Volume 2009, Article ID 149563,26pages doi:10.1155/2009/149563
Research Article
Bifurcation Analysis of a Van der Pol-Duffing
Circuit with Parallel Resistor
Denis de Carvalho Braga,
1Luis Fernando Mello,
2and Marcelo Messias
31Instituto de Sistemas El´etricos e Energia, Universidade Federal de Itajub´a, Avenida BPS 1303, Pinheirinho, 37500-903 Itajub´a, MG, Brazil
2Instituto de Ciˆencias Exatas, Universidade Federal de Itajub´a, Avenida BPS 1303, Pinheirinho, 37500-903 Itajub´a, MG, Brazil
3Departamento de Matem´atica, Estat´ıstica e Computac¸˜ao, Faculdade de Ciˆencias e Tecnologia, UNESP—Univ Estadual Paulista, Cx. Postal 266, 19060-900 Presidente Prudente, SP, Brazil Correspondence should be addressed to Marcelo Messias,[email protected]
Received 29 April 2009; Accepted 16 August 2009
Recommended by Dane Quinn
We study the local codimension one, two, and three bifurcations which occur in a recently proposed Van der Pol-Duffing circuitADVPwith parallel resistor, which is an extension of the classical Chua’s circuit with cubic nonlinearity. The ADVP system presents a very rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations. Aiming to contribute to the understand of the complex dynamics of this new system we present an analytical study of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given. Then, by studying the continuation of such periodic orbits, we numerically find a sequence of period doubling and symmetric homoclinic bifurcations which leads to the creation of strange attractors, for a given set of the parameter values.
Copyrightq2009 Denis de Carvalho Braga et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and Statement of the Main Results
In this paper we study the local codimension one, two, and three bifurcations and the respective qualitative changes in the dynamics of the following system of nonlinear equations:
˙
x dx
dτ −ν
x3−µx−y, y˙ dy
dτ x−αy−z, z˙
dz
N C1 C2 RP L
V1 R V2
Figure 1:ADVP circuit with parallel resistorRp.
where x, y, z ∈ R3 are the state variables and µ ∈ R, α ≥ 1, ν > 0, and β > 0 are real
parameters.
As far as we know, system1.1was proposed and firstly studied in1, and it can be
obtained from the system
c1v′1−iN v1−v2/R, c2v2′ v1−αv2/R−iL, i′Lv2/L, 1.2
by the following changes in variables, parameters, and rescaling in time:
xbRv1, y
bRv2, z
bR3iL,
µ−1aR, νc2/c1, βc2R2/L, τ t/Rc2.
1.3
In1.2v1andv2are the voltages across the capacitorsC1andC2, respectively,iN is
the current through the nonlinear diodeN,iLis the current through the inductor,c1andc2
are the capacitances,L is the inductance,R is the linear resistor, Rp is the parallel resistor,
α 1 R/Rp, and the prime denotes derivatives with respect to the timet. It is assumed
that the current through the nonlinear diode is given byiN iNv1 av1bv13, witha <0
andb >0. SeeFigure 1and1for more details.
Despite the simplicity of the electronic circuit shown inFigure 1, the related system
1.1 has a rich dynamical behavior, ranging from stable equilibrium points to periodic
and even chaotic oscillations, depending on the parameter values. The study of simple nonlinear electronic circuits presenting chaotic behavior was initiated by Chua in the
mid-1980s see 2 and the interest on the subject has grown in the last decades. In fact,
it has useful applications like chaos synchronization which is applied in industry and
secure communications, beyond other areas of sciences see 1 and references therein
and also 3, 4. System1.1 with α 1 reduces to a system equivalent to the classical
Chua’s differential equations with cubic nonlinearity. The authors have recently developed
a complete study of degenerate Hopf bifurcations for this case in5, answering a challenge
proposed by Moiola and Chua in6.
System1.1has the equilibrium pointE0 0,0,0, which exists for any parameter
values. Forµ > 0 it also has the symmetric equilibriaE± ±√µ,0,±√µ. Codimension one
Hopf bifurcation which occurs at the equilibrium point E0 was studied in 1, by using a
−0.4 −0.2
0
µ
1 1.4 1.8 2.2
α U:l1βc>0
ν
S:l1βc<0 1 3 5
a
0 0.1
0.2 0.3
µ
1 1.8 2.6
α
U:l1βc>0
ν C2:l2βc>0
S:l1βc<0
C1:l2βc<0 1
3 5
7
T
C
R
b
Figure 2:Surfaces{l10}forE0aandE±b,C{l10} ∩ {l20}.
In this paper by using the classical projection method which allows one the calculation of the Lyapunov coefficients associated to the Hopf bifurcations we study all possible
bifurcationsgeneric and degenerate oneswhich occur at the equilibriaE0andE±of system
1.1. In this way the analyses presented in1are extended and completed. More precisely,
we prove the following statements.
aFor the equilibrium E0 0,0,0 the Hopf hypersurface H1 is obtained in the
space of parametersµ, ν, α, βand the first Lyapunov coefficientl1 is calculated.
It is shown that this coefficient vanishes on a 2-dimensional surface contained in
H1, giving rise to codimension two Hopf bifurcationsseeFigure 2a. Then the
second Lyapunov coefficientl2is calculated and it is established that this coefficient
is always negative on the surface{l10}.
bFor the symmetric equilibria E± ±√µ,0,±√µ the Hopf hypersurface H2 is
obtained in the space of parameters µ, ν, α, β, the first Lyapunov coefficientl1 is
calculated, and it is shown that this coefficient vanishes on a 2-dimensional surface
contained inH2seeFigure 2b, giving rise to codimension two bifurcations. The
second Lyapunov coefficientl2is calculated and it is established that this coefficient
also vanishes on a 2-dimensional surface contained inH2seeFigure 2b, giving
rise to codimension three bifurcations. The third Lyapunov coefficients along the
curveCgiven by the intersection of the surfaces{l10}and{l2 0}are obtained
and found to be positive.
The corresponding bifurcation diagrams are traced for the above bifurcations see
Figures3,4,7, and8ofSection 4. From statementait follows that the maximum number
of small periodic orbits bifurcating from the origin is 2. Furthermore, from statementbone
can deduce that there is a region in the parameter space for which an attracting periodic orbit
and two other unstable periodic orbits coexist with the attracting equilibrium pointsE±. See
Figures7and8inSection 4.
We go further and perform a numerical study of the continuation of periodic orbits
period doubling and homoclinic bifurcations which seems to lead to the creation of strange
attractors of system1.1, for some parameter valuesseeSection 5.
