)

2 (11988620+ 11988622) = (


1198862

)(

minus119886119894


)

times (

2119894119886radic119886 minus 1

(119886 minus 1) (119886 minus 4)

) =

1

radic119886 minus 1

1205722= 120590 (4Re (119886

21) + 2 (119886

20+ 11988622) + 3120590

1)

(52)




4Re (11988621) + 2 (119886

20+ 11988622) + 3120590

1

=


119886 (119886 minus 1)

+

1

radic119886 minus 1

minus

3 (1198862

minus 8119886 + 8)


=

4 (119886 + 2)


1205722= (



)(


1198862

)

times (

4 (119886 + 2)


)

=


1198863radic119886 minus 1

(53)


Re (1205722) =

4 (119886 + 2)

1198863

(54)



6

4

2

minus2

minus4

minus6

(a)

minus4 minus2 2 4

4

2

minus2

minus4

(b)


5

4

3

2

1

0540 545 550 555 560 565 570


= 12





2 minus 119886

119886

lt 1


2 minus 119886

119886

)

2

= plusmn

2

119886

radic119886 minus 1

1205741= plusmn

2 (119886 + 2)

1198862radic119886 minus 1

(55)







119868 (119909119899 119909119899minus1

) = 1199092

1198991199092

119899minus1+ 1199092

119899+ 1199092


(56)




119911119899+1


119899 = 0 1 (57)





References



119899+2119906119899= 120595(119906

119899+1) in R+




119899+2+ 119906119899

= 120595(119906119899+1






















6

4

2

minus2

minus4

minus6

(a)

minus4 minus2 2 4

4

2

minus2

minus4

(b)


5

4

3

2

1

0540 545 550 555 560 565 570


= 12





2 minus 119886

119886

lt 1


2 minus 119886

119886

)

2

= plusmn

2

119886

radic119886 minus 1

1205741= plusmn

2 (119886 + 2)

1198862radic119886 minus 1

(55)







119868 (119909119899 119909119899minus1

) = 1199092

1198991199092

119899minus1+ 1199092

119899+ 1199092


(56)




119911119899+1


119899 = 0 1 (57)





References



119899+2119906119899= 120595(119906

119899+1) in R+




119899+2+ 119906119899

= 120595(119906119899+1























References



119899+2119906119899= 120595(119906

119899+1) in R+




119899+2+ 119906119899

= 120595(119906119899+1




















Birkhoff Normal Forms and KAM Theory for Gumowski … · Gumowski-Mira Equation ... [12] for the...

Documents

Transcript of Birkhoff Normal Forms and KAM Theory for Gumowski … · Gumowski-Mira Equation ... [12] for the...