Nonlinear Dyn (2007) 48:55–76DOI 10.1007/s11071-006-9051-y

O R I G I NA L A RT I C L E

A degenerate Hopf–saddle-node bifurcation analysisin a family of electronic circuits

A. Algaba · E. Gamero · C. Garcıa · M. Merino

Received: 5 December 2005 / Accepted: 10 April 2006 / Published online: 29 November 2006C© Springer Science + Business Media B.V. 2006

Abstract This paper presents a study of a three-parameter unfolding of a degenerate case in the Hopf–saddle-node singularity. This analysis shows that thisnonlinear degeneracy is a source of interesting bifur-cations of periodic orbits as well as global bifurcationsof equilibria. The results achieved are applied to thestudy of a simple autonomous electronic circuit, whichhas just only one nonlinearity. The numerical results in-clude the analysis of interesting resonance behaviors.

Keywords Bifurcations . Hopf–saddle-node .

Resonances . Invariant torus

1 Introduction

The local bifurcation theory is a very important tool inthe understanding of the dynamical behavior of familiesof dynamical systems. It is based on several techniquesthat look for the simplification of a given dynamicalsystem (in the dimension and also in the analytical ex-pression), leading to canonical systems whose analysiswill provide interesting information for the original sys-

A. Algaba · C. Garcıa · M. MerinoDepartment of Mathematics, Facultad de Ciencias,University of Huelva, Spain

E. Gamero (�)Department of Applied Mathematics II, E. S. I., Universityof Sevilla, Spainemail:estanis@us.es

tem. Our study reaches these simplifications using themethod of center manifolds and normal forms for thereduced system. The local bifurcation analysis providesa first insight of the bifurcation set around equilibriumpositions, and it is very useful as a guide in the use ofthe adequate numerical methods.

In the case of codimension-two local bifurcationsfor equilibria of tridimensional systems having a zeroand a pair of pure imaginary eigenvalues, the bifurca-tion analysis is based on the rotational symmetry ofthe normal form. This allows us to reduce the studyto a planar system, which can be understood as an ap-proximating local Poincare map of the full system byregarding the two-dimensional flow in the right half-plane.

To be more precise, in the study of the nondegen-erate Hopf–saddle-node bifurcation, one reduces theproblem to the following two-parameter unfolding ofthe normal form for the Hopf–saddle-node singularityin cylindrical coordinates (see, e.g., Guckenheimer andHolmes [1, Section 7.4]):

ρ = μ1ρ + a1ρz + . . . ,

z = μ2 + b1ρ2 + b2z2 + . . . ,

θ = ω0 + . . . ,

and some conditions on the vector field are imposed.Namely, the coefficients a1, b1, b2 are assumed to bedifferent from zero. In this paper, we follow the notationof Guckenheimer and Holmes [1, Section 7.4]), and the

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56 Nonlinear Dyn (2007) 48:55–76

four nondegenerate cases that can appear are labelledwith roman numerals.

Several cases of nonlinear degeneracies have beenalready considered: Langford [2], Dangelmayr andWegelin [3], Krauskopf and Rousseau [4]. The case ofthe family we plan to study (a1 = 0) can be embeddedinto the one considered in Dangelmayr and Wegelin [3],where a codimension-four case is described. Neverthe-less, this work does not cover all the possible situationsthat can appear. It is remarkable that our theoreticalstudy of the codimension-three degenerate case showsthe existence and location of several global bifurca-tions not described by these authors (although in thedegenerate case considered in [3], some of these globalbifurcations are numerically detected).

We include a case study which corresponds to theelectronic circuit represented in Fig. 3, which is de-signed by coupling two simple subsystems: one of themis linear with two branches, and the other one is one-dimensional with a nonlinear element, which we modelby a third-degree polynomial.

This idea of coupling simple systems will lead usto systems with complex dynamical behavior: a widescenario of bifurcations appear because the appearanceof several degeneracies of equilibria; which range froma single-zero to a triple-fold zero degeneracy of equi-libria (see Algaba et al. [5]).

This paper is organized as follows. In Section 2 weconsider the theoretical study of a degenerate case inthe Hopf–saddle-node bifurcation: firstly, we show thatthe singularity is determined under higher-order per-turbations; later we study a codimension-three unfold-ing of the planar singularity (obtained by disregard-ing the azimuthal component), where we connect thenondegenerate cases I←→IIa, III←→IVa,b (follow-ing the nomenclature of Guckenheimer and Holmes[1, Section 7.4]). It is remarkable that some problemsappearing in the nondegenerate case IIa are overcomein this degenerate case: the periodic orbit born in theHopf bifurcation (which grows without local bound inthe nondegenerate case) is limited in the degeneratecase due to the presence of a Takens-Bogdanov bifur-cation.

Beyond the bifurcations of the nondegenerate(codimension-two) cases, we point out the appearanceof several additional bifurcations: the above-mentionedTakens-Bogdanov bifurcation, and several kinds ofglobal connections. In Appendix A we include thetechnical proofs for different bifurcations of Section

2. The implications for the three-dimensional flow areconsidered in a standard way in Subsection 2.4. Here,we return to the tridimensional scenario (including theazimuthal component) by using the well known rela-tion that exists between the two- and three-dimensionalflows: equilibria on the z-axis remain equilibria, equi-libria outside the z-axis become periodic orbits andperiodic solutions turn out into invariant tori.

Finally, in Section 3, we address a case study corre-sponding to a model of an autonomous tridimensionalelectronic circuit, made up by a inductance L1, two ca-pacitors C , C2, two resistors R1, R2 and a nonlinearconductance of current-voltage characteristic i(V ).

The numerical study performed provides numericalverification of the bifurcations theoretically detected inSection 2.

We remark that the ideas presented in this paper pro-vide an analytical procedure to detect several kinds ofdegeneracies of periodic orbits, including several typesof Takens-Bogdanov bifurcation of periodic orbits.

Also, our study allows us to predict resonance phe-nomena on the invariant torus that appears in the sec-ondary Hopf bifurcation of periodic orbits. In particu-lar, we are able to detect angular degeneracies. Also,some aspects related to the shape of the saddle-nodebifurcation of periodic orbits curves that bound the res-onance zones are derived.

2 A nonlinear degeneracy in the

Hopf–saddle-node bifurcation

Our first goal is the analytical study of a degenerate casein the Hopf–saddle-node bifurcation which involves thenondegenerate cases I←→IIa, III←→IVa,b (follow-ing the nomenclature of Guckenheimer and Holmes [1,Section 7.4]). Recall that, in the nondegenerate case IIa,the periodic orbit born in the Hopf bifurcation growsand leaves any small neighborhood of the equilibrium.Consequently, the local analysis of the associated pla-nar system is incomplete. However, in our degeneratecase, the existence of this periodic orbit is limited due tothe presence of a homoclinic orbit related to a Takens-Bogdanov bifurcation, which completes the local anal-ysis.

We start by reviewing some facts on the nondegener-ate Hopf–saddle-node bifurcation, which can be found,e.g., in Guckenheimer and Holmes [1], Chow et al. [6],Kuznetsov [7], Wiggins [8].

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Nonlinear Dyn (2007) 48:55–76 57

A basic idea in the study is the use of the simplestnormal form for the Hopf–saddle-node singularity,which can be found in Algaba et al. [9], together withcomputer algebra algorithms for its computation. Inthat paper, the following simplest normal form in cylin-drical coordinates (X = ρ cos θ, Y = ρ sin θ, Z = z) isderived:

ρ = a1ρz + a2ρz2 + a3ρ3 + . . . ,

z = b1ρ2 + b2z2 + b3z3 + . . . , (2.1)

θ = ω0 + c1z + c2ρ2 + c3z2 + . . . .