The rest of this paper is organized as follows. InSection 2through a linear analysis
of system1.1we obtain the Hopf hypersurfaces forE0andE±. InSection 3following8–
10we present a brief review of the methods used to study codimension one, two, and three
Hopf bifurcations, describing in particular how to calculate the Lyapunov coefficients related to the stability of the equilibrium point as well as of the periodic orbits which appear in these bifurcations. In general the Lyapunov coefficients are very difficult to be obtained.
These methods are used inSection 4 to prove the main results of this paper, described in
statementsaandb. In Section 5we present numerical simulations for particular values
of the parameters which illustrate and corroborate the analytical results obtained. Finally, in
Section 6we make some concluding remarks.
2. Linear Analysis of System
1.1
In a vectorial notation which will be useful in the next calculations system1.1can be written
as
x′fx,ζ −νx3−µx−y, x−αy−z, βy, 2.1
where
x
x, y, z
∈R3, ζ
µ, ν, α, β
∈−∞,∞×0,∞×1,∞×0,∞. 2.2
The originE0is an equilibrium point of system1.1for all values of the parameters.
The characteristic polynomial of the Jacobian matrix of the functionfgiven in2.1atE0is
given by
pλ λ3
α−µν
λ2
β−ν−αµν
λ−µβν. 2.3
Ifµ >0 then the term−µβνin the above polynomial is negative and this implies thatE0is an
unstable equilibrium point.
For the sake of completeness we state the following lemmaRouth-Hurwitz stability
criterionwhose proof can be found in11, page 58.
Lemma 2.1. The polynomialLλ p0λ3p1λ2p2λp3,p0 > 0, with real coefficients, has all roots with negative real parts if and only if the numbersp1,p2,p3 are positive and the inequality
p1p2> p0p3is satisfied.
The following proposition is a direct consequence ofLemma 2.1.
Proposition 2.2. Consider µ < 0 in system 1.1. If αµ ≤ −1 then the equilibrium point E0 is
asymptotically stable for allβ >0andν >0. Ifαµ >−1and
β > β1c
ν
α−µν
1αµ
then system1.1has an asymptotically stable equilibrium point atE0. Ifαµ >−1and0< β < β1c thenE0is unstable.
Forµ 0 the origin becomes a nonhyperbolic equilibrium point and the situation is
highly degenerate. This case, which will not be considered in this work, can be studied by
using the methods presented for instance in12.
For µ > 0 two new equilibria E± ±√µ,0,±√µappear. System1.1is invariant
under the change of coordinatesx, y, z → −x,−y,−z. So the stability of the equilibrium
E−can be obtained from the stability ofE.
The characteristic polynomial of the Jacobian matrix of the functionfgiven in2.1at
Eis given by
pλ λ3
α2µν
λ2
β−ν2αµν
λ2µβν. 2.5
The following proposition is also a direct consequence ofLemma 2.1.
Proposition 2.3. Consider µ > 0 in system 1.1. If 2αµ ≥ 1 then the equilibrium point E is
asymptotically stable for allβ >0andν >0. If2αµ <1and
β > β2c
ν
α2µν
1−2αµ
α , 2.6
then system1.1has an asymptotically stable equilibrium point atE. If2αµ <1and0< β < β2c thenEis unstable.
The equationsβ β1c andβ β2c in2.4and2.6give the equations of the Hopf
hypersurfacesH1andH2in the parameter spaceµ, ν, α, β. These equations will be used in
Section 4in the study of degenerate Hopf bifurcations which occur at the equilibriaE0and
E±of system1.1.
3. Outline of the Hopf Bifurcation Methods
This section is a review of the projection method described in 8 for the calculation of
the first and second Lyapunov coefficients associated to Hopf bifurcations, denoted byl1
and l2, respectively. This method was extended to the calculation of the third and fourth
Lyapunov coefficients in9,10. Other equivalent definitions and algorithmic procedures to
write the expressions of the Lyapunov coefficients for two-dimensional systems can be found
in Andronov et al.13and Gasull and Torregrosa14, among others, and can be adapted to
the three-dimensional system of this paper if it is restricted to the center manifold. See also
15.
Consider the differential equation
wherex ∈R3andζ ∈R4are, respectively, vectors representing phase variables and control
parameters. Assume thatfis of classC∞inR3×R4. Suppose that3.1has an equilibrium
pointxx0atζζ0and, denoting the variablex−x0also byx, write
Fx fx,ζ0 3.2
as
Fx Ax1
2 Bx,x
1
6 Cx,x,x
1
24 Dx,x,x,x
1
120 Ex,x,x,x,x
1
720 Kx,x,x,x,x,x
1
5040 Lx,x,x,x,x,x,x O
x8,
3.3
whereAfx0,ζ0and, fori1,2,3,
Bix,y
3
j,k1
∂2F
iξ
∂ξj∂ξk
ξ0
xjyk, Cix,y,z
3
j,k,l1
∂3F
iξ
∂ξj∂ξk∂ξl
ξ0
xjykzl, 3.4
and so on forDi,Ei,Ki, andLi.
Suppose thatx0,ζ0is an equilibrium point of3.1where the Jacobian matrixAhas
a pair of purely imaginary eigenvaluesλ2,3 ±iω0,ω0 >0, and admits no other eigenvalue
with zero real part. LetTcbe the generalized eigenspace ofAcorresponding toλ
2,3. By this it
is meant the largest subspace invariant byAon which the eigenvalues areλ2,3.
Letp, q∈C3be vectors such that
Aqiω0q, A⊤p−iω0p, p, q
3
i1
piqi1, 3.5
whereA⊤is the transpose of the matrixA. Any vectory∈Tccan be represented asywq
w q, wherewp, y ∈C. The two-dimensional center manifold associated to the eigenvalues
λ2,3 ±iω0can be parameterized by the variableswandwby means of an immersion of the
formxHw, w, whereH:C2 → R3has a Taylor expansion of the form
Hw, w wqw q
2≤jk≤7
1
j!k!hjkw
jwkO
|w|8, 3.6
withhjk ∈C3andhjk hkj. Substituting this expression into3.1we obtain the following
differential equation:
Hww′Hww′FHw, w, 3.7
where F is given by 3.2. The complex vectors hij are obtained solving the system of
see9, Remark 3.1, page 27, so that system 3.7, on the chartw for a central manifold, writes as follows:
w′iω0w1
2G21w|w|
2 1
12G32w|w|
4 1
144G43w|w|
6O
|w|8, 3.8
withGjk ∈C.
Thefirst Lyapunov coefficientl1is defined by
l1 1
2ReG21, 3.9
whereG21 p,H21, andH21 Cq, q, q Bq, h20 2Bq, h11.
Thesecond Lyapunov coefficientis defined by
l2 1
12ReG32, 3.10
whereG32 p,H32. The expression forH32can be found in9, equation36, page 28.
Thethird Lyapunov coefficientis defined by
l3 1
144ReG43, 3.11
whereG43 p,H43. The expression forH43can be found in9, equation44, page 30.
AHopf pointx0,ζ0of system3.1is an equilibrium point where the Jacobian matrixA
has a pair of purely imaginary eigenvaluesλ2,3±iω0,ω0>0, and the other eigenvalueλ1/0.
From the Center Manifold Theorem, at a Hopf point a two-dimensional center manifold is
well defined, it is invariant under the flow generated by 3.1 and can be continued with
arbitrary high class of differentiability to nearby parameter valuessee8.
A Hopf point is calledtransversal if the parameter dependent complex eigenvalues
cross the imaginary axis with nonzero derivative. In a neighborhood of a transversal Hopf
point withl1/0 the dynamic behavior of the system3.1, reduced to the family of
parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to the
following complex normal formw′ ηiωwl1w|w|2, wherew∈C,η,ω, andl1are real
functions having derivatives of arbitrary higher order, which are continuations of 0,ω0, and
the first Lyapunov coefficient at the Hopf point. See8,9for details. Whenl1 <0l1 >0one
family of stableunstableperiodic orbits can be found on this family of manifolds, shrinking
to an equilibrium point at the Hopf point.
AHopf point of codimension 2is a Hopf point wherel1 vanishes. It is calledtransversal
if {η 0} and {l1 0} have transversal intersections, whereη ηζ is the real part of
the critical eigenvalues. In a neighborhood of a transversal Hopf point of codimension 2
with l2/0 the dynamic behavior of the system 3.1 reduced to the family of
parameter-dependent continuations of the center manifold is orbitally topologically equivalent tow′
η iω0w τw|w|2 l2w|w|4, where η and τ are unfolding parameters. See 8, 9. The
bifurcation diagrams forl2/0 can be found in8, page 313and in16.
AHopf point of codimension 3is a Hopf point of codimension 2 wherel2 vanishes. It
neighborhood of a transversal Hopf point of codimension 3 withl3/0 the dynamic behavior
of the system3.1, reduced to the family of parameter-dependent continuations of the center
manifold, is orbitally topologically equivalent tow′ ηiω
0wτw|w|2νw|w|4l3w|w|6,
whereη,τ, andνare unfolding parameters. The bifurcation diagram forl3/0 can be found
in Takens16.
4. Hopf Bifurcations in System
1.1
4.1. Hopf Bifurcations at
E
0In this subsection we study the stability of E0 under the conditions β β1c and −1 <
αµ < 0, that is, on the Hopf hypersurface H1 complementary to the range of validity of
Proposition 2.2.
The Jacobian matrix of the functionfgiven in2.1atE0is given by
ADfE0
⎛
⎜ ⎜ ⎝
µν ν 0
1 −α −1
0 β1c 0
⎞
⎟ ⎟
⎠. 4.1
Then, using the notation of the previous section see expression 3.3 the multilinear
symmetric functions corresponding tofcan be written as
Bx,y Dx,y,z,u Ex,y,z,u,v 0,0,0,
Cx,y,z
−6νx1y1z1,0,0.
4.2
The eigenvalues ofAare
λ1−αµν <0, λ2,3±iω0±iν
−µ
1αµ
α , 4.3
and from3.5one has
q
−µνiω0ν −iω0
, iω0, β1c
,
p 1
ρ0
iω0µν−iω0,−iω0µν−iω02,µν−iω02
,
4.4
where
ρ0β1cµν−iω02
ν µν−iω02
ω20. 4.5
Using these calculations we prove the next two theorems, from which statementaof
Theorem 4.1. Consider the four-parameter family of differential equations1.1. The first Lyapunov coefficient associated to the equilibriumE0is given by
l1µ, ν, α, β1c−
3α
αµ1
ν3
α2−2µνα−ν
2
α3−2µνα2−µν2 . 4.6
If
s
µ, ν, α
−α22µναν 4.7
is different from zero then system1.1has a transversal Hopf point atE0forββ1cand−1< αµ <0. More precisely, ifµ, ν, α, β1c∈S∪U(seeFigure 2(a)), then system1.1has a Hopf point of codimension 1 atE0. Ifµ, ν, α, β1c∈SthenE0is asymptotically stable and for eachβ < β1c, but close toβ1c, there exists a stable periodic orbit near the unstable equilibrium pointE0. Ifµ, ν, α, β1c∈U thenE0is unstable and for eachβ > β1c, but close toβ1c, there exists an unstable periodic orbit near the asymptotically stable equilibrium pointE0.
Proof. As the function Bx,y 0,0,0 then the expression G21 in 3.9 reduces to
G21p, Cq, q, q. Performing the calculations we get
G21−
6ν4ω4
0
µ2ν2ω2
0
β1cµν−iω02ω20
µ2ν2−2iµω
0νν−ω20
. 4.8
Substituting the expressions ofω0 and β1c into the above expression ofG21 and taking the
real part we have, from3.9, the expression of the first Lyapunov coefficientl1given in4.6.
Note that the sign ofl1is given by the sign of the functionsdefined in4.7.
It remains only to verify the transversality condition of the Hopf bifurcation. In order
to do so, consider the family of differential equations1.1regarded as dependent on the
parameter β. The real part, η ηβ, of the pair of complex eigenvalues at the critical
parameterββ1cverifies
η′
β1cRe
p, dA
dβ
ββ1c
q
− α
2
2
α3−2µνα2−µν2 <0. 4.9
Therefore, the transversality condition at the Hopf point holds.
The setsSandUillustrated inFigure 2aare defined implicitly as{l1 <0}and{l1 >
0}, respectively. The theorem is proved.
Theorem 4.2. Consider the four-parameter family of differential equations1.1restricted toββ1c
and−1< αµ <0. Then the second Lyapunov coefficientl2associated to the equilibriumE0is negative
Proof. Following the notation introduced in Section 3, we have Bx,y Dx,y,z,u
Ex,y,z,u,v 0,0,0.Then
H323C
q, q, h21
6C
q, q, h213Cq, q, h30. 4.10
The expressions of the complex vectors h21 and h30 are too long to be put in print. The
interested author can encounter such expressions and all the details of the calculations
presented in this paper in the website17.