Taking the angular variable as time, we reduce the prob-lem to the following planar one:

ρ ′ = a1ρz + a2ρz2 + a3ρ3 + . . . ,

(2.2)z′ = b1ρ

2 + b2z2 + b3z3 + b4ρ2z + . . . .

The nondegenerate case assumes some hypothesis onthe normal form (2.1): a1 �= 0, b1 �= 0, b2 �= 0, a1 �= b1,and up to four cases for the two-parameter unfoldingarise.

There are some papers devoted to analyze some de-generate situations: the case b2 = 0 has been consid-ered by Langford [2] in the Hopf-hysteresis bifurca-tion; Krauskopf and Rousseau [4] analyzed the caseb1 = 0; and a codimension-four case, corresponding toa1 = b2 = 0 has been considered by Dangelmayr andWegelin [3].

We focus here in the degenerate case a1 = 0 (thatglues together the nondegenerate cases I ←→ IIa,III ←→ IVa,b following the classification of [1, Sec-tion 7.4]). This case is embedded in the one consideredby Dangelmayr and Wegelin [3], but this work does notcover all the possible situations that can appear and it ismainly devoted to numerical aspects. In particular, weshow the presence of several bifurcations not detectedby these authors (corresponding to different kinds ofheteroclinic connections), and also we provide theoret-ical results about existence, uniqueness and localizationof the bifurcation loci.

In our analysis, we assume some non-degeneracyconditions on system (2.2). Namely, we will assumeb1 �= 0, b2 �= 0. It is easy to show that, by meansof a suitable rescaling, we can achieve b1 = b = ±1,b2 = −1. Moreover, by taking the variable r = ρ2, we

arrive to:

r ′ = 2a2r z2 + 2a3r2 + . . . ,

z′ = br − z2 + b3z3 + b4r z + . . . .

The simplest formal normal form under equivalencefor this kind of systems has been considered in Algabaet al. [10], following ideas of quasi-homogeneous ex-pansions of the vector fields. The quoted normal formis:

r =∞∑

l=2

clr l +∞∑

l=2

alr l z,

(2.3)z = br − z2.

Under the additional hypothesis c2 �= 0, we can furthersimplify the above normal form, obtaining:

r = r2 +∞∑

l=2

alr l z,

(2.4)z = br − z2.

2.1 Study of the singularity

Here, we study the determinacy of system (2.4) trun-cated to second order, i. e., we plan to show that

r = r2,

(2.5)z = br − z2,

and system (2.4) are qualitatively equivalent.Axis r = 0 is invariant for system (2.4), and the dy-

namics on this axis is locally given by z = − z2, whichis determined.

To study the dynamics in a neighborhood of theorigin with r > 0, let us perform the r -directionalquasi-homogeneous blow-up (see Andronov et al. [11],Dumortier [12]): r = u, z = uz. System (2.4) becomes:

u = u2 +∞∑

l=2

alul z,

˙z = b − uz − uz2 −∞∑

l=2

alul−1 z2.

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58 Nonlinear Dyn (2007) 48:55–76

As b = ±1, we conclude that there are no equilibria onthe axis u = 0.

Next, we perform a z-directional quasi-homogeneous blow-up: r = u2r , z = u. Then, weget

˙r = r2u2 +∞∑

l=2

al r lu2l−1 − 2br2u + 2ru,

u = bru2 − u2,

which is equivalent to:

˙r = r

(−2(br − 1) + ru +

∞∑l=2

al r l−1u2l−2

),

u = u (br − 1) .

Here, both axes are invariant. The dynamics on u = 0is governed by ˙r = −2r (br−)1; and the dynamics onr = 0 is given by u = − u. On the axis r = 0 there isonly one equilibrium: the origin. With respect to theaxis u = 0, beyond the equilibrium at the origin, anotherone (located at (1, 0)), appears in the case b = +1.

The study of these equilibria is easy to perform:� The equilibrium at the origin is a hyperbolic saddle.� In the case b = +1, the linearization matrix at the

equilibrium (1, 0) is (−2 10 0

). Then, it is a semi-

hyperbolic equilibrium, whose center manifold is lo-cally given by r = 1 − 1

2 u + O(u2). The reduced sys-tem on the center manifold is u = − 1

2 u2 + O(u3).Hence, this equilibrium is determined with respectto higher-order perturbation terms.

In summary, the system (2.5) is determined for per-turbations leaving the axis r = 0 invariant. The differentphase portraits for the singularity, obtained by means ofthe corresponding blow-down, are presented in Fig. 1.

2.2 Study of a codimension-three unfolding

Next, we plan to analyze the following three-parameterunfolding of system (2.5):

r = r (μ1 + μ3z + r ), (2.6)

z = μ2 + br − z2.

Firstly, we summarize its equilibria:

Fig. 1 Different configurations for the singularities of (2.5)

� On the axis r = 0, the equilibria are given by:

(0, Z+) = (0, +√μ2), (0, Z−) = (0, −√

μ2),

whenever μ2 ≥ 0.� Outside the axis r = 0, the equilibria are

(R1, Z1)

=(

−μ1−μ3 Z1,−bμ3 +

√μ2

3 + 4(μ2 − bμ1)

2

),

(2.7)

(R2, Z2)

=(

−μ1−μ3 Z2,−bμ3 −

√μ2

3 + 4(μ2 − bμ1)

2

).

These equilibria must be considered only for pa-rameter values satisfying μ2

3 + 4(μ2 − bμ1) ≥ 0 andleading to non-negative coordinates R1, R2.

To describe the stability and the different local bifur-cations exhibited by these equilibria, we consider theirlinearization matrices:� For the equilibrium (0, Z±), the quote matrix is:(

μ1 ± μ3√

μ2 0

b ∓2√

μ2

).

Its eigenvalues are λ1 = μ1 ± μ3√

μ2, λ2 = ∓2√

μ2, so that:

– On the surface SN1 = {μ2 = 0} there is a saddle-node bifurcation where both equilibria collapse.There are no equilibria for μ2 < 0, whereas forμ2 > 0 we find that:

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Nonlinear Dyn (2007) 48:55–76 59

* (0, Z+) is a stable node for μ1 + μ3√

μ2 < 0,or a saddle otherwise.

* (0, Z−) is an unstable node for μ1 − μ3√

μ2 >

0, or a saddle otherwise.– On the surface T± = {μ2

1 − μ23μ2 = 0} there is

a transcritical bifurcation. The center mani-fold around the equilibrium (0, Z±) is givenby z = ∓ √

μ2 − bμ3

2μ1r + O(r, ε±)2, where ε± =

μ1 ± μ3√

μ2, and the reduced system on the

center manifold becomes r = r (ε± + 2μ1−bμ23

2μ1r ),

which has an equilibrium at r = 2μ1ε∓2μ1−bμ2

3> 0, pro-

vided that 2μ1 − bμ23 �= 0.

Consequently:

* The equilibrium (0, Z+) undergoes a transcriti-cal bifurcation at

T+ = {μ2

1 − μ23μ2 = 0, μ1μ3 < 0,

2μ1 − bμ23 �= 0

}.

It is supercritical if sign (2μ1 − bμ23) �=

sign (μ1), and subcritical if sign (2μ1 − bμ23) =

sign (μ1). A degenerate transcritical bifurcationoccurs at

DT+ = {μ2

1 − μ23μ2 = 0, μ1μ3 < 0,

2μ1 − bμ23 = 0

}.