By direct calculations the second Lyapunov coefficient l2 associated to E0 can be
written as
l2µ, ν, α, β1c−
3α2αµ12ν5N
µ, ν, α, β1c
2µ
−α32µνα2µν23−α32µνα28µ2ν2α9µν2, 4.11
whereNµ, ν, α, β1chas the form
α10−8µνα9ν16µ2ν−1α82µν28µ2ν−7α72µ2ν361−40µ2να6
2µν332ν2µ4−112νµ213α5−µ2ν464νµ219α440µ3ν510µ2ν−9α3
µ2ν5−192ν2µ4724νµ2−121α2−8µ3ν643µ2ν−34α−142µ4ν7.
4.12
Along the surface{l10}we have, from4.7ofTheorem 4.1,sµ, ν, α −α22µναν0,
or equivalently,να2/12αµ. Substituting this expression into4.11one has
l2|{l10}−
9α11
αµ1
4αµ−1
2αµ148αµ−1 <0. 4.13
Here the transversality condition is equivalent to the transversality of the
hypersur-facesββ1candl1µ, ν, α, β 0. The bifurcation diagram for a typical pointRon the surface
{l1 0}is depicted inFigure 3.
4.2. Hopf Bifurcations at
E
±In this subsection we study the stability of E± under the conditions β β2c and 0 <
2αµ < 1, that is, on the Hopf hypersurfaceH2 complementary to the range of validity of
Proposition 2.3.
The Jacobian matrix of the functionfgiven in2.1calculated atEis given by
ADfE
⎛
⎜ ⎜ ⎝
−2µν ν 0
1 −α −1
0 β2c 0
⎞
⎟ ⎟
N1
N2
N3
N1 N6
N5
N4
N3 β
l1 R
a
N1 N2 N3
N4 N5 N6
b
Figure 3:aTypical pointsN1,N2,N3, N4,N5, andN6in the parameter space near the pointR.b
Phase portraits of system1.1for the flow restricted to the center manifold and its continuation. For parameters atN1the equilibrium is asymptotically stable; for parameters atN2the equilibrium is a weak
stable focusHopf point withl1negative; for parameters atN3the equilibrium is unstable and a stable
limit cycle appears from a Hopf bifurcation; for parameters atN4the equilibrium is an weak unstable
focusHopf point withl1positiveand there is a stable limit cycle; for parameters atN5the equilibrium
is asymptotically stable and an unstable limit cycle appears from a Hopf bifurcation, so there are two limit cycles encircling the equilibrium; for parameters atN6the equilibrium is asymptotically stable and the
two cycles collide, giving rise to a nondegenerate fold bifurcation of the cycles.
Then, using the notation of Section 3 see expression 3.3 the multilinear symmetric
functions corresponding tofcan be written as
Bx,y
−6ν√µx1y1,0,0, Cx,y,z −6νx1y1z1,0,0,
DEKL 0,0,0.
4.15
The eigenvalues ofAare
λ1−α−2µν <0, λ2,3±iω0±iν
2µ
1−2αµ
α , 4.16
and from3.5one has
q
iω
0ν
2µνiω0, iω0, β2c
, 4.17
p 1
ρ0
−iω02µνiω0,−iω02µνiω02,2µνiω02
, 4.18
where
ρ0β2c2µνiω02
ν 2µνiω02
ω20. 4.19
Using these calculations we prove the next three theorems, from which statementb
Theorem 4.3. Consider the four-parameter family of differential equations1.1. The first Lyapunov coefficient associated to the equilibriumEis given by
l1µ, ν, α, β2c−
3α
2αµ−1
ν3g
µ, ν, α
α34µνα22µν2
α34µνα2−12µ2ν2α8µν2, 4.20
where
g
µ, ν, α
α58µνα4ν4µ2ν−1α32µν25−24µ2να280µ2ν3α−20µν3. 4.21
Ifgµ, ν, αis different from zero then system1.1has a transversal Hopf point atEforββ2cand
0<2αµ <1.
More precisely, ifµ, ν, α, β2c ∈S∪U(seeFigure 2(b)) then system1.1has a Hopf point of codimension 1 atE. That is, ifµ, ν, α, β2c ∈ SthenE is asymptotically stable and for each
β < β2c, but close toβ2c, there exists a stable periodic orbit near the unstable equilibrium pointE; if
µ, ν, α, β2c∈ UthenE is unstable and for eachβ > β2c, but close toβ2c, there exists an unstable
periodic orbit near the asymptotically stable equilibrium pointE.
Proof. The theorem follows by expanding the expressions in definition of the first Lyapunov
coefficient3.9. It relies on extensive calculations involving the vectorqin4.17, the vector
p in 4.18 and the multilinear functions B and C. The inclusion here of the enormous
expressions obtained from the mentioned calculations would not bring any contribution in clarifying the reading of the text. Then the detailed calculations related to this proof,
corroborated by Computer Algebra, have been posted in17, including its implementation
using the software MATHEMATICA 518.
Notice that the sign of the first Lyapunov coefficient is given by the sign of the function
−gdefined in4.21since the denominator ofl1and the expression 32αµ−1ν3are negative.
InFigure 2bthe surface{l1 0}and the regionsSandUare illustrated. These regions are
defined implicitly as{l1 <0}and{l1>0}, respectively.
It remains only to verify the transversality condition of the Hopf bifurcation. In order
to do so, consider the family of differential equations1.1regarded as dependent on the
parameter β. The real part, η ηβ, of the pair of complex eigenvalues at the critical
parameterββ2cverifies
η′
β2cRe
p, dA
dβ
ββ2c
q
− α
2
2
α34µνα22µν2 <0. 4.22
Therefore, the transversality condition holds at the Hopf point. The theorem is proved.
Theorem 4.4. Consider the four-parameter family of differential equations1.1restricted toββ2c
and0<2αµ <1. Then the second Lyapunov coefficientl2associated to the equilibriumEis given by
l2µ, ν, α, β2c
α2
2αµ−1
ν5h
µ, ν, α
4µ
α34µνα2−32µ2ν2α18µν2
d
where the functionhµ, ν, αis given by
21µα20588µ2ν−4α193µν1876µ2ν−53α184ν3192ν2µ4−294νµ21α17
µν2−121296ν2µ414156νµ2−65α16−12µ2ν359248ν2µ4−18704νµ2473α15
µν3114240ν3µ6716976ν2µ4−62932νµ2199α14
16µ2ν4477120ν3µ6−165272ν2µ4769νµ2119α13
4µ3ν51634304ν3µ6−4121136ν2µ4751264νµ2−16945α12
−16µ2ν52225664ν4µ8−327552ν3µ6−585624ν2µ453587νµ2−262α11
−4µ3ν66967296ν4µ8−29565184ν3µ67648944ν2µ481592νµ2−11129α10
16µ4ν74644864ν4µ8773376ν3µ6−8268704ν2µ41627004νµ2−41405α9
4µ3ν7−80474112ν4µ832219456ν3µ613655344ν2µ4−1903712νµ217531α8
16µ4ν85308416ν4µ834114176ν3µ6−13584144ν2µ4−51152νµ239713α7
32µ5ν91327104ν4µ8−10505088ν3µ6−13899000ν2µ44382138νµ2−130059α6
−16µ4ν911446272ν4µ8−35076672ν3µ6−10290192ν2µ42504248νµ2−37769α5
32µ5ν1010175616ν3µ6−15588272ν2µ4−317442νµ2129593α4
−64µ6ν114720704ν2µ4−3846404νµ2142349α3
128µ5ν111199432ν2µ4−495365νµ213321α2
−256µ6ν12157944µ2ν−25841α4308992µ7ν13,
4.24
and the functiondµ, ν, αin the denominator is given by
α68µνα54µ2ν2α42µν25−24µ2να340µ2ν3α2−24µ3ν4α16µ2ν43. 4.25
If µ, ν, α, β2c ∈ C1 ∪ C2 (see Figure 2(b)), then system 1.1 has a transversal Hopf point of
M3
M4 M5
M6 M1
M3
M2
M1 β
l1 T
a
M1 M2 M3
M4 M5 M6
b
Figure 4:aTypical pointsM1,M2,M3,M4,M5, andM6in the parameter space near the pointT.b
Phase portraits of system1.1for the flow restricted to the center manifold and its continuation. For parameters atM1the equilibrium is unstable; for parameters atM2the equilibrium is an weak unstable
focusHopf point withl1positive; for parameters atM3the equilibrium is stable and an unstable limit
cycle appears from a Hopf bifurcation; for parameters at M4 the equilibrium is an weak stable focus
Hopf point withl1negativeand there is an unstable limit cycle; for parameters atM5the equilibrium is
unstable and a stable limit cycle appears from a Hopf bifurcation, so there are two limit cycles encircling the equilibrium; for parameters atM6the equilibrium is unstable and the two cycles collide corresponding
to a nondegenerate fold bifurcation of the cycles.
Proof. The theorem follows by expanding the expressions in definition3.10of the second
Lyapunov coefficient. It relies on extensive calculation involving the vectorqin4.17, the
vectorpin4.18and the multilinear functionsB,C,D, andE. The calculations in this proof,
corroborated by Computer Algebra, have been posted in17.
In Figure 2 we present a geometric synthesis interpreting the long calculations
involved in this proof. The sign of−hµ, ν, αgives the sign of the second Lyapunov coefficient
since the functiondα, µ, νin the denominator ofl2 is positive. The graph ofhµ, ν, α 0
on the surface{l1 0}, which determines the curveC, and the signs of the first and second
Lyapunov coefficients are also illustrated. As follows,l2 <0 on the open region on the surface
{l1 0}, denoted byC1. On this region a typical reference pointRis depicted. Alsol2 > 0
on the open region on the surface{l1 0}, denoted byC2. This region contains the typical
reference point, denoted byT.
The bifurcation diagrams of system1.1at the pointsRandTare illustrated in Figures
3and4, respectively. The theorem is proved.
Remark 4.5. There are numerical evidences that the sign of the third Lyapunov coefficient
is always positive along the curveCgiven by the intersection of the surfaces{l1 0}and
{l2 0}seeFigure 2b. Forα1 see the article in5. Forα2 seeTheorem 4.6in what
follows.Figure 5shows the behavior of the third Lyapunov coefficient as a function ofα, for
µ, ν, βon the curveC. So the bifurcation diagrams depicted in Figures7and8are essentially
the same for all values ofα≥1.
All the calculations presented here are illustrated inFigure 6for the casesα 2a
andα3b. In this figure are depicted the surfacesβ β2c, the solid curves wherel1 0,
the regionsSandUwherel1<0 andl1>0, respectively, the dashed curves wherel20 and
the arcsC1andC2wherel2 <0 andl2>0, respectively. The bifurcation diagrams for typical
pointsR1andR2are depicted inFigure 3while the bifurcation diagrams for typical pointsT1
0 0.5 1 1.5 2 2.5 3 3.5 ×1012
l3
1 1.5 2 2.5
α
Figure 5:Graph ofl3with respect toα.
3.9 3.95 4 4.05
β
0 0.03
µ
4 4.05
4.1
ν U:l1βc>0
C2:l2βc>0
S:l1βc<0
C1:l2βc<0
R1
Q1
T1
a
8.8 9 9.2
β
0 0.02
µ
8.9 9
9.1 9.2
ν U:l1βc>0
C2:l2βc>0
S:l1βc<0
C1:l2βc<0
R2
Q2
T2
b
Figure 6:aHopf surfaceβ βcand curves{l1 0}solidand{l2 0}dashedforα 2.bHopf
surfaceββcand curves{l10}solidand{l20}dashedforα3.
coefficient at the pointQ1, that is, for the caseα 2 or equivalently forRRp. For the case
α3 the calculations are very similar.
Theorem 4.6. Considerα2in system1.1. Then there is a unique pointQ1Qα2 µ, ν, β2c
where the surfaces{l1 0}and {l2 0}intersect transversally on the Hopf hypersurfaceH2. For the parameter values at the pointQ1, system1.1has a transversal Hopf point of codimension 3 at
E which is unstable sincel3Q1 > 0. The bifurcation diagram of system1.1at the pointQ1 is illustrated inFigure 7. InFigure 8the bifurcation diagram of system1.1at a typical pointR1 of
β
l1
l2 T Q1
R1
Figure 7:Bifurcation diagram of system1.1forα2 at the pointQ1.