* The equilibrium (0, Z−) undergoes a transcriti-cal bifurcation at

T− = {μ2

1 − μ23μ2 = 0, μ1μ3 > 0,

2μ1 − bμ23 �= 0

}.

It is supercritical if sign (2μ1 − bμ23) �=

sign (μ1), and subcritical if sign (2μ1 − bμ23) =

sign (μ1). A degenerate transcritical bifurcationoccurs at

DT− = {μ2

1 − μ23μ2 = 0, μ1μ3 > 0,

2μ1 − bμ23 = 0

}.� For the equilibria (Rk, Zk), k = 1, 2, the linearization

matrix is(Rk μ3 Rk

b −2Zk

).

Its determinant and trace are:

Det = −Rk(2Zk + bμ3)

= (−1)k Rk

√μ2

3 + 4(μ2 − bμ1),

Tr = Rk − 2Zk .

Then:

– The equilibrium (R1, Z1), whenever it exists, isa saddle.

– The equilibrium (R2, Z2), whenever it exists, isa node or focus. It is unstable for R2 − 2Z2 = −μ1 − (μ3 + 2)Z2 > 0, or stable otherwise.

Moreover, the following bifurcations are present:

– There is a saddle-node bifurcation at SN2 ={μ2 = bμ1 − μ2

34 , μ1 <

bμ23

2 } (the inequalityarise from the condition Rk > 0). Here, theequilibria (R1, Z1) and (R2, Z2) collapse.

– There is a Hopf bifurcation for the equilibrium(R2, Z2) at:

h ={μ2 = μ1

μ3 + 2

(μ1

μ3 + 2+ 2b

),

bμ3(μ3 + 2)

2< μ1 < 0

}.

The linearization matrix, for these criticalvalues, has eigenvalues ±ω0i , with ω2

0 =−R2(R2 + bμ3) = μ2

3 + 4(μ2 − bμ1). In par-ticular, the condition bμ3 < −R2 < 0 must befulfilled. A standard normal form procedureyields the expressions for the first Hopf normalform coefficient:

bμ1

2ω20(μ3 + 2)2

(μ1 + b(μ3 + 2)) .

It is easy to show that, taking the parametersclose to zero, this coefficient is negative andconsequently, the Hopf bifurcation is supercrit-ical, giving rise to a stable limit cycle (see, e.g.,Guckenheimer and Holmes [1]). Moreover, theuniqueness of limit cycles for this class of sys-tems follows from Chow et al. [13] or Zhanget al. [14].

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60 Nonlinear Dyn (2007) 48:55–76

– There is a Takens-Bogdanov bifurcation at

TB ={μ1 = μ3(μ3 + 2)

2,

μ2 = μ3(μ3 + 4)

4, bμ3 < 0

}.

Notice that, for these values, R1 = R2 = −bμ3,Z1 = Z2 = − bμ3

2 .In this case, we can also perform a standardnormal form procedure (translating the equi-librium to the origin, later doing a linear trans-formation leading the linear part to Jordannormal form and using computer algebra al-gorithms [15]). The Takens-Bogdanov normalform reads as:

x1 = x2,

x2 = −bμ3x21 − (μ3 + 2)x1x2 + . . . .

As bμ3 < 0, we obtain that the Takens-Bogdanov is non-degenerate. In particular, wecan assert that three surfaces in the tridimen-sional parameter space: the saddle-node SN2,the Hopf bifurcation h and a surface of homo-clinic connections meet in the curve TB. Theanalysis of global connections is the subject ofthe next subsection.

2.3 Global bifurcations

Beyond the local bifurcations previously reported,there are several kinds of global connections inthe three-parameter unfolding (2.6). Namely, inAppendix A we will show the following:� Let us consider system (2.6) with b = −1. Then,

there exists a function μ2 = �hetdeg(μ3), such thatalong the curve

A = {μ2 = �hetdeg(μ3),

μ1 = −μ3

√�hetdeg(μ3), μ3 > 0},

there is a heteroclinic orbit that arises when thecenter manifold (with repulsive dynamics) of thesemi-hyperbolic saddle (0, Z+) agrees with the stablemanifold of the hyperbolic saddle (0, Z−). Moreover,μ2

3(μ3+2)2

4(μ3+1)2 < �hetdeg(μ3) < 4μ23 (see Lemma A.1).� Let us consider system (2.6) with b = −1. Then,

there exists a function μ1 = �het(μ2, μ3) such that

on the surface

R = {μ1 = �het(μ2, μ3), μ3 > 0,

0 < μ2 < �hetdeg(μ3), },

there is a heteroclinic orbit between the twohyperbolic saddles (0, Z+) and (0, Z−). More-over, −μ3

√μ2 < �het(μ2, μ3) < μ3

√μ2 (see

Lemma A.2).� Let us consider system (2.6) with b = +1. Then,there exists a function μ1 = �hom+(μ2, μ3), such thaton the surface

L ={μ1 = �hom+(μ2, μ3),

μ3 < 0,μ3(μ3 + 4)

4< μ2 < 0

},

there is a homoclinic connection for (R1, Z1).Moreover, this function satisfies μ2 −μ2

3(μ3 + 1) < �hom+(μ2, μ3) < b(μ3 + 2)(−1 +√1 + μ2). These results, among others, can be

found in Lemma A.4.� Let us consider system (2.6) with b = −1. Then,there exists a function μ1 = �hom−(μ2, μ3), such thaton the surface

L ={μ1 = �hom−(μ2, μ3), μ3 > 0,

�hetdeg(μ3) < μ2 <μ3(μ3 + 4)

4

},

there is a homoclinic connection for (R1, Z1).Moreover, this function satisfies −μ2 − μ2

34 < �hom−

(μ2, μ3) < −μ3√

μ2. These results, among others,can be found in Lemma A.5.� Let us consider system (2.6) with b = −1. Then,there exists a function μ1 = �J(μ2, μ3), such that onthe surface

J ={μ1 = �J(μ2, μ3), μ3 > 0,

�hetdeg(μ3) < μ2 <μ3(μ3 + 4)

4

},

there is a heteroclinic orbit connecting (R1, Z1)with the saddle (0, Z−). Moreover, this functionsatisfies �hom−(μ2, μ3) < �J(μ2, μ3) < −μ3

√μ2.

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Nonlinear Dyn (2007) 48:55–76 61

Fig. 2 Planar representations of the bifurcation sets for system (2.6), obtained by intersecting the three-parameter space with a sphere,and deforming it in a plane. The phase portraits in each zone are included below

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62 Nonlinear Dyn (2007) 48:55–76

Fig. 3 Schematic picture ofthe electronic circuit and itsrepresentation as a pair ofcoupled simple subsystems

These results, among others, can be found in LemmaA.6.

The above information allows us to drawn the bi-furcation sets of Fig. 2. These bifurcation sets are ob-tained by intersecting the three-parameter space with asphere (centered in the codimension-three singularity),and deforming it in a plane.

2.4 Implications for the 3D system

We take into account the corresponding changes thatarise when including the azimuthal component: equi-libria on the Z -axis remain equilibria, equilibria outsidethe Z -axis become periodic orbits, periodic solutionsturn out into invariant tori, and global connections be-tween equilibria become global connections for equi-libria and/or periodic orbits.