V3
V4
V5
V8 V9
V7 V2
V1 V3
V6 β
l1 R1
a
V1 V2 V3
V4 V5 V6
V7 V8 V9
b
Figure 8:aTypical pointsV1,V2,V3,V4,V5,V6,V7,V8, andV9in the parameter space near the point
R1seeFigure 7.bPhase portraits of system1.1for the flow restricted to the center manifold and its
Computer Assisted Proof
The pointQ1is the intersection of the surfaces{l10}and{l20}on the Hopf hypersurface
ββ2ctaking into accountα2. It is defined and obtained as the solution of the equations
gµ, ν, α 0 see Theorem 4.3 and hµ, ν, α 0 see Theorem 4.4. The point Q1 has
coordinates
µ0.01528952519719747820. . . , ν4.03634524181150646828. . . ,
β2c4.02335387131139691747. . . .
4.26
The existence and uniqueness of Q1 with the previous coordinates has been established
numerically with the software MATHEMATICA 518 see also 17and can be checked
by the Newton-Kantorovich Theorem19.
For the pointQ1take five decimal round-offcoordinatesµ0.01528,ν4.03634, and
β2c4.02335 for simplicity. For these values of the parameters one has
p −0.105420.46290i,0.210840.10811i,0.22356−0.43599i,
q 3.789490.96718i,0.48359i,4.02335,
h11 −371.10203,0,−371.10203,
h20 82.2586660.23037i,−1.9571026.99100i,112.278298.14125i,
h30
⎛
⎜ ⎜ ⎝
1209.864653423.09545i
−366.528491469.19120i
4074.400901016.46677i
⎞
⎟ ⎟ ⎠,
G21−2878.05753i,
h21
⎛
⎜ ⎜ ⎝
13925.763642288.71956i
−623.13603−1297.11421i
13152.944765184.29327i
⎞
⎟ ⎟ ⎠,
h40
⎛
⎜ ⎜ ⎝
−83816.60065166465.53895i
−47712.5468379760.69093i
165895.8842099238.29710i
⎞
⎟ ⎟ ⎠,
h31
⎛
⎜ ⎜ ⎝
2.26342·1061.12088·106i
−87119.52533376583.80975i
2.56884·106435080.93606i
⎞
⎟ ⎟ ⎠,
h22
−8.42248·106,0,−8.42248·106,
h32 ⎛
⎜ ⎜ ⎝
5.84180·1081.09060·108i
−1.70175·107−6.11162·107i
5.51875·1082.34141·108i
⎞
⎟ ⎟ ⎠,
h41
⎛
⎜ ⎜ ⎝
1.25475·1081.68063·108i
−1.64672·1076.39292·107i
2.25787·1085.77662·107i
⎞
⎟ ⎟ ⎠,
h42
⎛
⎜ ⎜ ⎝
1.93748·10119.38389·1010i
−7.94067·1092.60031·1010i
2.15396·10114.67300·1010i
⎞
⎟ ⎟ ⎠,
h33
−7.28682·1011,0,−7.28682·1011,
G431.17553·1010−1.14289·1013i.
4.27
From3.9,3.10,3.11, and4.27one has
l1Q1 0, l2Q1 0, l3Q1
1
144ReG43
1 144
1.17553·1010. 4.28
The previous calculations have also been corroborated with 20 decimals round-offprecision
performed using the software MATHEMATICA 518; see17.
The gradients of the functionsl1, given in4.20, andl2, given in4.23, at the pointQ1
are, respectively,
181.95647,−14.19675, −5.26376·106,4.12653·106. 4.29
The transversality condition at Q1 is equivalent to the nonvanishing of the determinant
of the matrix whose columns are the above gradient vectors, which was evaluated and
has furnished the value 6.76121·108 > 0. The transversality condition being satisfied, the
bifurcation diagrams in Figures7 and 8 follow from the work of Takens 16, taking into
account the orientation and signs.
Definel0 β−β2c. The open region in the parameter space where system1.1has
three small periodic orbits bifurcating from the equilibria E± can be described byl2 < 0,
l1 > 0, andl0 > 0 with|l2| ≫ l1 ≫ l0 > 0. SeeFigure 7. The phase portraits of system1.1
for the flow restricted to the center manifold and its continuation are shown inFigure 8b.
For parameters at V1 the equilibrium is unstable; for parameters atV2 the equilibrium is
an weak unstable focusHopf point with positivel1; for parameters atV3 the equilibrium
is stable and an unstable limit cycle appears from the Hopf bifurcation; for parameters at
V4 the equilibrium is an weak stable focus Hopf point with negative l1 and there is an
unstable limit cycle; for parameters atV5the equilibrium is unstable and a stable limit cycle
appears from the Hopf bifurcation, so there are two limit cycles encircling the equilibrium;
a nondegenerate fold bifurcation of the cycles; for parameters atV7the equilibrium is stable
and two cycles collide corresponding to a nondegenerate fold bifurcation of the cycles;
for parameters at V8 the equilibrium is stable and two cycles collide corresponding to a
nondegenerate fold bifurcation of the cycles; for parameters atV9 the equilibrium is stable
and there are three limit cycles encircling the equilibrium.
5. Numerical Simulations
In this section we present some numerical simulations of system1.1for several values of
the parametersν,µ,α, andβ. The main purpose is to illustrate the creation of stable and
unstable limit cycles through the Hopf bifurcations at the equilibria E0 and E±, proved to
occur in the previous sections. The simulations were developed using the Software MAPLE 8, under the Runge-Kutta method of fourth order with several different stepsizes. We also perform a numerical study of the continuation of periodic orbits which arise from the Hopf
bifurcations at the pointE±. Throughout this analysis we have found period doubling and
homoclinic bifurcations which seems to lead to the creation of strange attractors for1.1,
for some parameter values. For the study of homoclinic bifurcations in three-dimensional
systems see20,21.
5.1. Bifurcations at
E
0Forµ <0, system1.1has the originE0 0,0,0as its unique equilibrium point. According
toTheorem 4.1, the system undergoes a Hopf bifurcation when the parameterβcrosses the
critical valueβ1cνα−µν1αµ/αwith−1< αµ <0. This type of bifurcation is illustrated
in Figures9 and10. To draw these figures we have takenν 3,µ −0.25,α 2 and have
varied the parameterβ, whose values are presented in the captions of the figures. Observe
that for this parameter values we haveβ1c2.0625. One can observe that when the parameter
βtends to the critical valueββ1cfrom above the spiralling behavior of the solution becomes
slower as expected, corroborating the correctness of the calculations presented in the previous
sectionssee Figures9aand9b; the limit cycle which arises when the parameterβcrosses
β β1c is very small observe the range of the state variablesx, y, z in Figure 10. The
numerical analysis performed suggests that such small limit cycle which birth in the Hopf
bifurcation forβ β1c dies forβ 0; that is, the cycle exists forβ ∈ 0,2.0625. One can
observe that as the parameterβstay away from the critical valueβ1cthe solutions tend faster
to the limit cycle see Figure 11. The shape and behavior of the limit cycles in Figure 11
resemble the one of the Van der Pol system, see22, page 267.