With these changes in mind, we can summarize thebifurcations for the tridimensional system:� The saddle-node of equilibria SN1 corresponds to

another saddle-node of equilibria which we denoteSN.� The transcritical bifurcations T± correspond to Hopfbifurcations of equilibria h±.� The degenerate transcritical bifurcations DT± corre-spond to degenerate Hopf bifurcations of equilibriaDh±.� The saddle-node of equilibria SN2 corresponds to asaddle-node of periodic orbits sn.� The Hopf bifurcation of equilibria h corresponds toa secondary Hopf (or torus) bifurcation of periodicorbits: HH.� The Takens-Bogdanov bifurcation of equilibria TB

corresponds to a Takens-Bogdanov bifurcation of pe-riodic orbits: TBPO.

� With respect to the global connections in the planarfamily, they will provide several kinds of connectionsbetween equilibria and/or periodic orbits.

Another important fact is related to the effect of thehigher-order terms truncated in the normal form pro-cedure. They affect the flow on the tori, and also theglobal connections that appear in the planar family.Generically, the behavior on the invariant torus will bedifferent when including the truncated terms, due tophase-locking. Moreover, in the truncated system, thetorus breaks in a spheroidal surface filled with orbitsjoining two equilibria or in a homoclinic orbit for aperiodic orbit. This situation will disappear by addinghigher-order terms. Instead, we must expect complexheteroclinic structure and also homoclinic behavior re-lated to equilibria and periodic orbits (T-points, Hopf-Shil’nikov, etc.).

3 Numerical study of an autonomous electronic

circuit

In this section, we plan to achieve numerical veri-fication of the bifurcations previously reported, in acase study corresponding to an autonomous electronicsystem. This circuit is schematically represented inFig. 3(a). It is obtained by coupling of two simple cir-cuits. One of them is linear with two branches (thefirst one is made up by an inductance L1 and a resistorR1, and the second one by a capacitor C2 and a resistorR2), and the other one is one-dimensional with a capac-itor C and a nonlinear conductance of current-voltagecharacteristic i(V ), which we model by a third-degreepolynomial:

i(V ) = A0 − A1V + A3V 3, with A0, A1, A3 > 0.

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Nonlinear Dyn (2007) 48:55–76 63

This corresponds to the cubic nonlinearity of the van derPol circuit, where a constant term has been included inorder to take into account imperfections in the device.Applying Kirchhoff’s laws, and taking the voltages onthe capacitors and the current across the inductance L1

as states variables, we obtain the equations:

CV ′ = −i(V ) − IL1 − V − VC2

R2,

L1 I ′L1

= V − R1 IL1 ,

C2V ′C2

= V − VC2

R2.

Let us take dimensionless variables and parameters by

t = ωT, V = κx, IL1 = κ

L1ωy1,

VC2 = κy2, being ω = 1

R2C2, κ3 = A0

2A3.

Denoting′ = d/dt , = d/dT , we get the system

x = f (x) − βy1 + r y2,

y1 = x − αy1, (3.8)

y2 = x − y2,

where

r = C2

C≥ 0, α = R1

L1ω≥ 0, β = 1

Cω2L1> 0,

f (x) = −γ0 + μx − γ0

2x3, with

γ0 = A0

Cωκ, μ = r (A1 R2 − 1).

This last system can be embedded in a wide familyof systems of the form:

x = f (x) + ct y,

(3.9)y = bx + My,

where x ∈ R, y ∈ R2, M is a 2 × 2 matrix and b, c ∈R2. As shown in Fig. 3(b), this kind of systems repre-sents a pair of subsystems (one of them linear) linearlyinterconnected, where the outputs of both subsystems:

u = ct y and v = bx , depend linearly on the state vari-ables (see Algaba et al. [5]).

Notice that, in our case, we have

b =(

11

), c =

(−β

r

), M =

(−α 0

0 −1

).

Notice that � = det(M) = α. From now on, we will as-sume α �= 0, so that we have isolated equilibria (x, y),obtained from

f (x) = (ct M−1b

)x, y = −x M−1b. (3.10)

Let us denote σ = β

α− r − μ + 3

2γ0. Then, the equi-libria of system (3.8) are (x, x/α, x) where x satisfies:

γ0 +(

β

α− r − μ

)x + γ0

2x3 = σ + σ (x − 1)

+ 3γ0(x − 1)2

2!+ 3γ0

(x − 1)3

3!= 0. (3.11)

So, we obtain that a saddle-node of equilibria SN takesplace at σ = 0.

Taking σ < 0, there are two equilibria Eq±, whichcollapse at σ = 0 and disappear for σ > 0 in a neigh-borhood of (x, y1, y2) = (

1, 1α, 1

).

As x is obtained by intersecting the cubic z = f (x)with a straight line passing through the origin with slopect M−1b, we can warrant the presence of another equi-librium, which is far from the above ones and will beomitted in our local study.

The following study is valid not only for the system(3.8), but also for the family of coupled subsystems(3.9). Translating the equilibrium (x, y) to the origin,and using adequate linear transformations (see Algabaet al. [5]), we get⎛⎜⎝ x

y1

y2

⎞⎟⎠ =

⎛⎜⎝ a 1 0

ct b τ 1

−�ct M−1b −� 0

⎞⎟⎠⎛⎜⎝ x

y1

y2

⎞⎟⎠

+ g(x)

⎛⎜⎝ 1

0

0

⎞⎟⎠ , (3.12)

where

a = f ′(x), τ = trace(M), g(x)

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64 Nonlinear Dyn (2007) 48:55–76

Fig. 4 (a): Partial bifurcation set for system (3.8) with α = 3.5,r = 2, near the degenerate Hopf–saddle-node point DHZ. (b):Zoom of the Hopf–saddle-node point HZ

= f ′′(x)

2!x2 + f ′′′(x)

3!x3.

The characteristic polynomial of the linearization ma-trix at the equilibrium is

λ3−(a + τ )λ2 + (� + aτ − ct b)λ − �(a − ct M−1b),

so that there is a Hopf–saddle-node degeneracy if

a + τ = 0, a = ct M−1b, � + aτ − ct b > 0.

(3.13)

In summary, the equilibrium (x, y) undergoes a Hopf–saddle-node degeneracy for the critical values:

a = ct M−1b, τ = −ct M−1b, (3.14)

Fig. 5 (a): Partial bifurcation set for system (3.8) with α = 2.29,r = 2, near the degenerate Hopf–saddle-node point DHZ. (b):Zoom of the Hopf–saddle-node point HZ

Fig. 6 Evolution of the argument of the Floquet multiplier (indegrees) when moving along HH in system (3.8) with α = 2.29,r = 2

whenever � − a2 − ct b > 0, or equivalently, � >

ct(I + M−1bct M−1

)b. Under these assumptions, the

linearization matrix for the equilibrium has eigenvalues0, ±iω0, being

ω0 =√

� − a2 − ct b > 0.

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Nonlinear Dyn (2007) 48:55–76 65

Fig. 7 Partial bifurcation set for system (3.8) with α = 2.29,r = 2, showing the curve of secondary Hopf bifurcations of peri-odic orbits and the curves of saddle-node bifurcations of periodicorbits corresponding to 1:2, 1:3, 1:4, 1:5, 1:6, 1:7 and 1:8 reso-nances

Fig. 8 (a): Bifurcation diagram for system (3.8) with α = 2.29,r = 2, β = 12.75 of the saddle-node bifurcation of periodic orbitssn6. (b): Projection on the x − y2 plane of a periodic orbit forsystem (3.8) with α = 2.29, r = 2, β = 12.75, μ = 6.631593776,inside the 1:6 parabolic resonance zone, near the homoclinicity

Fig. 9 (a): Partial bifurcation set for system (3.8) with α = 2.29,r = 2, showing the curve of secondary Hopf bifurcations ofperiodic orbits and the curve of saddle-node bifurcations of7T periodic orbits. (b): Qualitative picture of the bifurcationset

In order to compute the normal form for these pa-rameter values, we reduce the linearization matrix in(3.12) to Jordan normal form by means of the lineartransformation:⎛⎜⎝ x

y1

y2

⎞⎟⎠ =

⎛⎜⎝ 0 1 1

ω0 −a −a

0 −� ω20 − �

⎞⎟⎠⎛⎜⎝ X

Y

Z

⎞⎟⎠ ,

obtaining⎛⎜⎝ X

Y

Z

⎞⎟⎠ =

⎛⎜⎝ 0 −ω0 0

ω0 0 0

0 0 0

⎞⎟⎠⎛⎜⎝ X

Y

Z

⎞⎟⎠

+ g(Y + Z )

⎛⎜⎝ k1

k2

k3

⎞⎟⎠ , (3.15)

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66 Nonlinear Dyn (2007) 48:55–76

Fig. 10 (a): Partial bifurcation set for system (3.8) withα = 2.29,r = 2, showing the curve of secondary Hopf bifurcations of pe-riodic orbits and the curve of saddle-node bifurcation of 8T pe-riodic orbits. (b): Qualitative picture of the bifurcation set

Fig. 11 Bifurcation diagram for system (3.8) with α = 2.29,r = 2, β = 12.75 (inside the 1:8 banana split resonance zone)

where

k1 = aω0

, k2 = � − ω20

ω20

, k3 = �

ω20

.

Once the linear part has been normalized, we use thealgorithm presented in Algaba et al. [9], which lead usto the normal form (2.1), where the expressions for thesecond-order coefficients are:

a1 = ω20 − �

2ω20

f ′′(x), b1 = �

4ω20

f ′′(x),

b2 = �

2ω20

f ′′(x), c1 = − a2ω0

f ′′(x).

Notice that b1 �= 0, and also b2 �= 0, but a nonlineardegeneracy, corresponding to the vanishing of a1, arisesfor

� = ω20 ⇔ ct b + a2 = 0.

Now, we are able to classify the nondegenerate casesin the Hopf–saddle-node bifurcation. Following the no-tation of Guckenheimer and Holmes [1], we find for� �= ω2

0:� If sign(ct b + a2) = sign�, the case III [1, Section7.4] arises.� If sign(ct b + a2) �= sign�, the case IV [1, Section7.4] occurs. In fact, we find the case IVa for −1 <ct b+a2

�< 0 or the case IVb for ct b+a2

�< −1.

The study of bifurcations in the degenerate valuefor � = ω2

0 requires the computation of a higher-ordernormal form. Next, we present the expressions for thethird-order coefficients corresponding to the simplestnormal form under C∞-equivalence (see also Algabaet al. [9]):

a1 = 0, a2 = 0, a3 = 7a32ω2

0

f ′′(x)2,

b1 = 1

4f ′′(x), b2 = 1

2f ′′(x),

b3 = 2ω20 f ′′′(x) + 3a f ′′(x)2(2 + ω0)

12ω20

,

c1 = − a2ω0

f ′′(x), c2 = 0, c3 = 0.

In order to get information about bifurcations aroundthis degenerate bifurcation, we consider the follow-ing three-parameter unfolding for the planar systemobtained by truncating the above normal form to

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Nonlinear Dyn (2007) 48:55–76 67

Fig. 12 (a), (b): Zooms of the bifurcation sets for system (3.8) with α = 2.29, α = 2.289 respectively (r = 2 is fixed), showing theevolution of the 1:7 resonances zones (from banana to banana split). (c), (d): Qualitative pictures of the bifurcation sets

Fig. 13 Poincare section in the plane y2 = 1.015 of a 7T invariant torus obtained for parameter values α = 2.29, β =12.174999, μ = 6.325548995, on HH7

third-order and removing the azimuthal component:

ρ = ρ(ε1 + ε3z + a3ρ2),

z = ε2 + b1ρ2 + b2z2 + b3z3,

where

ε1 = (a + τ )ω20 − 2�(a − ct M−1b)

2ω20

+

O(|a + τ, a − ct M−1b|3),

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68 Nonlinear Dyn (2007) 48:55–76

Fig. 14 Relative positions of the center manifold of (0, Z+) and the stable manifold of (0, Z−): (a) taking μ2 >μ2

34 and μ2 ≈ μ2

34 , (b)

taking μ2 = 4μ23

ε2 = �2(a − ct M−1b)2

2 f ′′(x)ω40

+O(|a + τ, a − ct M−1b|3),

ε3 = ω20 − �

2ω20

f ′′(x) + O(|ε1, ε2|).

These expressions are useful in order to get numericalresults near this codimension-three degenerate Hopf–saddle-node bifurcation for this case-study.

In summary, for the electronic system (3.8),the equilibrium (x, y1, y2) = (

1, 1α, 1

)undergoes a

Hopf–saddle-node bifurcation at

μc = α + 1 + 3

2γ0,

βc = α (α + r + 1) , αr − r − 1 > 0.

In this case, the equilibrium has eigenvalues 0, ±iω0,with ω2

0 = αr − r − 1. Finally, the Hopf–saddle-node bifurcation becomes degenerate at the criticalvalues

μc = 2rr − 1

+ 3

2γ0, αc = r + 1

r − 1,

βc = r (r + 1)2

(r − 1)2, r > 1,

being ω20 = r+1

r−1 .The numerical computations we will present have

been performed with AUTO97 [16], and they have been

Fig. 15 Relative position of the unstable manifold of (0, Z+) andthe stable manifold of (0, Z−) for μ1 < μ3

√μ2, μ1 ≈ μ3

√μ2

done keeping fixed

r = 2 > 0, γ0 = 2,

so that the critical values are:

μc = 7, αc = 3, βc = 18.

We present two parameter plane bifurcation sets nearthe codimension-three degenerate Hopf–saddle-nodebifurcation by fixing α = 3.5 (see Fig. 4) and α = 2.29(see Fig. 5), respectively. These sections are taken onboth sides of the critical value αc.

Notice that the numerical study detects all the ele-ments that have been previously predicted in the ana-lytical study.

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Nonlinear Dyn (2007) 48:55–76 69

Fig. 16 Relative position of stable and unstable manifoldsof equilibrium (x+

0 , 0) for parameter values close to thehomoclinicity

In the first case (α = 3.5), we can see� The saddle-node bifurcation SN where the equilibriaEq± emerge.� The subcritical Hopf bifurcation h+ of equilibriumEq+.� The supercritical Hopf bifurcation h− of equilibriumEq−.� The degenerate Hopf bifurcation Dh−. Af-ter this point, the Hopf bifurcation h− issubcritical.� The saddle-node bifurcation of periodic orbits sn,arising from Dh−.

In the second case (α = 2.29), the following bifurca-tions appear:� The saddle-node bifurcation SN.