5.2. Bifurcations at
E
±Forµ >0, system1.1has the originE0and also the symmetric equilibriaE± ±√µ,0,±√µ.
According toTheorem 4.3, the system undergoes a Hopf bifurcation when the parameter β
crosses the critical valueβ β2c να2µν1−2αµ/α, for 0 < αµ < 1/2. This type of
bifurcation is illustrated in Figures12and13, where we have takenν 10,µ 0.1,α 2
and have varied the parameterβ, whose values are presented in the captions of the figures.
Observe that for these values of the parametersν,µ, andαwe haveβ2c 12. In the figures
−2 0 2 4 ×10−4
z t 4 2 0 −2 ×10−4 x
t
1 0.5 0 −0.5 − 1 ×10−4 yt
a −6 −4 −2 0 2 4 6 ×10−4
z t 5 0 −5 ×10−4 x
t
2 1 0 −1 −
2 ×10−4 yt
b
Figure 9:Phase portrait of system1.1near the equilibrium pointE0for the parameter values ν 3,
µ−0.25,α2, andaββ1c0.52.5625;bββ1c0.052.1125. Time of integration is5,350
fora 5,450forb. Initial condition is0.001,−0.001,0.001. Stepsize0.1. In both cases the pointE0is
a stable focus, but the convergence in the casebis slower than inasince the parameterβis closer to the critical valueβ1c2.0625. Eigenvalues atE0areaλ1−2.624,λ2,3−0.062±0.853i;bλ1−2.737,
λ2,3−0.006±0.760i.
−0.1 −0.05 0 0.05 0.1
z
t
0.1 0
−0.1
x t
0.04 0.02 0 −0.02 − 0.04
yt
a
−0.1 −0.05 0 0.05 0.1
z
t
0.1 0
−0.1
x t
0.04 0.02 0 −0.02 − 0.04
yt
b
Figure 10:Limit cycle created in the Hopf bifurcation for system1.1at the pointE0. The parameter values
areν3,µ−0.25,α2, andββ1c−0.11.9625. Time of integration is450,750inaand900,950
inb. Initial condition is0.001,−0.001,0.001. Stepsize0.1. The pointE0is an unstable focus and a limit
cycle arises around it. Eigenvalues atE0areλ1−2.774,λ2,30.012±0.728i.
β > β2c. Observe that when the parameterβtends to the critical valueβ β2cthe spiralling
behavior of the solutions becomes slowerFigure 12b; forβ < β2cthe unstable manifolds
spiral to the symmetric limit cycles arise in the Hopf bifurcation forββ2cFigure 13.
We observe that when the parameter βmoves away from the critical value β2c, the
numerical simulations performed suggest that firstly a duplication of period occurs with
the limit cycles and then two double symmetric homoclinic loops to the originE0 appear
see Figure 14. After this, a strange attractor seems to exist near these homoclinic loops Figure 15. This is one of the mechanisms through which system 1.1enters into chaotic regime. Observe that it begins with the creation of the limit cycles in the Hopf bifurcations
which takes place at the pointsE±for the critical value of parameterββ2c. For the caseα1
−0.2 −0.1 0 0.1 0.2 0.3
z
t
0.4 0.2
0 −0.2
−0.4
x t
0.15 0.05 −
0.05 −0.15
yt
a
−0.1 −0.05 0 0.05 0.1
z
t
0.5 0
−0.5
x t
0.3 0.2 0.1 0 −
0.1−0.2−0.3
yt
b
Figure 11:Limit cycle created in the Hopf bifurcation for system1.1at the pointE0. The parameter values
areν3,µ−0.25,α2, andββ1c−11.0625 ina;ββ1c−20.0625 inb. Time of integration is
0,150inaand0,100inb. Initial condition is0.001,−0.001,0.001. Stepsize0.1. As the parameter
βstay away from the critical valueβ1cthe solutions tend faster to the limit cycle. Eigenvalues atE0area
λ1−2.985,λ2,30.117±0.502i;bλ1−3.203,λ20.418,λ30.034.
−0.4 −0.2 0 0.2 0.4
z
t
0.02 0.01 0
−0.01 −0.02
yt
−0.2 0.2
xt
a
−0.4 −0.2 0 0.2 0.4
z
t
0.03 0.01
−0.01 −0.03
yt
−0.2 0.2
xt
b
Figure 12:Phase portraits of system1.1before the Hopf bifurcation at the pointsE±. The parameter
values areν10,µ0.1,α2, andββ2c0.812.8 ina;ββ2c0.212.2 inb. Time of integration
is−4.5,50fora;−4.5,270forb. Initial conditions are0.0001,0,0.0001 and−0.0001,0,−0.0001. Stepsize0.05. Eigenvalues atE0areaλ11.951,λ2,3−1.475±2.093i;bλ11.987,λ2,3−1.493±
1.976i.Eigenvalues atE±areaλ1−3.928,λ2,3−0.035±2.552i;bλ1−3.981,λ2,3−0.009±2.475i.
5.3. Bifurcation at the Critical Point
Q
1in the Parameter Space
As stated inTheorem 4.6, the most complex Hopf bifurcation at the pointsE± occurs at the
pointQ1 µ, ν, β2cin the parameter space, when we considerα2 fixed. In this subsection
we present some numerical simulations of the solutions of system1.1forµ, ν, βnear and
at the pointQ1 µQ1, νQ1, βQ1 0.015289525,4.036345241,4.023353871. InFigure 16we
show an approximation of the unstable manifolds of the equilibrium E0. These manifolds
spiral to the equilibriaE±ifβ > βQ14.023353871 and this spiralling behavior become slower
asβtends toβQ1β2cfrom aboveforµµQ1andννQ1fixed.
Forµ, α, βat the pointQ1seeTheorem 4.6the system presents a highly degenerate
behavior. In fact, due to the vanishing of the first and second Lyapunov coefficients,l1and
−0.4 −0.2 0 0.2 0.4
z
t
0.03 0.01
−0.01 −0.03
yt
−0.2 0.2
xt
a
−0.4 −0.2 0 0.2 0.4
z
t
0.03 0.01
−0.01 −0.03
yt
−0.2 0.2
xt
b
Figure 13: Phase portraits of system 1.1 after the Hopf bifurcation at the points E±. The parameter
values areν 10,µ 0.1,α 2, andβ β2c−0.2 11.8. The equilibria become unstable and two
symmetric limit cycles arise. Time of integration is−4.5,270fora;500,550forb. Initial conditions are0.0001,0,0.0001,−0.0001,0,−0.0001,0.5,0.01,0.5, and−0.5,0.01,−0.5fora;0.0001,0,0.0001
and−0.0001,0,−0.0001forb. Stepsize0.05. Eigenvalues atE0areλ1 2.012,λ2,3 −1.506±1.895i.
Eigenvalues atE±areλ1−4.018,λ2,30.009±2.423i.
−0.4 −0.2 0 0.2 0.4
z
t
0.04 0.02 0
−0.02 −0.04
yt
−0.4 0
0.4
xt
a
−0.4 −0.2 0 0.2 0.4
z
t
0.04 0.02 0
−0.02 −0.04
yt
−0.4 0
0.4
xt
b
Figure 14:aPeriod doubling bifurcation for the limit cycle created in the Hopf bifurcation at the point
E±. The parameter values areν 10,µ 0.1,α2, andβ β2c−1.13 10.87.bDouble symmetric
homoclinic loop to the origin. Parameter valuesν10,µ0.1,α2, andββ2c−1.21432210.785678.
Initial conditions are0.0001,0,0.0001and−0.0001,0,−0.0001. Eigenvalues atE0areaλ12.073,λ2,3 −1.536±1.697i;bλ12.079,λ2,3−1.539±1.678i.Eigenvalues atE±areaλ1−4.104,λ2,30.052±2.300i;
bλ1−4.111,λ2,30.055±2.289i.
integration. This behavior is shown inFigure 17. Although we have proved theoretically the
instability of the equilibriaE±inTheorem 4.6, sincel3Q1>0, the solutions move away from
these points very slowly, with repeller rate as 10−9. Then, it is not detected numerically and
the solutions near the equilibriaE± seem to be periodic as shown in Figure 17. In addition
there is also a “big” periodic orbit encircling the equilibriaE±. It would be very interesting
to observe how the real electronic circuit behave for the parameter values at the bifurcation
pointQ1.
Forβ < βQ1, the equilibria become unstable and the solutions starting near them tend
to a “big” periodic orbitseeFigure 18, which seems to be the unique attractor of the system
−0.4 −0.2 0 0.2 0.4
z
t
0.04 0.02
0 −0.02
−0.04
yt
−0.4 0
0.4
xt
Figure 15:Strange attractor created after the existence of the double homoclinic loop shown inFigure 14. Parameter values are ν 10, µ 0.1,α 2, andβ β2c−1.743 10.257. Initial conditions are
0.0001,0,0.0001 and −0.0001,0,−0.0001. Eigenvalues atE0 are λ1 2.115, λ2,3 −1.557±1.556i;
eigenvalues atE±areλ1−4.161,λ2,30.080±2.218i.
−0.1 −0.05 0 0.05 0.1
z
t
0.004 0.002 0
−0.002 −0.004
yt
−0.1 0
0.1
xt
a
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
z
t
0.006 0.002
−0.002 −0.006
yt
−0.1 0 0.1
xt
b
Figure 16:Phase portraits of system1.1before the Hopf bifurcation at the pointsE±. The parameter values
are near the pointQ1ofTheorem 4.6in the parameter space:µµQ1 0.015289525,ννQ14.036345241, andββQ0.54.523353871 ina;ββQ0.014.033353871 inb. Time of integration is−4,50 a;
−4,1380 b. Initial conditions are0.001,0,0.001and−0.001,0,−0.001. Stepsize0.01. Eigenvalues at
E0areaλ10.280,λ2−1.602,λ3−0.617;bλ10.357,λ2−1.937,λ3−0.358.Eigenvalues atE±
areaλ1−1.891,λ2,3−0.115±0.530i;bλ1−2.119,λ2,3−0.002±0.484i.
6. Concluding Remarks
In this paper we analyze the Lyapunov stability of the equilibria E0 and E± of a Van der
Pol-Duffing system with a parallel resistor given by1.1. Through these analyses we obtain
the hypersurfaces in the 4-dimensional parameter space for which the system presents Hopf
bifurcations at these equilibria Figure 2. Then we make an extension of the analysis to
the more degenerate cases, happening in the locus on the Hopf hypersurfaces where the Lyapunov coefficients associated to the Hopf bifurcations vanish. The bifurcation analysis at
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
z
t
0.008 0.004 0
−0.004
yt −0.15
−0.05 0.05 0.15
xt
Figure 17:Phase portrait of system1.1for the parameters at the pointQ1 µQ1, νQ1, βQ1ofTheorem 4.6 in the parameter space:µ, ν. β.The solutions near the equilibriaE±behave like periodic solutions. Time
of integration is10500,12500. There are several initial conditions. Stepsize0.1. There is also a “big” periodic orbit encircling the equilibriaE±. Eigenvalues atE0 areλ1 0.359, λ2 −1.942, λ3 −0.355.
Eigenvalues atE±areλ1−2.123,λ2,3−0.312×10−10±0.483i≈ ±0.483i.
−0.2 −0.1 0 0.1 0.2
z
t
0.03 0.01 −0.01
−0.03
yt
−0.1 0.1 xt
a
−0.2 −0.1 0 0.1 0.2
z
t
0.03 0.01 −0.01
−0.03
yt
−0.1 0.1 xt
b
Figure 18:Phase portrait of system1.1forµ µQ1,ν νQ1, andβ 3.973353871 < βQ1. The three equilibria are unstable and there exists a “big” stable periodic orbit encircling them. Eigenvalues atE0are
λ10.368,λ2−1.969,λ3−0.337.Eigenvalues atE±areλ1−2.144,λ2,30.010±0.478i.
Lyapunov coefficients, which makes possible the determination of the Lyapunov stability at
the equilibria as well as the largest number of small periodic orbits bifurcating of these points. With the analytic data provided in the analysis performed here, the bifurcation
diagrams are established. Figures3,4, and8provide a qualitative synthesis of the dynamical
conclusions achieved for the parameter values where system 1.1 has the most complex
equilibria. For parameters inside the open “tongue” see Figures 7 and 8 there are one
attracting periodic orbit and two other unstable periodic orbits coexisting with the attracting
equilibrium pointsE±.
The calculations of the Lyapunov coefficients, being extensive, rely on Computer