Fig. 17 Relative positions of the stable and unstable manifolds of (R1, Z1): (a) for μ1 = μ2 − μ23(μ3 + 1), (b) for μ1 = b(μ3 + 2)(−1 +√

1 + μ2), (c) for μ2 = 0, μ1 < 0

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70 Nonlinear Dyn (2007) 48:55–76

Fig. 18 Relative positions of the stable and unstable manifolds (R1, Z1): (a) for μ1 = − μ3√

μ2, μ2 < �hetdeg(μ3), (b) for μ1 <

−μ3√

μ2, μ1 ≈ −μ3√

μ2, (c) for μ1 > −μ2 − μ23

4 , μ1 ≈ −μ2 − μ23

4

� The supercritical Hopf bifurcation h+ of equilibriumEq+.� The subcritical Hopf bifurcation h− of equilibriumEq−.� The degenerate Hopf bifurcation Dh+. Afterthis point, the Hopf bifurcation h+ becomessubcritical.� The saddle-node bifurcation of periodic orbits sn.� The secondary Hopf bifurcation of periodic orbitsHH.� The Takens-Bogdanov bifurcation of periodic orbitsTBPO.

Also, in Figs. 4 and 5, there is a Takens-Bogdanov bi-furcation of equilibria TB (not related to the degenerateHopf–saddle-node bifurcation we are analyzing), fromwhich a curve of homoclinic connections ho emerges.In both cases, the curve ho ends when it reaches theHopf bifurcation h− and h+ respectively, giving rise toa Hopf-Shil’nikov bifurcation HS (see Hirschberg andKnobloch [17]). Notice that the Hopf bifurcation curveconnects the points TB and HZ.

Next, we focus on bifurcation behaviors relatedto the resonance phenomena on the torus that ap-pears in the secondary Hopf bifurcation HH. The

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Nonlinear Dyn (2007) 48:55–76 71

Fig. 19 Relative positions of the unstable manifold of (R1, Z1) and the stable manifold of (0, Z−): (a) for μ1 = − μ3√

μ2, �hetdeg(μ3) <

μ2,(b) for μ1 = �hom−(μ2, μ3)

periodic orbit that undergoes this bifurcation has pe-riod T ≈ 2π

ω0. It has a conjugate pair of Floquet mul-

tiplier on the unit circle. When moving along thecurve HH (which is bounded by the points HZ andTBPO), the argument of the Floquet multiplier variesas shown in Fig. 6 (here, we have fixed α = 2.29,r = 2).

This allows us to predict resonance phenomena onHH (that occur when the Floquet multiplier movesthrough a root of the unity). In particular, as in thisparticular case the argument is always less than 52 de-grees, we can expect the presence of 1:7 and higherresonances.

In the parameter plane μ − β, there is a point D

≈ (6.3375, 12.1822), where the argument fails to varymonotonically (see Fig. 6). This angular degeneracydetermines some aspects related to the shape of thesaddle-node bifurcation of periodic orbits curves thatbound the resonance zones: they have a particular shapecalled “banana” and “banana split” by Peckham et al.[18].

In Fig. 7 (where α = 2.29, r = 2 are kept fixed), wehave drawn the saddle-node bifurcations of periodicorbits curves sn2, . . ., sn8 corresponding to resonances1:2 to 1:8. Several of these curves are not related tothe secondary Hopf bifurcation HH. In this particularcase, the resonances 1:2 to 1:6 are related to the ho-moclinicity and the curves sn2, . . . , sn6 have parabolicshapes. For large parameter values, these curves have

a common asymptote related to homoclinic with dif-ferent pulses corresponding to equilibrium Eq− (seeFig. 8).

The banana resonance zone arises in our case in theresonance 1:7 (see Fig. 9). Notice that the saddle-nodeof 7T periodic orbits bifurcation curve sn7 has twobranches joining a pair of points (tips) located on HH

(each tip located on a side of the angular degeneracy D).In Fig. 9 we can see a third branch of sn7, with parabolicshape which is related to the homoclinicity. Inside thebanana resonance zone there is a pair of 7T periodicorbits, one saddle and the other one stable. Numericalevidence shows that these orbits do not undergo anybifurcations.

The banana split resonance zone arises for reso-nances 1:8, 1:9, . . . (see Fig. 10) corresponds to theresonance 1:8. Here, we can see three branches of thesaddle-node of 8T periodic orbits bifurcation curvesn8: one joining both tips, and two open branches,each one starting from one tip. Inside this bananasplit resonance zone there are 8T periodic orbits withdifferent stabilities: stable, saddle and completely un-stable. They undergo different bifurcations, in partic-ular, the merging of different resonance zones occurs(see Kirk [19]). A bifurcation diagram is presented inFig. 11.

In Fig. 12 we show how the resonance zone, in thecase of the 1:7 resonance, changes from banana to ba-nana split by moving the parameter α.

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72 Nonlinear Dyn (2007) 48:55–76

As commented before, for α = 2.29, the saddle-node of 7T periodic orbits bifurcation curve sn7 hasseveral branches. A deep analysis shows that, on thebranch with parabolic shape related to the homoclinic-ity, there are two codimension-two points TBPO7 cor-responding to Takens-Bogdanov of 7T periodic orbits(this degeneracy corresponds to a non-diagonalizabledouble +1 Floquet multiplier). From TBPO7, a sec-ondary Hopf bifurcation curve of the 7T periodic or-bits HH7 emerges. The curve HH7 ends when it meets aperiod-doubling curve pd7. The intersection point is an-other Takens-Bogdanov bifurcation of periodic orbitsTBPO′

7, corresponding to non-diagonalizable double−1 Floquet multiplier.

These additional bifurcations (TBPO7 and pd7)are not related to the Hopf–saddle-node degeneracywe are dealing with, but to a triple-zero degeneracy(see [5]).

Decreasing a bit the parameter α (we have takenα = 2.289), the parabolic branch of sn7 has joined withthe upper branches joining the tips, giving rise to apair of open branches, each one starting from one tip.Notice that for this value of α, the Takens-Bogdanov of7T periodic orbits TBPO7, TBPO′

7, the secondary HopfHH7 and the period-doubling pd7 still exist.

A detailed study of this procedure, in the case of adegenerate codimension-three Hopf–pitchfork bifurca-tion in a Z2-symmetric electronic circuit, can be foundin Algaba et al. [20].

In Fig. 13 we present a Poincare section inthe plane y2 = 1.015 of a 7T invariant torus ob-tained for parameter values near HH7 (α = 2.29, β =12.174999, μ = 6.325548995).

Appendix A: Proofs

In this Appendix we include the proofs of the resultssummarized in Subsection 2.3, concerning to differentkinds of global connections in the three-parameter un-folding (2.6).

Appendix A.1: Degenerate heteroclinic connections

Here, we consider system with b = − 1 and μ3 > 0.Along the transcritical bifurcation (μ1 = − μ3

√μ2,

μ1 ≤ 0), system (2.6) becomes:

r = P(r, z) = r (−μ3√

μ2 + μ3z + r ), (A.16)

z = Q(r, z) = μ2 − r − z2.

Its equilibria are (0, Z±) = (0, ±√μ2) and (R2, Z2) =

(μ3(2√

μ2 − μ3), μ3 − √μ2) (assuming

√μ2 >

μ3/2). They undergo the following bifurcations:� A saddle-node bifurcation for equilibria (0, Z±) atSN1 ≡ {μ2 = 0},� A transcritical bifurcation for equilibria (0, Z+) =(R2, Z2) at T− ≡ {√

μ2 = μ3

2

}, and� A Hopf bifurcation of equilibrium (R2, Z2) at h ≡{

μ2 = μ23(μ3+2)2

4(μ3+1)2

}.

Next result shows the existence of heteroclinic connec-tions in this case:

Lemma A.1. Let us consider system (A.16) with μ2 >

0. Then, there exists a function μ2 = �hetdeg(μ3), de-fined for μ3 > 0, such that for the parameter val-ues μ2 = �hetdeg(μ3), the system presents a hetero-clinic orbit connecting the center manifold with repul-sive dynamics of the semi-hyperbolic saddle (0, Z+)

and the hyperbolic saddle (0, Z−). Moreover, μ23

4 <

�hetdeg(μ3) < 4μ23.

Proof: Let us consider the straight line of vertical iso-clines −μ3

√μ2 + μ3z + r = 0. On this straight line,

we consider the points A = (r+, z+) and B = (r−, z−)corresponding, respectively, to the intersection with thecenter manifold of the equilibrium (0, Z+), and the sta-ble manifold of (0, Z−) (see Fig. 14). Let us denoted = z+ − z−.

We have the following properties:� For the parameter value μ2 = μ23

4 , the equilibrium(R2, Z2) collapses with (0, Z+). Then, taking μ2 >μ2

34 but close enough, both equilibria are connected,

so that d > 0 (see Fig. 14(a)).� For the parameter value μ2 = 4μ23, the straight

lines r = μ2, z = − √μ2 intersect the straight line

of vertical isoclines z = − 1μ3

r + √μ2 in a point

P = (4μ23, −2μ3). As shown in Fig. 14(b), the point

A is located below P, whereas the point B is locatedabove it. Then, we have d < 0.� For μ2 > 0 fixed, due to the continuity with respectto the parameters, we can assure the existence of a

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Nonlinear Dyn (2007) 48:55–76 73

value of μ3, satisfying 12

√μ2 < μ3 < 2

√μ2, such

that d = 0.� To show that the above value of μ3 is unique, we usethat∣∣∣∣∣ P Q

∂ P∂μ3

∂ Q∂μ3

∣∣∣∣∣ = −Q∂ P∂μ3

= −r (z − √μ2)

(μ2 − r − z2) < 0.

Then, we deal with a one-parameter family of rotatedvector fields (see Andronov et al.[21], Perko [22]).This means that increasing μ3 (keeping μ2 fixed), theequilibria (0, Z±) remain fixed and the vector fieldrotates clockwise. Then, function d(μ3) is strictlymonotone and its root is unique.

To obtain the estimation for the function μ2 =�hetdeg(μ3), we use that the heteroclinic bifurcation

takes places above the Hopf bifurcation, so that μ23

4 <

μ23 ≈ μ2

3(μ3+2)2

4(μ3+1)2 < �hetdeg(μ3) < 4μ23. �

Appendix A.2: (Nondegenerate heteroclinic

connections)

Here, we are concerned with the existence of non-degenerate heteroclinic connections joining equilibria(0, Z±) in system (2.6) with b = − 1.

r = P(r, z) = r (μ1 + μ3z + r ),

z = Q(r, z) = μ2 − r − z2.

Lemma A.2. Let us consider system (2.6) with b = −1. Then, there exists a function μ1 = �het(μ2, μ3), de-fined for μ3 > 0, 0 < μ2 < �hetdeg(μ3), such that forthe parameter values μ1 = �het(μ2, μ3), the systempresents a heteroclinic orbit connecting the hyperbolicsaddles (0, Z+) and (0, Z−). Moreover, −μ3

√μ2 <

�het(μ2, μ3) < μ3√

μ2.

Proof: As in the proof of the above lemma, let usconsider the vertical isoclines straight line μ1 + μ3z +r = 0. On this line, we take the points A = (r+, z+) andB = (r−, z−) corresponding to the unstable manifold of(0, Z+) and to the stable manifold of (0, Z−), respec-

tively (see Fig. 15). Let us denote d = z+ − z−. Then,we have the following:� From Lemma A.1, we obtain d > 0 for the parameter

values μ1 = − μ3√

μ2, μ2 < �hetdeg(μ3).� Taking μ1 < μ3√

μ2 but close enough, the stablemanifold of (0, Z−) is connected with the equilib-rium (R2, Z2), and consequently d < 0 (see Fig. 15).� For each μ3 > 0 and μ2 > 0, due to the continuitywith respect to the parameters, we deduce that there isμ1, satisfying −μ3

√μ2 < μ1 < μ3

√μ2, for which

d = 0.� To prove that this value of μ1 is unique, we use thatthe stable and unstable manifolds of the equilibria(0, Z±) are located into the zone where Q < 0, sothat

∣∣∣∣∣ P Q∂ P∂μ1

∂ Q∂μ1

∣∣∣∣∣ = −Q∂ P∂μ1

= −r(μ2 − r − z2) > 0.

Then, we deal with a one-parameter family of ro-tated vector fields: keeping μ2 > 0 and μ3 > 0 fixedand increasing μ1, the equilibria (0, Z±) remain fixedand the vector field rotates counterclockwise. In sum-mary, d is a strictly monotone function of μ1, and itsroot is unique.

�

Appendix A.3: Homoclinic connections

Here we study the homoclinic bifurcation for equilib-rium (R1, Z1). Initially, we show that at most, this bifur-cation can occurs for one value of parameter μ1 (withμ2, μ3 fixed), in both cases b = ± 1. Later, we analyzeits existence.

Lemma A.3. Let us consider system (2.6) with bμ3 <

0. If there is a value of μ1 for which the system under-goes a homoclinic bifurcation for equilibrium (R1, Z1),then this value is unique.

Proof: Let us perform on system (2.6) the transforma-tion

x = z + ‘bμ3

2, y = μ2 + br − z2,

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74 Nonlinear Dyn (2007) 48:55–76

and take as parameters

μ1 = ν1, μ2 = bν1 + ν2, μ3 = ν3.

Then, we obtain:

x = P(x, y) = y,

y = Q(x, y) = R(x, y)S(x, y) − 2xy + bν3 y,

where

R(x, y) = by + bx2 − ν3x − ν1 − bν2 + bν23

4,

S(x, y) = y + x2 − ν2 − ν23

4.

� The invariant axis r = 0 has been moved to theparabola R(x, y) = 0. The equilibria outside the

invariant axis becomes (x±0 , 0) = ± (

√ν2 + ν2

34 , 0),

and they are located in the region R(x, y) > 0. Itis a simple matter to show that (x+

0 , 0) is always asaddle, whereas (x−

0 , 0) is a focus or a node.� It is easy to show that the negative eigenvalue of thesaddle equilibrium (x+

0 , 0) is less than the slope ofthe curve S(x, y) = 0 at this point. Then, the stablemanifold of the equilibrium (x+

0 , 0) is located on thezone S(x, y) > 0 for y > 0.

Consequently, taking the parameter close enoughto those where the homoclinic exists, the stable mani-fold of equilibrium (x+

0 , 0) will intersect the straight-line of vertical isoclines y = 0 in a point labelled B

in Fig. 16. Analogously, it can be shown that the un-stable manifold of (x+

0 , 0) will intersect the quotedstraightline in a point labelled A.� As we have bν3 < 0, we find

sign

((∂S∂x

,∂S∂y

))· (P, Q) = sign (2xy + Q)

= sign (−bν3 y)

= sign(y),

at S(x, y) = 0. Then, when crossing the curveS(x, y) = 0, the flow points as represented in Fig. 16:in the portion of the curve S(x, y) = 0, y > 0, itpoints to the zone S(x, y) < 0; whereas in the portionS(x, y) = 0, y < 0 it points to the zone S(x, y) > 0.

� Using that:∣∣∣∣∣ P Q∂ P∂ν1

∂ Q∂ν1

∣∣∣∣∣ = −yS(x, y),

we deduce: if yS(x, y) > 0, the vector field rotatesclockwise by increasing ν1; whereas the rotation iscounterclockwise in the zone yS(x, y) < 0.� Hence, the x-component of the point B decreasesand the one of point A increases when increasing ν1.Therefore, the distance between the points A and B,as a function of ν1 has, at most, one root. �

Lemma A.4. Let us consider system (2.6)with b = +1. Then, there exists a func-tion μ1 = �hom+(μ2, μ3), defined for μ3 < 0,μ3(μ3+4)

4 < μ2 < 0, such that for the parameter valuesμ1 = �hom+(μ2, μ3), the system presents a homoclinicorbit for the saddle equilibrium (R1, Z1). Moreover,this function satisfies:

μ2 − μ23(μ3 + 1) < μ1 < μh

1, (A.17)

where μh1 = b(μ3 + 2)(−1 + √

1 + μ2) is the value ofμ1 corresponding to the Hopf bifurcation h. Also,�hom+

(μTB

2 , μ3) = μTB

1 , and �hom+(0, μ3) = 0.

Proof: As shown in Fig. 17, let us consider on the ver-tical isoclines straight line μ1 + μ3z + r = 0 the pointsA = (r+, z+) and B = (r−, z−) corresponding, respec-tively, to the unstable and to the stable manifolds of(R1, Z1). Let us denote d = r+ − r−. Then, we have:� For the parameter value μ1 = μ2 − μ2

3(μ3 + 1), thetwo lines r = − μ2 and z = − Z1 intersect the ver-tical isoclines straight line z = − 1

μ3z + √

μ2 in apoint P = (−μ2,

μ2

μ3+ √

μ2). As shown in Fig. 17(a),the point A is located below P, whereas B is locatedabove it. Then, we have d < 0.� At μ1 = μh

1, the unstable manifold of the saddle(R1, Z1) is connected with the counterclockwise at-tractive weak focus (R2, Z2).

As shown in Fig. 17(b), the point B must be locatedbelow A. Then, d > 0.� Due to the continuity with respect to the parameters,for each μ2 < 0, μ3 < 0 fixed, there exists μ1 satis-fying (A.17) such that d = 0. Moreover, this value isunique because Lemma A.3.

Springer

Nonlinear Dyn (2007) 48:55–76 75� In the μ2–μ3 plane, from the Takens-Bogdanov bi-furcation point, a homoclinic curve starts. Due to theuniqueness, we obtain �hom+(μTB),μ3

2 ) = μTB1 .� If we take parameters values on SN1 ≡ {μ2 = 0},

with μ1 < 0, the equilibria (0, Z±) collapse in theorigin, which is a saddle-node (with a stable mani-fold). On the other hand, (R1, Z1) is a saddle equi-librium and (R2, Z2) is a repulsive focus or node. Asshown in Fig. 17(c), the stable manifold of (R1, Z1) isconnected with (R2, Z2). Also, the unstable manifoldis connected with the origin.

Then, the points A and B are located as shown inFig. 17(c), so that d < 0.

As the homoclinic bifurcation only occursin the saddle-node surface in the singularity(i.e., for μ1 = μ2 = 0), we get �hom+(0, μ3) =0.

�

Lemma A.5. Let us consider system 2.6 with b = − 1.Then, there exists a function μ1 = �hom−(μ2, μ3), de-fined for μ3 > 0, �hetdeg(μ3) < μ2 < μTB

2 , such thatfor the parameter values μ1 = �hom−(μ2, μ3) the sys-tem presents a homoclinic orbit for (R1, Z1). Moreover,this function satisfies:

−μ2 − μ23

4< �hom−(μ2, μ3) < −μ3

√μ2. (A.18)

Also, �hom−(μTB2 , μ3) = μTB

1 , and �hom−(�hetdeg(μ3), μ3

) = − μ3√

�hetdeg(μ3).

Proof: Let us consider, on the straight line of verti-cal isoclines μ1 + μ3z + r = 0, the points A = (r+, z+)and B = (r−, z−) corresponding to the unstable and tothe stable manifolds of the saddle equilibrium (R1, Z1),respectively. Let us denote d = z+ − z−. Then:� As shown in Lemma A.1, for parameter values on

the transcritical bifurcation surface μ1 = − μ3√

μ2,μ2 < �hetdeg(μ3), the point A is located abovethe point B (because in this case, the equilibria(0, Z−) and (R1, Z1) collapse, see Fig. 18(a)). Hence,for μ1 < −μ3

√μ2 and close enough, the stable

manifold of (R1, Z1) approaches the union of theinvariant axis and the stable manifold of (0, Z−) (seeFig. 18(b)). Then, d > 0.

� For μ1 > μSN21 = − μ2 − μ2

34 close enough, the sta-

ble manifold of (R1, Z1) is connected with the equi-librium (R2, Z2) (see Fig. 18(c)), and then d < 0.� Using the continuity with respect to the parameters,we obtain that, for each μ3 > 0, μ2 > 0, there existsa value of μ1 such that d = 0. Moreover, from LemmaA.3, we deduce that this value is unique.� There is a homoclinic bifurcation starting from theTakens-Bogdanov bifurcation. Due to the unique-ness, we get �hom−

(μTB

2 , μ3) = μTB

1 . The last equal-ity in the statement follows because the uniquenessof the heteroclinic locus, and using that this ariseswhen the homoclinic locus reaches the transcriticalbifurcation. �

Appendix A.4: Type J global connections

Finally, we consider the heteroclinic orbit concerningthe equilibria (R1, Z1), (0, Z−).

Lemma A.6. Let us consider system (2.6) with b = −1. Then, there exists a function μ1 = �J(μ2, μ3), de-fined for μ3 > 0, �hetdeg(μ3) < μ2 < μTB

2 , such thatfor the parameter values μ1 = �J(μ2, μ3), the sys-tem presents a heteroclinic orbit connecting the sad-dle equilibrium (R1, Z1) with the saddle equilibrium(0, Z−). Moreover, this function satisfies

�hom−(μ2, μ3) < �J(μ2, μ3) < −μ3√

μ2. (A.19)

Moreover, �J(�hetdeg(μ3), μ3) = − μ3√

�hetdeg(μ3).

Proof: On the straight line of vertical isoclines μ1 +μ3z + r = 0, let us consider the points A = (r+, z+) andB = (r−, z−) located respectively on the unstable man-ifold of (R1, Z1) and on the stable manifold of (0, Z−).Let us denote d = z+ − z−.� On the transcritical bifurcation surface μ1 = −

μ3√

μ2, whenever �hetdeg(μ3) < μ2, the equilibrium(R1, Z1) collapses with (0, Z+). As shown in LemmaA.1, the point A is located below B (see Fig. 19(a)).Hence, d < 0.� For parameter values on the homoclinic bifurcationsurface μ1 = �hom−(μ2, μ3), the stable manifold of(0, Z−) is exterior to the homoclinic orbit of (R1, Z1)(see Fig. 19(b)). Then, d > 0.

Springer

76 Nonlinear Dyn (2007) 48:55–76� Using the continuity with respect to the parameters,we obtain that, for eachμ3 > 0,μ2 > 0, there exists avalue ofμ1 such that d = 0. Moreover, the uniquenessof this value can be obtained in the same way as inLemma A.2. �

Acknowledgements This work has been partially supported bythe Ministerio de Ciencia y Tecnologıa, Plan Nacional I+D+I,co-financed with FEDER funds in the frame of the projectsMTM2004-04066, BFM2001-2608; and by the Consejerıa de In-novacion, Ciencia y Empresa de la Junta de Andalucıa (projectsFQM-276, TIC-0130).

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